{\HugeAnalogue Gravity}
% {\LargeCarlos \Barcelo}
Instituto de Astrof{\'\i}sica de Andaluc{\'\i}a
Granada
Spain
http://www.iaa.csic.es/
{\LargeStefano Liberati}
International School for Advanced Studies and INFN
Trieste
Italy
http://www.sissa.it/\~{}liberati
and
{\LargeMatt Visser}
New Zealand
http://www.mcs.vuw.ac.nz/\~{}visser
{(Friday 13 May 2005; Updated 1 June 2005; \LaTeX-ed \today) }
Analogue models of (and for) gravity have a long and distinguished history dating back to the earliest years of general relativity. In this review article we will discuss the history, aims, results, and future prospects for the various analogue models. We start the discussion by presenting a particularly simple example of an analogue model, before exploring the rich history and complex tapestry of models discussed in the literature. The last decade in particular has seen a remarkable and sustained development of analogue gravity ideas, leading to some hundreds of published articles, a workshop, two books, and this review article. Future prospects for the analogue gravity programme also look promising, both on the experimental front (where technology is rapidly advancing) and on the theoretical front (where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity).
add-ons.bib.tex

 
@Article{Paddy1,
     author    = "Srinivasan, K. and  Padmanabhan, T.",
     title     = "Particle Production and Complex Path Analysis",
     journal   = "Phys. Rev. D",
     volume    = "60",
     year      = "1999",
     pages     = "24007",
     eprint    = "gr-qc/9812028",
     SLACcitation  = "%%CITATION = GR-QC 9812028;%%"
}

 @Article{Paddy2,
     author    = "Shankaranarayanan,  S. and Padmanabhan, T.   and Srinivasan, K.",
     title     = "Hawking radiation in different coordinate settings: Complex paths approach",
     journal   = "Class. Quantum Grav.",
     volume    = "19",
     year      = "2002",
     pages     = "2671",
     eprint    = "gr-qc/0010042",
     SLACcitation  = "%%CITATION = GR-QC 0010042;%%"
}

@Article{Paddy3,
     author    = "Padmanabhan, T.",
     title     = "Entropy of Horizons, Complex Paths and Quantum Tunneling",
     journal   = "Mod. Phys. Lett. A",
     volume    = "19",
     year      = "2004",
     pages     = "2637-2643",
     eprint    = "gr-qc/0405072",
     SLACcitation  = "%%CITATION = GR-QC 0405072;%%"
}

@Article{Paddy4
     author    = "Sriramkumar,  L. and Padmanabhan, T.",
     title     = "Initial state of matter fields and trans-{Planckian} physics: 
                    Can {CMB} observations disentangle the two?",
     eprint    = "gr-qc/0408034",
     SLACcitation  = "%%CITATION = GR-QC 0408034;%%"
}


@Article{Piotr,
     author    = "Chrusciel, P.",
     title     = "Black holes",
     journal   = ".",
     volume    = "",
     year      = "2004",
     pages     = "277-283",
     eprint    = "",
     SLACcitation  = "%%CITATION = GR-QC ;%%"
}

@Article{Savage,
     author    = "Crusciel, P.",
     title     = "Black holes",
     journal   = ".",
     volume    = "",
     year      = "2004",
     pages     = "277-283",
     eprint    = "",
     SLACcitation  = "%%CITATION = GR-QC ;%%"
}




@incollection{Trautman,
     author    = "Trautman, A.",
     title     = "Comparison of Newtonian and relativistic theories of space-time",
     booktitle = "",
     publisher = "Indiana University Press, Bloomington",
     year      = "1966",
     pages     = "413-425",
     editor = "Hoffman, B."
    }
    
     @Article{Laschkarew,
     author    = "Laschkarew, W.",
     title     = "Zur {Theorie} der {Gravitation}",
     journal   = "Z. fur Physik",
     volume    = "35",
     year      = "1926",
     pages     = "473-476"
}

    
analogue.tex
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% Living Reviews
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\usepackage[hypertex,backref,colorlinks=true]{hyperref}
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\RequirePackage{longtable}
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\newcommand{\mathbold}[1]{\mbox{\boldmath $#1$}}
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\begin{document}
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\def\Barcelo{{Barcel\'o}}
\title{{\Huge\bf 
Analogue Gravity}}
\author{%
{\Large\bf Carlos \Barcelo},\\
Instituto de Astrof{\'\i}sica de Andaluc{\'\i}a\\
Granada, Spain\\
e-mail: carlos@iaa.es\\
http://www.iaa.csic.es/
\\
\\
{\Large\bf Stefano Liberati}, \\
International School for Advanced Studies and INFN\\
Trieste, Italy\\
e-mail: liberati@sissa.it\\
http://www.sissa.it/\~{}liberati
\\
\\
and  
\\
\\
{\Large\bf Matt Visser},\\
Victoria University of Wellington\\
New Zealand\\
e-mail: matt.visser@mcs.vuw.ac.nz\\
http://www.mcs.vuw.ac.nz/\~{}visser
\\
\\
\small{(Friday 13 May 2005; Updated 1 June 2005; \LaTeX-ed \today) }}

\date{ }
\maketitle

\clearpage

\begin{abstract}
Analogue models of (and for) gravity have a long and distinguished
history dating back to the earliest years of general relativity. In
this review article we will discuss the history, aims, results, and
future prospects for the various analogue models. We start the
discussion by presenting a particularly simple example of an analogue
model, before exploring the rich history and complex tapestry of
models discussed in the literature. The last decade in particular has
seen a remarkable and sustained development of analogue gravity ideas,
leading to some hundreds of published articles, a workshop, two books,
and this review article. Future prospects for the analogue gravity
programme also look promising, both on the experimental front (where
technology is rapidly advancing) and on the theoretical front (where
variants of analogue models can be used as a springboard for radical
attacks on the problem of quantum gravity).
\end{abstract}

\keywords{Analogies, models, gravity}

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\clearpage
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\tableofcontents % definitely to be used for development purposes
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\listoffigures
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\def\Lemaitre{Lema\^\i{}tre}
\def\Painleve{Painlev\'e}

%-----------------------------------------------------------------------
\begin{figure}[t]
\vbox{
%\vskip 20 pt
% width 7in, height 5in
\centerline{{\includegraphics[width=7in,height=5in]{F-conos9}}}

\caption[Cascading sound cones]{ {\sl Artistic impression of cascading sound cones (in the geometrical acoustics limit) forming an acoustic black hole when supersonic  flow tips the sound cones past the vertical.  \smallskip \label{F:cascade}}} }

\vskip 2 in
\end{figure}
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\vskip 2 in

\vfill

\clearpage
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\begin{figure}[t]
\vbox{
%\vskip 20 pt
% width 7in, height 5in
\centerline{{\includegraphics[width=7in,height=5in]{F-ondas9}}}

\caption[Trapped waves]{ {\sl Artistic impression of trapped waves (in the physical acoustics limit) forming an acoustic black hole when supersonic  flow forces the waves to move downstream.  \smallskip \label{F:ondas}}} }

\vskip 2 in
\end{figure}
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\vskip 2 in

\vfill

\clearpage
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\section{Introduction}
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\label{S:introduction}

\input{introduction}

\clearpage
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\section{The simplest example of an analogue model}
%----------------------------------------------
\label{S:simple}

\input{simple-fluid}

\clearpage
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\section{History and motivation}
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\label{S:history}

\input{history}

\clearpage
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\section{A catalogue of models}
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\label{S:catalogue}

\input{catalogue}

\clearpage
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\section{Lessons from analogue models}
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\label{S:lessons}

\input{lessons-from-analogue}

\clearpage
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\section{Future directions}
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\label{S:future-directions}

\input{future-directions}

\clearpage
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\section{Conclusions}
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\label{S:conclusions}

\input{conclusions}

\clearpage
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% REFERENCES
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\input{references}


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\end{document}
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% EOF
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catalogue.tex
%-----------------------------------
% catalogue.tex
%------------------------------------
% BOF
%------------------------------------
% This is the longest single chapter
%------------------------------------

In this chapter we will attempt to categorize the very many analogue
models researchers have investigated. Perhaps the most basic subdivision
is into classical models and quantum models, but even then many other levels
of refinement are possible. Consider for instance the
following list:
%
\begin{itemize}

\item Classical models

\begin{itemize}

\item Classical sound.
\item Water waves (gravity waves).
\item Classical refractive index.
\item Normal modes.

\end{itemize}

\item Quantum models:

\begin{itemize}

\item Bose--Einstein condensates [BECs]. 
\item The Heliocentric universe. \\ 
(Helium as an exemplar for just about anything.)

\item Slow light.


\end{itemize}

\end{itemize}
%
We will now provide a few words on each of these topics.

%----------------------------------------------
%----------------------------------------------
%----------------------------------------------
%----------------------------------------------
%----------------------------------------------
%----------------------------------------------
%----------------------------------------------

%----------------------------------------------
\subsection{Classical models}
%----------------------------------------------


%----------------------------------------------
\subsubsection{Classical sound}
%----------------------------------------------

Sound in a moving fluid has already been extensively discussed in
section (\ref{S:simple}), and we will not repeat such discussion here.
In contrast, sound in a solid exhibits its own distinct and interesting
features, notably in the existence of a generalization of the normal
notion of birefringence --- longitudinal modes travel at a different
speed (typically faster) than do transverse modes. This may be viewed as
an example of an analogue model which breaks the ``light cone'' into two
at the classical level; as such this model is not particularly useful if
one is trying to simulate special relativistic kinematics with its
universal speed of light, though it may be used to gain insight into yet
another way of ``breaking'' Lorentz invariance.

%----------------------------------------------
\subsubsection{Shallow water waves (gravity waves)}
%----------------------------------------------


A wonderful example of the occurrence of an effective metric in nature
is that provided by gravity waves in a shallow basin filled with
liquid~ \cite{Schutzhold:2002xm}.\footnote{Of course we now mean ``gravity wave'' in the
traditional fluid mechanics sense of a water wave whose restoring force
is given by ordinary Newtonian gravity. Waves in the fabric of spacetime
are more properly called ``gravitational waves'', though this usage
seems to be in decline within the general relativity community. Be very
careful in any situation where there is even a possibility of confusing
the two concepts. } (See figure~\ref{F:fig1}). If one neglects the
viscosity and considers an irrotational flow,
$\v=\bnabla\phi$, one can write Bernoulli's equation in
the presence of Earth's gravity as %

\begin{eqnarray}
\partial_t \phi + {1 \over 2} (\bnabla\phi)^2=
-{p \over \rho}-gz-V_{\parallel}.
\label{bernoulli-gravity}
\end{eqnarray}
%

Here $\rho$ is the density of the fluid, $p$ its pressure, $g$ the
gravitational acceleration and $V_{\parallel}$ a potential associated
with some external force necessary to establish an horizontal flow in
the fluid.  We denote that flow by $\v_B^{\parallel}$. We must
also impose the boundary conditions that the pressure at the surface,
and the vertical velocity at the bottom, both vanish: $p(z=h_B)=0$,
$v_\perp(z=0)=0$.

Once a horizontal background flow is established, one can see that the
perturbations of the velocity potential satisfy
%
\begin{eqnarray}
\partial_t \,\delta \phi + 
\v_B^{\parallel} \cdot \bnabla_{\parallel} \delta \phi
=- {p \over \rho}.
\end{eqnarray}
%
If we now expand this perturbation potential in a Taylor series 
%
\begin{eqnarray}
\delta \phi =
\sum_{n=0}^{\infty} {z^n \over n!} \; \delta \phi_n(x,y),
\end{eqnarray}
%
it is not difficult to prove \cite{Schutzhold:2002xm} that surface
waves with long wavelengths, (long compared with the depth of the basin,
$\lambda \gg h_B$), can be described to a good approximation by $\delta
\phi_0(x,y)$ and that this field ``sees'' an effective metric of the
form
%
\begin{eqnarray}
ds^2={1 \over c^2} \left[-(c^2 - v_B^{\parallel 2})\; \d t^2
-2\v_B^{\parallel} \cdot \mathbf{dx} \; \d t
+\mathbf{dx} \cdot \mathbf{dx}\right],
\end{eqnarray}
%
where $c \equiv \sqrt{gh_B}$. The link between small variations of the
potential field and small variations of the position of the surface is
provided by the following equation
%
\begin{eqnarray}
\delta v_{\perp}=-h_B \;\nabla_{\parallel}^2 \delta \phi_0=
\partial_t \,\delta h + 
\v_B^{\parallel} \cdot \bnabla_{\parallel} \delta h
=
{\d\over\d t} \,\delta h.
\end{eqnarray}
%
The entire previous analysis can be generalized to the case in which
the bottom of the basin is not flat, and the background flow not purely
horizontal \cite{Schutzhold:2002xm}. Therefore, one can create
different effective metrics for gravity waves in a shallow fluid basin
by changing (from point to point) the background flow velocity and the
depth, $h_B(x,y)$.




%====================================================
\begin{figure}[htb]
  \vbox{
  \centerline{{\includegraphics[width=5in]{F-gravity-waves}}}
  \caption[Gravity waves in a shallow fluid basin]{Gravity waves in a 
  shallow fluid basin with
a background horizontal flow.
  \label{F:fig1}}}
\end{figure}
%====================================================

The main advantage of this model is that the velocity of the surface
waves can very easily be modified by changing the depth of the basin. 
This velocity can be made very slow, and consequently, the creation of
ergoregions should be relatively easier than in other models. As
described here, this model is completely classical, so cannot serve us
to study Hawking evaporation an other quantum phenomena. However, it can
certainly serve to investigate the classical phenomena of mode mixing
that underlies the quantum processes.




%----------------------------------------------
\subsubsection{Classical refractive index}
\label{sec:cri}
%----------------------------------------------



%---------------------------------------------------------------------
\def\mathbfs#1{\mbox{\boldmath$#1$}} 
\def\tr{{\mathrm{tr}}}
%---------------------------------------------------------------------

The macroscopic Maxwell equations inside a dielectric take the well-known form
%
\begin{eqnarray}
&\mathbf{\nabla} \cdot \mathbf{B}=0,
&\mathbf{\nabla} \times \mathbf{E} +\partial_t \mathbf{B}=0, \\ 
&\mathbf{\nabla} \cdot \mathbf{D}=0, 
&\mathbf{\nabla} \times \mathbf{H} - \partial_t \mathbf{D}=0,
\end{eqnarray}
%
with the constitutive relations 
$\mathbf{H}=\mathbfs{\mu}^{-1}\cdot \mathbf{B}$ and 
$\mathbf{D}=\mathbfs{\epsilon}\cdot  \mathbf{E}$. Here, 
$\mathbfs{\epsilon}$ is the $3\times 3$ permittivity tensor and $\mathbfs{\mu}$ 
the $3\times 3$ permeability tensor of the medium. 
These equations can be written in a condensed way as 
%
\begin{eqnarray}
\partial_\alpha \left( Z^{\mu\alpha\nu\beta} \; F_{\nu\beta}\right)=0
\label{elec-equation}
\end{eqnarray}
%
where $F_{\nu\beta}= A_{[\nu,\beta]}$ is the electromagnetic tensor, 
%
\begin{eqnarray}
F_{0i}=-F_{i0}=-E_i,~~~~F_{ij}=\varepsilon_{ijk}B^k,
\end{eqnarray}
%
and (assuming the medium is at rest) the non-vanishing components of the 
4'th rank tensor $Z$ are given by
%
\begin{eqnarray}
&&Z^{0i0j}=-Z^{0ij0}=Z^{i0j0}=-Z^{i00j}=
-{1 \over 2}\epsilon^{ij}; \\
&&Z^{ijkl}={1 \over 2}\varepsilon^{ijm}\;\varepsilon^{kln}\; \mu^{-1}_{mn};
\end{eqnarray}
supplemented by the conditions that $Z$ is antisymmetric on its first
pair of indices and antisymmetric on its second pair of indices. Without
significant loss of generality we can ask that $Z$ also be symmetric
under pairwise interchange of the first pair of indices with the second
pair --- thus $Z$ exhibits most of the algebraic symmetries of the
Riemann tensor, though this appears to merely be accidental, and not
fundamental in any way.

If we compare this to the Lagrangian for electromagnetism in curved spacetime
\begin{equation}
\mathcal{L} = \sqrt{-g} \; g^{\mu\alpha} \; g^{\nu\beta}\; F_{\mu\nu} \; F_{\alpha\beta}
\end{equation}
we see that in curved spacetime we can also write the electromagnetic 
equations of motion in the form (\ref{elec-equation}) where now (for 
some constant $K$):
\begin{equation}
Z^{\mu\nu\alpha\beta} = K\; \sqrt{-g} 
\;\left\{  g^{\mu\alpha} \; g^{\nu\beta}  
-  g^{\mu\beta} \; g^{\nu\alpha}\right\} 
\end{equation}
If we consider a static gravitational field we can always re-write 
it as a conformal factor multiplying an ultra-static metric 
\begin{eqnarray}
g_{\mu\nu}=\Omega^2\; \left\{-1\oplus g_{ij} \right\}
\end{eqnarray}
 then
\begin{eqnarray}
&&
Z^{0i0j}=-Z^{0ij0}=Z^{i0j0}=-Z^{i00j}=
-K\;\sqrt{-g} \; g^{ij}; \\
&&Z^{ijkl}= K\; \sqrt{-g}  \left\{  g^{ik} \; g^{jl}  
-  g^{il} \; g^{jk}\right\} 
\end{eqnarray}

The fact that $Z$ is independent of the conformal factor $\Omega$ is
simply the reflection of the well-known fact that the Maxwell equations
are conformally invariant in (3+1) dimensions. Thus if we wish to have
the analogy (between a static gravitational field and a dielectric
medium at rest) hold \emph{at the level of the  wave equation} (physical
optics)  we must satisfy the two stringent constraints

\begin{eqnarray}
&&
K\;\sqrt{-g} \; g^{ij} = {1 \over 2}\;\epsilon^{ij};
\\
&&
K\; \sqrt{-g}  \left\{  g^{ik} \; g^{jl}  -  g^{il} \; g^{jk}\right\} = 
{1 \over 2}\;\varepsilon^{ijm}\;\varepsilon^{kln}\; \mu^{-1}_{mn}.
\end{eqnarray}
The second of these constraints can be written as
\begin{equation}
K\; \sqrt{-g}  \; \varepsilon_{ijm}\;\varepsilon_{kln} 
\left\{  g^{ik} \; g^{jl}  \right\} =  \mu^{-1}_{mn}.
\end{equation}
In view of the standard formula for $3\times3$ determinants
\begin{equation}
\varepsilon_{ijm}\;\varepsilon_{kln} \left\{  X^{ik} \; X^{jl}  \right\} 
= 2 \;\det{X}\; X^{-1}_{mn},
\end{equation}
this now implies
\begin{equation}
2 K {g_{ij}\over\sqrt{-g} } = \mu^{-1}_{ij},
\end{equation}
whence
\begin{equation}
{1\over2 K} \; \sqrt{-g} \; g^{ij} = \mu^{ij}.
\end{equation}
Comparing this with
\begin{equation}
{2 K} \; \sqrt{-g} \; g^{ij}  = \epsilon^{ij},
\end{equation}
we now have:
\begin{eqnarray}
\epsilon^{ij} &=& 4 \; K^2 \; \mu^{ij};
\\
g^{ij} &=& {4\;K^2\over\det\mathbfs{\epsilon}}\; \epsilon^{ij};
\\
g^{ij} &=& {1\over4\;K^2\;\det\mathbfs{\mu}}\; \mu^{ij}.
\label{dielectric-metric-physical}
\end{eqnarray}
%
To rearrange this, introduce the matrix square root
$[\mathbfs{\mu}^{1/2}]^{ij}$, which
always exists because $\mathbfs{\mu}$ is real positive definite and
symmetric. Then
%
\begin{equation}
g^{ij} = 
\left[\left\{ {
\mathbfs{\mu}^{1/2} \; \mathbfs{\epsilon} \; \mathbfs{\mu}^{1/2}
\over \det(\mathbfs{\mu}\;\mathbfs{\epsilon}) } \right\}^{1/2}
\right]^{ij}.
\end{equation}
%
Note that if you are given the static gravitational field (in the form
$\Omega$, $g_{ij}$) one can always solve to find an equivalent  analogue
in terms of permittivity/permeability (albeit an analogue that satisfies
the mildly unphysical constraint
$\mathbfs{\epsilon}\propto\mathbfs{\mu}$).~\footnote{The existence of
this constraint has been independently re-derived several times in the
literature. In contrast, other segments of the literature seem blithely
unaware of this important restriction on just when permittivity and
permeability are truly equivalent to an effective metric.} On the other
hand, if you are given permeability and permittivity tensors 
$\mathbfs{\epsilon}$ and $\mathbfs{\mu}$, then it is \emph{only} for
that subclass of media that satisfy
$\mathbfs{\epsilon}\propto\mathbfs{\mu}$ that one can perfectly mimic
\emph{all} of the electromagnetic effects by an equivalent gravitational
field. Of course this can be done provided one only considers
wavelengths that are sufficiently long for the macroscopic description
of the medium to be valid. In this respect it is interesting to note
that the behaviour of the refractive medium at high frequencies has been
used to introduce an effective cutoff for the modes involved in Hawking
radiation~\cite{Reznik:1997ag}. We shall encounter this model later on
when we shall consider the trans-Planckian problem for Hawking
radiation.


%----------------------------------------------
\paragraph{Eikonal approximation:}
%----------------------------------------------

With a bit more work this discussion can be extended to a medium in
motion, leading to an extension of the Gordon metric. Alternatively, one
can agree to ask more limited questions by working at the level of
geometrical optics (adopting the eikonal approximation), in which case
there is no longer any restriction on the permeability and permittivity
tensors. To see this construct the matrix

\begin{eqnarray}
C^{\mu\nu}=Z^{\mu\alpha\nu\beta} \; k_\alpha \, k_\beta.
\end{eqnarray}
%
The dispersion relations for the propagation of photons (and therefore
the sought for geometrical properties) can be obtained from the reduced
determinant of $C$ (notice that the [full] determinant of $C$ is identically
zero as $C^{\mu\nu}k_\nu=0$; the reduced determinant is that
associated with the three directions orthogonal to $k_\nu=0$). By
choosing the gauge $A_0=0$ one can see that this reduced determinant
can be obtained from the determinant of the $3\times3$ sub-matrix $C^{ij}$.
This determinant is
%
\begin{eqnarray}
\det( C^{ij})={1 \over 8} \det 
\left(-\omega^2 \epsilon^{ij} + 
\varepsilon^{ikm}\;\varepsilon^{jln}\; \mu^{-1}_{mn} k_k k_l\right).
\end{eqnarray}
%
or, after making some manipulations,
%
\begin{eqnarray}
\det( C^{ij})= {1 \over 8} \det 
\left[-\omega^2 \epsilon^{ij} + (\det \mathbfs{\mu})^{-1} 
(\mu^{ij} \mu^{kl} k_k k_l -\mu^{im} k_m \mu^{jl} k_l) \right].
\end{eqnarray}
%

To simplify this, again introduce the matrix square roots
$[\mathbfs{\mu}^{1/2}]^{ij}$ and $[\mathbfs{\mu}^{-1/2}]_{ij}$, which
always exist because the relevant matrices are real positive definite and
symmetric.
Then define
\begin{equation}
\tilde k^i = [\mathbfs{\mu}^{1/2}]^{ij} \; k_j
\end{equation}
%
and
\begin{equation}
[\mathbfs{\tilde\epsilon}]^{ij} =  \det(\mathbfs{\mu}) \; 
[\mathbfs{\mu}^{-1/2} \; \mathbfs{\epsilon} \;\mathbfs{\mu}^{-1/2}]_{ij} 
\end{equation}
%
so that
%
\begin{eqnarray}
&&\det( C^{ij}) \propto 
\det \left\{
-\omega^2 \;
[\mathbfs{\tilde\epsilon}]^{ij} 
+ \delta^{ij} \; [\delta_{mn} \; \tilde k^m \; \tilde k^n]
-\tilde k^i \; \tilde k^j 
\right\}.
\end{eqnarray}
%

The behaviour of this dispersion relation now depends critically on the
way that the eigenvalues of $\mathbfs{\tilde\epsilon}$ are distributed.

%----------------------------------------
\paragraph{3 degenerate eigenvalues:} 
%-----------------------------------------

If all eigenvalues are degenerate then $\mathbfs{\tilde \epsilon} =
\tilde \epsilon \;\mathbf{I}$,  implying
$\mathbfs{\epsilon}\propto\mathbfs{\mu}$ but now with the possibility of
a position dependent proportionality factor (in the case of physical
optics the proportionality factor was constrained to be a
position-independent constant). In this case we now easily evaluate

\begin{equation}
\mathbfs{\epsilon} =  
{\tr(\mathbfs{\epsilon})\over\tr(\mathbfs{\mu})} \; \mathbfs{\mu}
\qquad
\hbox{and}
\qquad
\tilde\epsilon = \det\mathbfs{\mu} \; 
{\tr(\mathbfs{\epsilon})\over\tr(\mathbfs{\mu})},
\end{equation}
while
\begin{equation}
\det( C^{ij}) \propto  \omega^2 \;
\left\{ \omega^2 -  [ {\tilde\epsilon}^{-1}\; \delta_{mn} \; 
\tilde k^m\;\tilde k^n]   \right\}^2.
\end{equation}
That is
\begin{equation}
\det( C^{ij}) \propto  \omega^2 
\left\{ \omega^2 - [g^{ij} \;  k_i\; k_j]   \right\}^2,
\end{equation}
with
\begin{equation}
\label{dielectric-metric}
g^{ij} =  {1\over\tilde\epsilon}\; [\mathbfs{\mu}]^{ij} = 
{ \tr(\mathbfs{\mu})\;[\mathbfs{\mu}]^{ij}\over \tr(\mathbfs{\epsilon}) \; 
\det\mathbfs{\mu} }
= 
{ \tr(\mathbfs{\epsilon})\;[\mathbfs{\epsilon}]^{ij}\over \tr(\mathbfs{\mu}) \; 
\det\mathbfs{\epsilon} }.
\end{equation}

This last result is compatible with but more general than the result
obtained under the more restrictive conditions of physical optics. In
the situation where both permittivity and permeability are isotropic,
($\epsilon^{ij} =\epsilon\;\delta^{ij}$ and $\mu^{ij} =
\mu\;\delta^{ij}$) this reduces to the perhaps more expected result

\begin{equation}
g^{ij} = {\delta^{ij}\over \epsilon\;\mu}.
\end{equation}

%--------------------------------------
\paragraph{2 degenerate eigenvalues:} 
%-------------------------------------

If $\mathbfs{\tilde\epsilon}$ has two distinct eigenvalues then the
determinant $\det(C^{ij})$  factorizes into a trivial factor of
$\omega^2$ and two quadratics. Each quadratic corresponds to a distinct
effective metric. This is the physical situation encountered in uni-axial
crystals, where the \emph{ordinary} and \emph{extraordinary} rays each
obey distinct quadratic dispersion relations~\cite{Born-Wolf}. From the point of view of
analogue models this corresponds to a two-metric theory.

%-------------------------------------
\paragraph{3 distinct eigenvalues:}  
%-------------------------------------

If $\mathbfs{\tilde\epsilon}$ has three distinct eigenvalues then the
determinant $\det(C^{ij})$  is the product of a trivial factor of
$\omega^2$ and a \emph{non-factorizable quartic}. This is the physical
situation encountered in bi-axial crystals~\cite{Born-Wolf,
Visser:2002sf}, and it seems that no meaningful notion of effective
Riemannian metric can be assigned to this case. (The use of Finsler
geometries in this situation is an avenue that may be worth
pursuing~\cite{EDM}.)


%-------------------------------------
\paragraph{Abstract linear electrodynamics:}
%-------------------------------------

Hehl and co-workers have championed the idea of using the linear constitutive relations of electrodynamics as the primary quantities, and then treating the spacetime metric (even for flat space) as a \emph{derived} concept. See~\cite{Obukhov:1999ug, Hehl:2004yk, Lammerzahl:2004ww, Hehl:2004tk}.



%-------------------------------------
\paragraph{Nonlinear electrodynamics:}
%-------------------------------------


In general, the permittivity and permeability tensors can be modified by
applying strong electromagnetic fields (this produces an effectively
non-linear electrodynamics). The entire previous discussion still
applies if one considers the photon as the linear perturbation of the
electromagnetic field over a background configuration
\begin{equation}
F_{\mu\nu}=F_{\mu\nu}^{\rm bg}+f_{\mu\nu}^{\rm ph}. 
\end{equation}
The background
field $F_{\mu\nu}^{\rm bg}$ sets the value of $\epsilon^{ij}(F^{\rm
bg})$, and $\mu^{ij}(F^{\rm bg})$. Equation (\ref{elec-equation}) then
becomes an equation for $f_{\mu\nu}^{\rm ph}$. This approach has been
extensively investigated by Novello and co-workers~\cite{Novello:1999pg, Novello:2001gk, DeLorenci:2002mi, Novello:2002ed, Novello:2003je, Novello:2003kh, Novello:2002iq, Fonseca-Barbatti:2002hq}.

%-------------------------------------
\paragraph{Summary:}
%-------------------------------------

The propagation of photons in a dielectric medium characterized by
$3\times3$ permeability and permittivity tensors constrained by
$\mathbfs{\epsilon} \propto \mathbfs{\mu}$ is equivalent (at the level
of geometric optics) to the propagation of photons in a curved spacetime
manifold characterized by the ultrastatic metric
(\ref{dielectric-metric}), provided one only considers wavelengths that
are sufficiently long for the macroscopic description of the medium to
be valid.  If in addition one takes a fluid dielectric, by controlling
its flow one can generalize the Gordon metric and again reproduce
metrics of the Painlev\'e--Gullstrand type,  and therefore geometries
with ergoregions. If the proportionality constant relating
$\mathbfs{\epsilon} \propto \mathbfs{\mu}$ is position independent, one
can make the stronger statement (\ref{dielectric-metric-physical}) which
holds true at the level of physical optics.


%----------------------------------------------
\subsubsection{Normal mode meta-models}
%----------------------------------------------
\def\aka{{\emph{aka}}}
%-------------------------------------


We have already seen how linearizing the Euler--Lagrange equations for a
single scalar field naturally leads to the notion of an effective
spacetime metric. If more than one field is involved the situation
becomes more complicated, in a manner similar to that of geometrical
optics in uni-axial and bi-axial crystals. (This should, with hindsight,
not be too surprising since electromagnetism, even in the presence of a
medium, is definitely a Lagrangian system and definitely involves more
than one single scalar field.) A normal mode analysis based on a general
Lagrangian (many fields but still first order in derivatives of those
fields) leads to a concept of \emph{refringence}, or more specifically
\emph{multi-refringence}, a generalization of the birefringence of
geometrical optics. To see how this comes about, consider a
straightforward generalization of the one-field case.

We want to consider linearized fluctuations around some background
solution of the equations of motion. As in the single-field case we
write (here we will follow the notation and conventions of
\cite{Barcelo:2001cp})
%
\begin{equation}
\phi^A(t,\vec x) = \phi_0^A(t,\vec x) + \epsilon  \;\phi_1^A(t,\vec x) + 
{\epsilon^2\over2}  \; \phi_2^A(t,\vec x) + O(\epsilon^3).
\end{equation}
%
Now use this to expand the Lagrangian
%
\begin{eqnarray}
{\L}(\partial_\mu \phi^A,\phi^A) 
&=& 
{\L}(\partial_\mu \phi^A_0,\phi^A_0)
+\epsilon \left[ 
{\partial \L\over\partial(\partial_\mu \phi^A)} \; \partial_\mu \phi^A_1
+
{\partial \L\over\partial\phi^A} \; \phi^A_1
\right]
\nonumber
\\
&+&
{\epsilon^2\over2} \left[ 
{\partial \L\over\partial(\partial_\mu \phi^A)} \; \partial_\mu \phi^A_2
+
{\partial \L\over\partial\phi^A} \;  \phi^A_2
\right]
\nonumber
\\
&+&
{\epsilon^2\over2} \Bigg[ 
{\partial^2 \L\over\partial(\partial_\mu \phi^A) \; \partial(\partial_\nu \phi^B)} 
\; \partial_\mu \phi^A_1 \; \partial_\nu \phi^B_1
\nonumber
\\
&&
+
2 {\partial^2 \L\over \partial(\partial_\mu \phi^A)\; \partial \phi^B} 
\; \partial_\mu \phi^A_1 \; \phi^B_1
+
{\partial^2 \L\over \partial\phi^A\; \partial\phi^B} 
\; \phi^A_1 \; \phi^B_1
\Bigg]
\nonumber
\\
&&
+
O(\epsilon^3).
\end{eqnarray}
%
Consider the action
%
\begin{equation}
S[\phi^A] = \int \d^{d+1} x \; \L(\partial_\mu\phi^A,\phi^A).
\end{equation}
%
Doing so allows us to integrate by parts. As in the single-field
case we can use the Euler--Lagrange equations to
discard the linear terms (since we are linearizing around a solution
of the equations of motion) and so get
%
\begin{eqnarray}
S[\phi^A] &=& S[\phi_0^A] 
\nonumber
\\
&+&
{\epsilon^2\over2}
\int \d^{d+1} x \Bigg[
\left\{
  {\partial^2 \L\over
   \partial(\partial_\mu \phi^A) \; \partial(\partial_\nu \phi^B)} 
\right\}
\; \partial_\mu \phi^A_1
\; \partial_\nu \phi^B_1
\nonumber
\\
&&
\quad
+
2 
\left\{
{\partial^2 \L\over \partial(\partial_\mu \phi^A)\; \partial \phi^B} 
\right\}
\; \partial_\mu \phi^A_1 
\; \phi^B_1
+
\left\{
{\partial^2 \L\over \partial\phi^A\; \partial \phi^B} 
\right\}
\; \phi^A_1 \; \phi^B_1
\Bigg]
\nonumber
\\
&+&
O(\epsilon^3).
\end{eqnarray}
%
Because the fields now carry indices ($AB$\/) we cannot cast the
action into quite as simple a form as was possible in the single-field
case.  The equation of motion for the linearized fluctuations are now
read off as
%
\begin{eqnarray}
&&  \partial_\mu \left(\left\{
  {\partial^2 \L\over\partial(\partial_\mu \phi^A) \; 
  \partial(\partial_\nu \phi^B)} 
   \right\}
 \partial_\nu \phi^B_1 \right)
+   \partial_\mu \left(
{\partial^2 \L\over \partial(\partial_\mu \phi^A)\; \partial \phi^B} 
\; \phi^B_1 \right)
\nonumber
\\
&&
\qquad
-  \partial_\mu \phi^B_1 \; 
{\partial^2 \L\over \partial(\partial_\mu \phi^B)\; \partial \phi^A} 
- \left(
{\partial^2 \L\over \partial\phi^A\; \partial \phi^B} 
\right)
\phi^B_1
= 0. 
\label{E:unmaked-up}
\end{eqnarray}
%
This is a linear second-order {\emph{system}} of partial differential
equations with position-dependent coefficients. This system of PDEs is
automatically self-adjoint (with respect to the trivial ``flat''
measure $\d^{d+1} x$).

To simplify the notation we introduce a number of definitions. First
%
\begin{equation}
f^{\mu\nu}{}_{AB} \equiv
{1 \over 2}\left(
{\partial^2 {\cal L}\over\partial(\partial_\mu \phi^A) \; 
\partial(\partial_\nu \phi^B)}+
{\partial^2 {\cal L}\over\partial(\partial_\nu \phi^A) \; 
\partial(\partial_\mu \phi^B)} 
\right).
\end{equation}
%
This quantity is independently symmetric under interchange of $\mu$,
$\nu$ and $A$, $B$.  We will want to
interpret this as a generalization of the ``densitized metric'', $f^{\mu\nu}$, but
the interpretation is not as straightforward as for the single-field
case.
Next, define
%
\begin{eqnarray}
\Gamma^\mu{}_{AB} &\equiv&
+
{\partial^2 {\cal L}\over \partial(\partial_\mu \phi^A)\; \partial \phi^B}
-
{\partial^2 {\cal L}\over \partial(\partial_\mu \phi^B)\; \partial \phi^A} 
\nonumber\\
&&
+
{1 \over 2}\partial_{\nu}\left(
{\partial^2 {\cal L}\over\partial(\partial_\nu \phi^A) \; 
\partial(\partial_\mu \phi^B)}-
{\partial^2 {\cal L}\over\partial(\partial_\mu \phi^A) \; 
\partial(\partial_\nu \phi^B)} 
\right).
\end{eqnarray}
%
This quantity is anti-symmetric in $A$, $B$. One might want to interpret
this as some sort of ``spin connexion'', or possibly as some
generalization of the notion of ``Dirac matrices''. Finally, define
%
\begin{equation}
K_{AB} =
-{\partial^2 \L\over \partial\phi^A\; \partial \phi^B} 
+
{1\over2}
\partial_\mu \left(
{\partial^2 \L\over \partial(\partial_\mu \phi^A)\; \partial \phi^B} 
\right)
+
{1\over2}
\partial_\mu \left(
{\partial^2 \L\over \partial(\partial_\mu \phi^B)\; \partial \phi^A} 
\right).
\end{equation}
%
This quantity is by construction symmetric in ($AB$\/). We will want
to interpret this as some sort of ``potential'' or ``mass matrix''.
Then the crucial point for the following discussion is to realize that
equation (\ref{E:unmaked-up}) can be written in the compact form
%
\begin{equation}
\partial_\mu \left( f^{\mu\nu}{}_{AB}\; \partial_\nu \phi^B_1 \right)
+ {1\over2} 
\left[ \Gamma^\mu_{AB} \; \partial_\mu \phi^B_1 
+ \partial_\mu (\Gamma^\mu_{AB} \; \phi^B_1) \right]
+ K_{AB} \; \phi^B_1 
= 0.
\label{E:system}
\end{equation}
%
Now it is more transparent that this is a formally self-adjoint
second-order linear {\em system} of PDEs. Similar considerations can be 
applied to the linearization of any hyperbolic system of second-order PDEs. 

Consider an eikonal approximation for an arbitrary direction
in field space, that is, take
%
\begin{equation}
\phi^A(x) = \epsilon^A(x) \; \exp[-i\varphi(x)],
\end{equation}
%
with $\epsilon^A(x)$ a slowly varying amplitude, and $\varphi(x)$ a
rapidly varying phase. In this eikonal approximation (where we neglect
gradients in the amplitude, and gradients in the coefficients of the
PDEs, retaining only the gradients of the phase) the linearized system
of PDEs (\ref{E:system}) becomes
%
\begin{equation}
\left\{
f^{\mu\nu}{}_{AB}\; \partial_\mu \varphi(x) \; \partial_\nu \varphi(x) 
+
\Gamma^\mu{}_{AB}\;   \partial_\mu \varphi(x)
+
K_{AB} 
\right\} \; 
\epsilon^B(x) = 0.
\label{E:eikonal}
\end{equation}
%
This has a nontrivial solution if and only if $\epsilon^A(x)$ is a
null eigenvector of the matrix 
%
\begin{equation}
f^{\mu\nu}{}_{AB}\; k_\mu \; k_\nu
+ \Gamma^\mu{}_{AB}\;k_\mu 
+ K_{AB},
\end{equation}
where $k_\mu=\partial_\mu\varphi(x)$.
%
Now, the condition for such a null eigenvector to exist is that
%
\begin{equation}
F(x,k) \equiv 
\det \left\{ f^{\mu\nu}{}_{AB}\; 
k_\mu  \; k_\nu 
+
\Gamma^\mu{}_{AB}\;   k_\mu 
+
K_{AB}
\right\} = 0,
\label{E:fresnel}
\end{equation}
%
with the determinant to be taken on the field space indices $AB$. This
is the natural generalization to the current situation of the Fresnel
equation of bi-refringent optics~\cite{Born-Wolf,Landau}.  Following the
analogy with the situation in electrodynamics (either nonlinear
electrodynamics, or more prosaically propagation in a bi-refringent
crystal), the null eigenvector $\epsilon^A(x)$ would correspond to a
specific ``polarization''. The Fresnel equation then describes how
different polarizations can propagate at different velocities (or in more geometrical
language, can see different metric
structures).  In particle physics language this determinant condition
$F(x,k)=0$ is the natural generalization of the ``mass shell''
constraint. Indeed it is useful to define the mass shell as a subset of
the cotangent space by
%
\begin{equation}
{\cal F}(x) \equiv
\left\{ 
k_\mu \; \bigg| \; F(x,k) = 0
\right\}.
\label{E:mass-shell}
\end{equation}
%
In more mathematical language we are looking at the null space of the
determinant of the ``symbol'' of the system of PDEs. By investigating
$F(x,k)$ one can recover part (not all) of the information encoded in
the matrices $f^{\mu\nu}{}_{AB}$, $\Gamma^\mu{}_{AB}$, and $K_{AB}$, or
equivalently in the ``generalized Fresnel equation'' (\ref{E:fresnel}).
(Note that for the determinant equation to be useful it should be
non-vacuous; in particular one should carefully eliminate all gauge and
spurious degrees of freedom before constructing this ``generalized
Fresnel equation'', since otherwise the determinant will be identically
zero.) We now want to make this analogy with optics more precise, by
carefully considering the notion of characteristics and characteristic
surfaces.  We will see how to extract from the the high-frequency
high-momentum regime described by the eikonal approximation all the
information concerning the causal structure of the theory.


One of the key structures that a Lorentzian spacetime metric provides is
the notion of causal relationships. This suggests that it may be
profitable to try to work backwards from the causal structure to
determine a Lorentzian metric. Now the causal structure implicit in the
system of second-order PDEs given in equation (\ref{E:system}) is
described in terms of the characteristic surfaces, and it is for this
reason that we now focus on characteristics as a way of encoding causal
structure, and as a surrogate for some notion of Lorentzian metric. Note
that via the Hadamard theory of surfaces of discontinuity the
characteristics can be identified with the infinite-momentum limit of
the eikonal approximation~\cite{Hadamard}. That is, when extracting the
characteristic surfaces we neglect subdominant terms in the generalized
Fresnel equation and focus only on the leading term in the symbol
($f^{\mu\nu}{}_{AB}$). In particle physics language going to the
infinite-momentum limit puts us on the light cone instead of the mass
shell; and it is the light cone that is more useful in determining
causal structure.  The ``normal cone'' at some specified point $x$,
consisting of the locus of normals to the characteristic surfaces, is
defined by %

\begin{equation}
{\cal N}(x) \equiv
\left\{ 
k_\mu \; \bigg| \; 
\det\left(f^{\mu\nu}{}_{AB} \;\; k_\mu \; k_\mu\right) = 0
\right\}.
\label{E:normal}
\end{equation}
%

As was the case for the Fresnel equation (\ref{E:fresnel}), the
determinant is to be taken on the field indices $AB$.  (Remember to
eliminate spurious and gauge degrees of freedom so that this determinant
is not identically zero.) We emphasise that the algebraic equation
defining the normal cone is the leading term in the Fresnel equation
encountered in discussing the eikonal approximation. If there are $N$
fields in total then this ``normal cone'' will generically consist of
$N$ nested sheets each with the topology (not necessarily the geometry)
of a cone. Often several of these cones will coincide, which is not
particularly troublesome, but unfortunately it is also common for some
of these cones to be degenerate, which is more problematic.


It is convenient to define a function $Q(x,k)$ on the co-tangent bundle
%
\begin{equation}
Q(x,k) \equiv \det\left(f^{\mu\nu}{}_{AB}(x) \; k_\mu \; k_\mu \right).
\end{equation}
The function $Q(x,k)$ defines a completely-symmetric spacetime tensor
(actually, a tensor density) with $2N$ indices
%
\begin{equation}
Q(x,k) = Q^{\mu_1\nu_1\mu_2\nu_2\cdots\mu_N\nu_N}(x) 
\;\; k_{\mu_1} \; k_{\nu_1} \; k_{\mu_2} \; k_{\nu_2} \cdots \; 
k_{\mu_N} \;  k_{\nu_N}. 
\end{equation}
%
(Remember that $f^{\mu\nu}{}_{AB}$ is symmetric in both $\mu\nu$ and
$AB$ independently.)  Explicitly, using the expansion of the
determinant in terms of completely antisymmetric field-space
Levi--Civita tensors
%
\begin{eqnarray}
Q^{\mu_1\nu_1\mu_2\nu_2\cdots\mu_N\nu_N} &=& {1\over N!} \;
\epsilon^{A_1 A_2 A_3 \cdots A_N} \; \epsilon^{B_1 B_2 B_3 \cdots B_N} \;
\\
&&
\qquad 
f^{\mu_1\nu_1}{}_{A_1 B_1} \;  f^{\mu_2\nu_2}{}_{A_2 B_2} \; 
\cdots  f^{\mu_N\nu_N}{}_{A_N B_N}.
\nonumber
\end{eqnarray}
%
In terms of this $Q(x,k)$ function the normal cone is
%
\begin{equation}
{\cal N}(x) \equiv
\left\{ 
k_\mu \; \bigg| \; Q(x,k) = 0
\right\}.
\end{equation}
%
In contrast, the ``Monge cone'' ({\aka} ``ray cone'', {\aka}
``characteristic cone'', {\aka} ``null cone'') is the envelope of the
set of characteristic surfaces through the point $x$.  Thus the
``Monge cone'' is dual to the ``normal cone'', its explicit
construction is given by (Courant and Hilbert~\cite{Courant}, volume
2, pp 583):
%
\begin{equation}
{\cal M}(x) = 
\left\{ 
t^\mu =  {\partial Q(x,k)\over\partial k_\mu} 
\;
\bigg| \; k_\mu \in {\cal N}(x)
\right\}.
\label{E:null}
\end{equation}


The structure of the normal and Monge cones encode all the information
related with the causal propagation of signals associated with the
system of PDEs. We will now see how to relate this causal structure with
the existence of effective spacetime metrics, from the experimentally
favoured single-metric theory compatible with the Einstein equivalence
principle to the most complicated case of pseudo-Finsler
geometries~\cite{EDM}.

%
\begin{itemize}
\item
Suppose that $f^{\mu\nu}{}_{AB}$ factorizes
%
\begin{equation}
f^{\mu\nu}{}_{AB} = h_{AB} \;  f^{\mu\nu}.
%=  h_{AB} \;  \sqrt{-g} \; g^{\mu\nu}.
\end{equation}
%
Then 
\begin{equation}
Q(x,k) = \det(h_{AB}) \; [f^{\mu\nu}\; k_\mu\; k_\nu]^N
\end{equation}

The Monge cones and normal cones are then true geometrical cones (with
the $N$ sheets lying directly on top of one another). The normal modes
all see the  same spacetime metric, defined up to an unspecified
conformal factor by $g^{\mu\nu}\propto f^{\mu\nu}$.  This situation is
the most interesting from the point of view of general relativity.
Physically it corresponds to a single-metric theory, and mathematically
it corresponds to a strict algebraic condition on the
$f^{\mu\nu}{}_{AB}$.

\item
The next most useful situation corresponds to the commutativity condition:
\begin{equation}
f^{\mu\nu}{}_{AB}\; f^{\alpha\beta}{}_{BC} = 
f^{\alpha\beta}{}_{AB}\; f^{\mu\nu}{}_{BC};
\qquad
\hbox{that is}
\qquad
[\; \bm{f}^{\mu\nu},  \bm{f}^{\alpha\beta} \,] = 0.
\end{equation}
%
If this algebraic condition is satisfied, then for all spacetime indices
$\mu\nu$ and $\alpha\beta$ the $f^{\mu\nu}{}_{AB}$ can be simultaneously
diagonalized in field space leading to
%
\begin{equation}
\bar f^{\mu\nu}{}_{AB} = 
{\mathrm{diag}}\{\bar f^{\mu\nu}_1, \bar f^{\mu\nu}_2, \bar f^{\mu\nu}_3, 
\cdots \bar f^{\mu\nu}_N\}
\end{equation}
and then
\begin{equation}
Q(x,k) = \prod_{A=1}^N \; [\bar f^{\mu\nu}_A \; k_\mu\; k_\nu]
\end{equation}
This case corresponds to an $N$-metric theory, where up to an 
unspecified conformal factor
$g^{\mu\nu}_A \propto \bar f^{\mu\nu}_A$. This is the natural 
generalization of the two metric situation in bi-axial crystals. 

\item If $f^{\mu\nu}{}_{AB}$ is completely general, satisfying no
special algebraic condition, then $Q(x,k)$ does not factorize and is in
general a polynomial of degree $2N$ in the wave vector $k_\mu$. This is
the natural generalization of the situation in bi-axial crystals. (And
for any deeper analysis of this situation one will almost certainly need
to adopt pseudo-Finsler techniques~\cite{EDM}.)

\end{itemize}
%

The message to be extracted from this rather formal discussion is that
effective metrics are rather general and mathematically robust objects
that can arise in quite abstract settings --- in the abstract setting
discussed here it is the algebraic properties of the object
$f^{\mu\nu}{}_{AB}$ that eventually leads to mono-metricity,
multi-metricity, or worse. The current abstract discussion also serves
to illustrate, yet again, 
\begin{enumerate}

\item that there is a significant difference between the levels of
physical normal modes (wave equations), and geometrical normal modes
(dispersion relations), and

\item that the densitized inverse metric is in many ways more
fundamental than the metric itself.

\end{enumerate}
%---------------------------------------------------------------------
% end of section on normal mode meta-models
%---------------------------------------------------------------------





%----------------------------------------------
\subsection{Quantum models}
%-------------------------------------
\label{S:quantum-models}
%----------------------------------------------


%----------------------------------------------
\subsubsection{Bose--Einstein condensates}
%----------------------------------------------

%------------------------------------------------------------------------- 
% Author-specific definitions 
%------------------------------------------------------------------------- 
\def\g{\kappa} 
\def\half{{1\over2}} 
\def\L{{\mathcal L}} 
\def\S{{\mathcal S}} 
\def\d{{\mathrm{d}}} 
\def\x{{\mathbf x}} 
\def\v{{\mathbf v}} 
\def\k{{\mathbf k}} 
\def\im{{\rm i}} 
\def\etal{{\emph{et al\/}}} 
\def\det{{\mathrm{det}}} 
\def\tr{{\mathrm{tr}}} 
\def\ie{{\em i.e.\/}}
\def\Schrodinger{Schr\"odinger} 
\def\bnabla{\mbox{\boldmath$\nabla$}} 
\def\Box{\kern0.5pt{\lower0.1pt\vbox{\hrule height.5pt width 6.8pt     
\hbox{\vrule width.5pt height6pt \kern6pt \vrule width.3pt}     
\hrule height.3pt width 6.8pt} }\kern1.5pt} 
\def\HRULE{{\bigskip\hrule\bigskip}} 
%------------------------------------------------------------------------- 

We have seen that one of the main aims of research in analogue models of
gravity is the possibility of simulating semiclassical gravity
phenomena, such as the Hawking radiation effect or cosmological particle
production. In this sense systems characterized by a high degree of
quantum coherence, very cold temperatures, and low speeds of sound offer
the best test field. Hence it is not surprising that in recent years
Bose--Einstein condensates (BECs) have become the subject of extensive
study as possible analogue models of general
relativity~\cite{Garay:1999sk,Garay:2000jj,Barcelo:2001cp,Barcelo:2001ca,
Barcelo:2003wu, Fedichev:2003bv, Fedichev:2003id}.

Let us start by very briefly reviewing the derivation of the acoustic
metric for a BEC system, and show that the equations for the phonons
of the condensate closely mimic the dynamics of a scalar field in a
curved spacetime.  In the dilute gas approximation, one can describe a
Bose gas through a quantum field ${\widehat \Psi}$ satisfying 
%
\begin{eqnarray} \im
\hbar \; \frac{\partial }{\partial t} {\widehat \Psi}= \left( -
{\hbar^2 \over 2m} \nabla^2 + V_{\rm ext}(\x) +\g(a)\;{\widehat
\Psi}^{\dagger}{\widehat \Psi} \right){\widehat \Psi}. 
\end{eqnarray}
% 
Here $\g$ parameterizes the strength of the interactions between the
different bosons in the gas. It can be re-expressed in terms of the
scattering length as 
% 
\begin{eqnarray} \g(a) = {4\pi a \hbar^2\over
m}. 
\end{eqnarray} 
% 
As usual, the quantum field can be separated into
a macroscopic (classical) condensate and a fluctuation: ${\widehat
\Psi}=\psi+{\widehat \varphi}$, with $\langle {\widehat \Psi}
\rangle=\psi $. Then, by adopting the self-consistent mean field
approximation (see for example~\cite{Griffin}) 
% 
\begin{eqnarray}
{\widehat \varphi}^{\dagger}{\widehat \varphi}{\widehat \varphi}
\simeq 
2\langle {\widehat \varphi}^{\dagger}{\widehat \varphi} \rangle \;
{\widehat \varphi} 
+ \langle {\widehat \varphi} {\widehat \varphi}
\rangle \; {\widehat \varphi}^{\dagger}, 
\end{eqnarray} 
% 
one can arrive
at the set of coupled equations: 
% 
\begin{eqnarray} \im \hbar \;
\frac{\partial }{\partial t} \psi(t,\x)&=& \left ( - {\hbar^2 \over
2m} \nabla^2 + V_{\rm ext}(\x) + \g \; n_c \right) \psi(t,\x)
\nonumber
\\
&&
\qquad + \g \left\{2\tilde n  \psi(t,\x)+ \tilde m \psi^*(t,\x) \right\};
\label{bec-self-consistent1}
\\
&&
\nonumber
\\ 
\im \hbar \; 
\frac{\partial }{\partial t} {\widehat \varphi}(t,\x) &=&  
\left( - {\hbar^2 \over 2m} \nabla^2  +   
V_{\rm ext}(\x)   +\g \;2 n_T \right){\widehat \varphi} (t,\x)
\nonumber
\\
&&
\qquad +  \g \; m_T \; {\widehat \varphi}^{\dagger}(t,\x).  
\label{quantum-field} 
\end{eqnarray} 
% 
Here  
% 
\begin{eqnarray} 
&&n_c \equiv \left| \psi(t,\x) \right|^2;  \hspace{2cm} m_c \equiv 
\psi^2(t,\x); \\ &&\tilde n \equiv  \langle 
{\widehat \varphi}^{\dagger}\,{\widehat \varphi} \rangle;  
\hspace{2.7cm} \tilde m \equiv  \langle {\widehat \varphi}\,
{\widehat \varphi} \rangle; \\ 
&&n_T=n_c+\tilde n;  \hspace{2.4cm} m_T=m_c+\tilde m.  
\end{eqnarray} 
% 
The equation for the classical wave function of the condensate is
closed only when the back-reaction effect due to the fluctuations are
neglected. (This back-reaction is hiding in the parameters $\tilde m$
and $\tilde n$.) This is the approximation contemplated by the
Gross--Pitaevskii equation. In general one will have to solve both
equations simultaneously.  Adopting the Madelung representation for
the wave function of the condensate
% 
\begin{equation} 
\psi(t,\x)=\sqrt{n_c(t,\x)} \; \exp[-{\rm i}\theta(t,\x)/\hbar],  
\end{equation} 
% 
and defining an irrotational ``velocity field'' by $\v\equiv
{\bnabla\theta}/{m}$, the Gross--Pitaevskii equation can be rewritten
as a continuity equation plus an Euler equation:
% 
\begin{eqnarray} 
&& \frac{\partial}{\partial t}n_c+\bnabla\cdot({n_c \v})=0, 
\label{E:continuity}\\ 
&& m\frac{\partial}{\partial t}\v+\bnabla\left(\frac{mv^2}{2}+ 
V_\mathrm{ext}(t,\x)+\g n_c- \frac{\hbar^2}{2m}
\frac{\nabla^{2}\sqrt{n_c}}{\sqrt{n_c}} \right)=0. 
\label{E:Euler1} 
\end{eqnarray} 
%
These equations are completely equivalent to those of an irrotational
and inviscid fluid apart from the existence of the so-called quantum
potential 
%
\begin{equation}
V_{\rm quantum}=
-\hbar^2\nabla^{2}\sqrt{n_c}/(2m\sqrt{n_c}),
\end{equation}
%
which has the dimensions of an energy. Note that 
% 
\begin{equation} 
n_c \; \nabla_i
V_{\rm quantum} \equiv n_c \; \nabla_i \left[ -{\hbar^2\over2m}
{\nabla^{2}\sqrt{n_c}\over\sqrt{n_c}} \right] = \nabla_j \left[
-{\hbar^2\over4m} \; n_c \; \nabla_i \nabla_j \ln n_c \right],
\end{equation} 
% 
which justifies the introduction of the so-called
quantum stress tensor 
% 
\begin{equation} 
\sigma_{ij}^{\rm quantum} =
-{\hbar^2\over4m} \; n_c \; \nabla_i \nabla_j \ln n_c. 
\end{equation}
% 
This tensor has the dimensions of pressure, and may be viewed as an
intrinsically quantum anisotropic pressure contributing to the Euler
equation.  If we write the mass density of the Madelung fluid as $\rho
= m \; n_c$, and use the fact that the flow is irrotational then the
Euler equation takes the form 
% 
\begin{equation} \rho \left[
\frac{\partial} {\partial t}\v+ (\v\cdot\bnabla) \v \right] + \rho \;
\bnabla \left[\frac{V_\mathrm{ext}(t,\x)}{m} \right] + \bnabla
\left[{\g \rho^2\over 2 m^2}\right] + \bnabla \cdot \sigma^{\rm
quantum} =0. \label{E:Euler2} 
\end{equation} 
% 
Note that the term
$V_\mathrm{ext}/m$ has the dimensions of specific enthalpy, while $\g
\rho^2/(2m)$ represents a bulk pressure.  When the gradients in the
density of the condensate are small one can neglect the quantum stress
term leading to the standard hydrodynamic approximation. Because the
flow is irrotational, the Euler equation is often more conveniently
written in Hamilton--Jacobi form: 
% 
\begin{equation} m
\frac{\partial}{\partial t}\theta+ \left( \frac{[\bnabla\theta]^2}{2m}
+V_\mathrm{ext}(t,\x)+\g n_c-
\frac{\hbar^2}{2m}\frac{\nabla^{2}\sqrt{n_c}}{\sqrt{n_c}}
\right)=0. \label{E:HJ} \end{equation} 
% 
Apart from the wave function
of the condensate itself, we also have to account for the [typically
small] quantum perturbations of the system
(\ref{quantum-field}). These quantum perturbations can be described in
several different ways, here we are interested in the ``quantum
acoustic representation'' 
% 
\begin{eqnarray} \widehat
\varphi(t,\x)=&&\hspace{-3mm} e^{-\im \theta/\hbar} \left({1 \over 2
\sqrt{n_c}} \; \widehat n_1 - \im \; {\sqrt{n_c} \over \hbar} \;\widehat
\theta_1\right), \label{representation-change} \end{eqnarray} 
% 
where
$\widehat n_1,\widehat\theta_1$ are real quantum fields.  By using
this representation equation (\ref{quantum-field}) can be rewritten as
% 
\begin{eqnarray} &&\partial_t \widehat n_1 + {1\over m}
\bnabla\cdot\left( n_1 \; \bnabla \theta + n_c \; \bnabla \widehat
\theta_1 \right) = 0, \label{pt1}
\\ &&\partial_t \widehat \theta_1   +  
{1\over m} \bnabla \theta \cdot \bnabla \widehat \theta_1  
+ \g(a) \; n_1 - {\hbar^2\over2 m}\; D_2 \widehat n_1 = 0. 
\label{pt2} 
\end{eqnarray} 
% 
Here $D_2$ represents a second-order differential operator obtained
from linearizing the quantum potential. Explicitly: 
% 
\begin{eqnarray}
D_2\, \widehat n_1 &\equiv& 
-\half n_c^{-3/2} \;[\nabla^2
(n_c^{+1/2})]\; \widehat n_1 
+\half n_c^{-1/2} \;\nabla^2
(n_c^{-1/2}\; \widehat n_1). 
\end{eqnarray} 
% 
The equations we have just written can be obtained easily by
linearizing the Gross--Pitaevskii equation around a classical
solution: $n_c \rightarrow n_c + \widehat n_1$, $\phi \rightarrow \phi
+ \widehat \phi_1$.  It is important to realize that in those
equations the back-reaction of the quantum fluctuations on the
background solution has been assumed negligible.  We also see in those
equations, (\ref{pt1}) and (\ref{pt2}), that time variations of
$V_\mathrm{ext}$ and time variations of the scattering length $a$
appear to act in very different ways.  Whereas the external potential
only influences the background equation~(\ref{E:HJ}) [and hence the
acoustic metric in the analogue description], the scattering length
directly influences both the perturbation and background equations.
{From} the previous equations for the linearized perturbations it is
possible to derive a wave equation for $\widehat \theta_{1}$ (or
alternatively, for $\widehat n_{1}$). All we need is to substitute in
equation~(\ref{pt1}) the $\widehat n_{1}$ obtained from
equation~(\ref{pt2}).  This leads to a PDE that is second-order in
time derivatives but infinite order in space derivatives --- to
simplify things we can construct the symmetric $4 \times 4$ matrix 
%
\begin{equation} 
f^{\mu\nu}(t,\x) \equiv \left[
\matrix{f^{00}&\vdots&f^{0j}\cr
\cdots\cdots&\cdot&\cdots\cdots\cdots\cdots\cr f^{i0}&\vdots&f^{ij}\cr
} \right]. \label{E:explicit} 
\end{equation} 
% 
(Greek indices run from
$0$--$3$, while Roman indices run from $1$--$3$.)  Then, introducing
$(3+1)$--dimensional space-time coordinates 
%
\begin{equation}
x^\mu \equiv (t;\, x^i)
\end{equation}
%
the wave equation for $\theta_{1}$ is easily rewritten as 
%
\begin{equation} \label{E:compact} \partial_\mu ( f^{\mu\nu} \;
\partial_\nu \widehat \theta_1) = 0. \label{weq-phys} 
\end{equation} 
%
Where the $f^{\mu\nu}$ are \emph{differential operators} acting on space
only: 
%
\begin{eqnarray} 
f^{00} &=& - \left[ \g(a) - {\hbar^2\over 2m}\;
D_2 \right]^{-1} \\ f^{0j} &=& -\left[ \g(a) - {\hbar^2\over 2m}\; D_2
\right]^{-1}\; {\nabla^j \theta_0\over m} \\ f^{i0} &=& - {\nabla^{i}
\theta_0\over m} \; \left[ \g(a) - {\hbar^2\over2 m}\; D_2
\right]^{-1} \\ f^{ij} &=& {n_c \; \delta^{ij}\over m} - {\nabla^{i}
\theta_0\over m} \; \left[ \g(a) - {\hbar^2\over2 m}\; D_2
\right]^{-1}\; {\nabla^{j} \theta_0\over m}. 
\end{eqnarray} 
% 
Now, if we make an spectral decomposition of the field $\widehat
\theta_1$ we can see that for wavelengths larger than $\hbar /mc_s$
(this corresponds to the ``healing length'', as we will explain
below), the terms coming from the linearization of the quantum
potential (the $D_2$) can be neglected in the previous expressions, in
which case the $f^{\mu\nu}$ can be approximated by numbers, instead of
differential operators. (This is the heart of the acoustic
approximation.)  Then, by identifying 
% 
\begin{equation} \sqrt{-g} \;
g^{\mu\nu}=f^{\mu\nu},
\end{equation} 
% 
the equation for the field $\widehat \theta_1$
becomes that of a (massless minimally coupled) quantum scalar field
over a curved background 
% 
\begin{equation}
\Delta\theta_{1}\equiv\frac{1}{\sqrt{-g}}\;
\partial_{\mu}\left(\sqrt{-g}\; g^{\mu\nu}\; \partial_{\nu}\right)
\widehat\theta_{1}=0, 
\end{equation} 
% 
with an effective metric of the
form 
% 
\begin{equation} g_{\mu\nu}(t,\x) \equiv {n_c\over m\;
c_s(a,n_c)} \left[ \matrix{-\{c_s(a,n_c)^2-v^2\}&\vdots& - v_j \cr
\cdots\cdots\cdots\cdots&\cdot&\cdots\cdots\cr -v_i
&\vdots&\delta_{ij}\cr } \right]. 
\end{equation} 
% 
Here the magnitude
$c_s(n_c,a)$ represents the speed of the phonons in the medium: 
%
\begin{equation} 
c_s(a,n_c)^2={\g(a) \; n_c \over m}. 
\end{equation} 
%
With this effective metric now in hand, the analogy is fully
established, and one is now in a position to start asking more specific
physics questions.

%----------------------------------------------------------------
\subsubsection{BEC models in the eikonal approximation}
%---------------------------------------------------------------- 
It is interesting to consider the case in which the above
``hydrodynamical" approximation for BECs does not hold. In order to
explore a regime where the contribution of the quantum potential
cannot be neglected we can use the so called {\emph{eikonal}}
approximation, a high-momentum approximation where the phase
fluctuation $\widehat \theta_1$ is itself treated as a slowly-varying
amplitude times a rapidly varying phase. This phase will be taken to
be the same for both $\widehat n_1$ and $\widehat \theta_1$
fluctuations. In fact, if one discards the unphysical possibility that
the respective phases differ by a time varying quantity, any
time-constant difference can be safely reabsorbed in the definition of
the (complex) amplitudes.  Specifically, we shall write 
%
\begin{eqnarray} 
{\widehat\theta}_1(t,\x) &=& {\rm Re}\left\{
{\mathcal A}_\theta \; \exp(-i\phi) \right\},\\ {\widehat n}_1(t,\x)
&=& {\rm Re}\left\{ {\mathcal A}_\rho \; \exp(-i\phi)
\right\}. 
\end{eqnarray} 
% 
As a consequence of     our starting assumptions, gradients  of    the
amplitude, and gradients of  the background fields, are systematically
ignored relative to gradients of $\phi$.  [Warning:  what we are doing
here is  not quite a  ``standard'' eikonal approximation, in the sense
that  it is  not  applied directly  on the  fluctuations  of the field
$\psi(t,\x)$ but separately on their  amplitudes and phases $\rho_{1}$
and $\phi_{1}$.]  We  adopt the notation  
% 
\begin{equation}  
\omega ={\partial\phi\over\partial t};
\qquad k_i = \nabla_i \phi. \end{equation} 
% 
Then the operator $D_2$
can be approximated as 
% 
\begin{eqnarray} 
D_2 \;{\widehat n}_1
&\equiv& -\half n_c^{-3/2} \;[\Delta (n_c^{+1/2})]\; {\widehat n}_1
+\half n_c^{-1/2} \;\Delta (n_c^{-1/2} {\widehat n}_1) 
\\ 
&\approx&
+\half n_c^{-1} \;[\Delta {\widehat n}_1] 
\\ 
&=& -\half n_c^{-1}
\;k^2 \;{\widehat n}_1. 
\end{eqnarray} 
% 
A similar result holds for
$D_2$ acting on ${\widehat \theta}_1$.  That is, under the eikonal
approximation we effectively replace the {\emph{operator}} $D_2$ by
the {\emph{function}} 
% 
\begin{equation} 
D_2 \to -\half n_c^{-1}k^2. 
\end{equation} 
% 
For the matrix $f^{\mu\nu}$ this effectively
results in the replacement 
% 
\begin{eqnarray} 
f^{00} &\to& - \left[
\kappa(a) + {\hbar^2 \; k^2\over4m\;n_c} \right]^{-1} \\ f^{0j} &\to&
-\left[ \kappa(a) + {\hbar^2 \; k^2\over4m\;n_c}\right]^{-1}\;
{\nabla^j \theta_0\over m} \\ f^{i0} &\to& - {\nabla^i \theta_0\over
m} \; \left[ \kappa(a) + {\hbar^2 \; k^2\over4m\;n_c} \right]^{-1} \\
f^{ij} &\to& {n_c \; \delta^{ij}\over m} - {\nabla^i \theta_0\over m}
\; \left[ \kappa(a) + {\hbar^2 \; k^2\over4m\;n_c} \right]^{-1}\;
{\nabla^j \theta_0\over m} 
\end{eqnarray} 
% 
(As desired, this has the net effect of making $f^{\mu\nu}$ a matrix
of numbers, not operators.) The physical wave equation
(\ref{weq-phys}) now becomes a nonlinear dispersion relation 
%
\begin{equation} 
f^{00} \;\omega^2 + (f^{0i} +f^{i0}) \;\omega \;k_i +
f^{ij} \;k_i \;k_j = 0. 
\end{equation} 
% 
After substituting the approximate $D_2$ into this dispersion relation
and rearranging, we see (remember: $k^2 = ||k||^2 = \delta^{ij}
\;k_i \;k_j$) 
% 
\begin{equation} 
-\omega^2 + 2 \; v_0^i \; \omega k_i
+ {n_c k^2\over m}\left[\kappa(a)+{\hbar^2\over4m n_c} k^2\right] -
(v_0^i \; k_i)^2 = 0. 
\end{equation} 
% 
That is: 
% 
\begin{equation}
\left(\omega - v_0^i \; k_i\right)^2 = {n_c k^2\over
m}\left[\kappa(a)+{\hbar^2\over4m n_c} k^2\right] 
\end{equation} 
%
Introducing the speed of sound $c_s$ this takes the form: 
%
\begin{equation} \omega= v_0^i \; k_i \pm \sqrt{c_s^2 k^2+\left({\hbar
\over 2 m}\;k^2\right)^2}. 
\label{eq:disprel} 
\end{equation} 
% 
%Notice how the previous anisotropic dispersion relation differs from
%this isotropic one: in that case there are three different physical
%sound velocities in three orthogonal principal directions.

\noindent 
At this stage some observations are in order: 

---(1) It is interesting to recognize that the dispersion relation
(\ref{eq:disprel}) is exactly in agreement with that found in 1947 by
Bogoliubov~\cite{Bogoliubov:1947} (reprinted in \cite{Pines:1962}; see also 
\cite{LLP}) for the collective excitations of a
homogeneous Bose gas in the limit $T\to 0$ (almost complete
condensation). In his derivation Bogoliubov applied a diagonalization
procedure for the Hamiltonian describing the system of bosons.  

---(2) It is easy to see that (\ref{eq:disprel}) actually interpolates
between two different regimes depending on the value of the wavelength
$\lambda= 2\pi / ||k||$ with respect to the ``acoustic Compton
wavelength'' $\lambda_c=h/(m c_s)$. (Remember that $c_s$ is the speed
of sound; this is not a standard particle physics Compton wavelength.)
%Furthermore $||k||$ is defined using the inverse 3-metric $h^{ij}$.)  
In particular, if we assume $v_{0}=0$ (no background velocity), then
for large wavelengths $\lambda \gg \lambda_c$ one gets a standard
phonon dispersion relation $\omega \approx c ||k||$. For wavelengths
$\lambda \ll \lambda_c$ the quasi-particle energy tends to the kinetic
energy of an individual gas particle and in fact $\omega \approx
\hbar^2 k^2/(2 m)$.  

We would like also to highlight that in relative terms, the
approximation by which one neglects the quartic terms in the dispersion
relation gets worse as one moves closer to a horizon where $v_0=-c_s$.
The non-dimensional parameter that provides this information is defined by
%
\begin{equation} 
\delta \equiv {\sqrt{1+{\lambda_c^2 \over 4\lambda^2}}-1 \over (1-v_0/c_s)}
\simeq {1 \over (1-v_0/c_s)}{\lambda_c^2 \over 8\lambda^2}.
\end{equation}
%
As we will discuss in subsection \ref{S:Horizon-stability}, this is 
the reason why sonic horizons in a BEC can exhibit different features from
those in standard general relativity.



---(3) The dispersion relation (\ref{eq:disprel}) exhibits a
contribution due to the background flow $ v_0^i \; k_i$, plus a
quartic dispersion at high momenta. The group velocity is 
%
\begin{equation} 
v_g^i = {\partial\omega\over\partial k_i} = v_0^i \pm
{ \left(c^2+{\hbar^2 \over 2 m^2}k^2\right) \over \sqrt{c^2
k^2+\left({\hbar \over 2 m}\;k^2\right)^2} } \; k^i 
\end{equation} 
%
Dispersion relations of this type (but in most cases with the sign of
the quartic term reversed) have been used by Corley and Jacobson in
analyzing the issue of trans-Planckian modes in the Hawking radiation
from general relativistic black
holes~\cite{Jacobson:1991gr,Jacobson:1993hn,Corley:1996ar}. The
existence of modified dispersion relations (MDR), that is, dispersion
relations that break Lorentz invariance, can be taken as a
manifestation of new physics showing up at high energies/ short
wavelengths. In their analysis, the group velocity reverses its sign
for large momenta. (Unruh's analysis of this problem used a slightly
different toy model in which the dispersion relation saturated at high
momentum~\cite{Unruh:1994zw}.) In our case, however, the group
velocity grows without bound allowing high-momentum modes to escape
from behind the horizon. Thus the acoustic horizon is not absolute in
these models, but is instead frequency dependent, a phenomenon that is
common once non-trivial dispersions are included.

Indeed, with hindsight the fact that the group velocity goes to
infinity for large $k$ was pre-ordained: After all, we started from the
generalized nonlinear Schr\"odinger equation, and we know what its
characteristic curves are. Like the diffusion equation the
characteristic curves of the Schr\"odinger equation (linear or
nonlinear) move at infinite speed. If we then approximate this
generalized nonlinear Schr\"odinger equation in any manner, for
instance by linearization, we cannot change the characteristic curves:
For any well behaved approximation technique, at high frequency and
momentum we should recover the characteristic curves of the system we
started with. However, what we certainly do see in this analysis is a
suitably large region of momentum space for which the concept of the 
effective metric both makes sense, and leads to finite propagation speed
for medium-frequency oscillations.

This type of superluminal dispersion relation has also been analyzed by
Corley and Jacobson~\cite{Corley:1998rk}. They found that this escape of
modes from behind the horizon often leads to self-amplified
instabilities in systems possessing both an inner horizon as well as an
outer horizon, possibly causing them to disappear in an explosion of
phonons. This is also in partial agreement with the stability analysis
performed by Garay {\etal} \cite{Garay:1999sk,Garay:2000jj} using the
whole Bogoliubov equations. Let us however leave further discussion
regarding these developments to the subsection \ref{S:Horizon-stability}
on horizon stability.


%----------------------------------------------
\subsubsection{The Heliocentric universe}
%----------------------------------------------

Helium is one of the most fascinating elements provided by nature. Its
structural richness confers on helium a paradigmatic character regarding
the emergence of many and varied macroscopic properties from the
microscopic world (see~\cite{Volovik:2003fe} and references therein). 
Here, we are interested in the emergence of effective geometries in
helium, and their potential use in testing aspects of semiclassical
gravity.

Helium four, a bosonic system, becomes superfluid at low temperatures
(2.17 K at vapour pressure).  This superfluid behaviour is associated
with the condensation in the vacuum state of a macroscopically large
number of atoms. A superfluid is automatically an irrotational and
inviscid fluid, so in particular one can apply to it the ideas worked
out in Section \ref{S:simple}. The propagation of classical acoustic
waves (scalar waves) over a background fluid flow can be described in
terms of an effective Lorentzian geometry: the acoustic
geometry. However, in this system one can naturally go considerably
further, into the quantum domain. For long wavelengths, the
quasiparticles in this system are quantum phonons. One can separate
the classical behaviour of a background flow (the effective geometry)
from the behaviour of the quantum phonons over this background. In
this way one can reproduce, in laboratory settings, different aspects
of quantum field theory over curved backgrounds. The speed of sound in
the superfluid phase is typically of the order of $cm/sec$. Therefore,
at least in principle, it should not be too difficult to establish
configurations with supersonic flows and their associated ergoregions.

Helium three, the fermionic isotope of helium, in contrast becomes
superfluid at very much lower temperatures (below 2.5 milli-K). The
reason behind this rather different behaviour is the pairing of fermions
to form effective bosons (Cooper pairing), which are then able to
condense. In the so-called $^3$He--A phase, the structure of the
fermionic vacuum is such that it possesses two Fermi points, instead of
the more typical Fermi surface. In an equilibrium configuration one can
choose the two Fermi points to be located at $\{p_x=0,p_y=0,p_z=\pm
p_F\}$ (in this way, the $z$ axis signals the direction of the angular
momentum of the pairs).  Close to either Fermi point the spectrum of
quasiparticles becomes equivalent to that of Weyl fermions. From the
point of view of the laboratory, the system is not isotropic, it is
axisymmetric. There is a speed for the propagation of quasiparticles
along the $z$ axis, $c_\parallel \simeq cm/sec$, and a different speed,
$c_\perp\simeq 10^{-5}\,c_\parallel$, for propagation perpendicular to the
symmetry axis.  However, from an internal observer's point of view this
anisotropy is not ``real'', but can be made to disappear by an
appropriate rescaling the coordinates. Therefore, in the equilibrium
case, we are reproducing the behaviour of Weyl fermions over Minkowski
spacetime. Additionally, the vacuum can suffer collective excitations.
These collective excitations will be experienced by the Weyl
quasiparticles as the introduction of an effective electromagnetic field
and a curved Lorentzian geometry.  The control of the form of this
geometry provides the sought for gravitational analogy.

Apart from the standard way to provide a curved geometry based on
producing non-trivial flows, there is also the possibility of creating
topologically non-trivial configurations with a built-in non-trivial
geometry. For example, it is possible to create a domain-wall
configuration \cite{Jacobson:1998ms,Jacobson:1998he} (the wall contains
the $z$ axis) such that the transverse velocity $c_\perp$ acquires a
profile in the perpendicular direction (say along the $x$ axis) with
$c_\perp$ passing through zero at the wall (see figure \ref{F:soliton}).
This particular arrangement could be used to reproduce a black
hole--white hole configuration only if the soliton is set up to move with
a certain velocity along the $x$ axis. This configuration has the
advantage than it is dynamically stable, for topological reasons, even
when some supersonic regions are created.
   
%====================================================
\begin{figure}[htb]
  \vbox{
  \centerline{{\includegraphics[width=5in]{F-domain-wall}}}
  \caption[Domain wall configuration in $^3$He]{
Domain wall configuration in $^3$He.
  \label{F:soliton}}}
\end{figure}
%====================================================

A third way in which superfluid Helium can be used to create analogues
of gravitational configurations is the study of surface waves (or
ripplons) on the interface between two different phases of $^3$He
\cite{Volovik:2002ci,Volovik:2001yh}.  In particular, if we have a
thin layer of $^3$He-A in contact with another thin layer of $^3$He-B,
the oscillations of the contact surface ``see'' an effective metric of
the form~\cite{Volovik:2002ci,Volovik:2001yh}
% 
\begin{eqnarray}
\mathrm{ds}^2&=&{1 \over (1- \alpha_A\alpha_B U^2)}
\Bigg[-\left(1-W^2-\alpha_A\alpha_B U^2\right)\; \mathrm{dt}^2
\nonumber\\ 
&& \qquad\qquad - 2 \mathbf{W} \cdot \mathbf{dx}\;
\mathrm{dt} +\mathbf{dx}\cdot \mathbf{dx} \Bigg],
\end{eqnarray} 
% 
where 
% 
\begin{eqnarray} 
\mathbf{W}\equiv \alpha_A \mathbf{v_A} +\alpha_B \mathbf{v_B}; 
\qquad
\mathbf{U}\equiv \mathbf{v_A} - \mathbf{v_B};
\end{eqnarray} 
%
and 
% 
\begin{eqnarray} 
\alpha_A \equiv {h_B \rho_A \over h_A \rho_B + h_B \rho_A};
\qquad
\alpha_B \equiv {h_A \rho_B \over h_A \rho_B + h_B \rho_A}. 
\end{eqnarray} 
% 
(All of this provided that we are looking at wavelengths larger than
the layer thickness, $k\,h_A\ll 1$ and $k\,h_B\ll 1$.)
      
%====================================================
\begin{figure}[htb]
  \vbox{
  \centerline{{\includegraphics[width=5in]{F-ripplons}}}
  \caption[Interface between two sliding superfluids]{Ripplons in the 
  interface between two sliding superfluids.
  \label{F:ripplon}}}
\end{figure}
%====================================================

The advantage of using surface waves instead of bulk waves in
superfluids is that one could create horizons without reaching
supersonic speeds in the bulk fluid. This could alleviate the appearance
of dynamical instabilities in the system, that in this case are
controlled by the strength of the interaction of the ripplons with bulk
degrees of freedom~\cite{Volovik:2002ci,Volovik:2001yh}.



%%% end of section on helium

%----------------------------------------------
\subsubsection{Slow light}
%----------------------------------------------


% this chapter is devoted to the work done in slow light. It is possible
% that not all the material in this file will end up in the catalogue
% section (some issue about suitability of the model for simulating
% Hawking radiation might be moved in the issue section).
 
 
The geometrical interpretation of the motion of light in dielectric
media leads naturally to conjecture that the use of flowing dielectrics
might be useful for simulating general relativity metrics with
ergoregions and black holes. Unfortunately, these types of geometry
require flow speeds comparable to the group velocity of the light. Since
typical refractive indexes in non-dispersive media are quite close to
unity, it is then clear that it is practically impossible to use them to
simulate such general relativistic  phenomena. However recent
technological advances have radically changed this state of affairs. In
particular the achievement of controlled slowdown of light, down to
velocities of a few meters per second (or even down to complete rest)
\cite{Hau,Kash,Budker,Koch,Phil,Turu,SZ}, has opened a whole new set of
possibilities regarding the simulation of curved-space metrics via
flowing dielectrics.

But how can  light be slowed down to these ``snail-like'' velocities?
The key effect used to achieve this takes the name of
Electromagnetically Induced transparency (EIT). A laser beam is coupled
to the excited levels of some atom and used to strongly modify its
optical properties. In particular one generally choses an atom with two
long-lived metastable (or stable) states, plus a higher energy state
that has some decay channels into these two lower states. The coupling of the
excited states induced by the laser light can affect the transition from
a lower energy state to the higher one, and hence the capability of the
atom to absorb light with the required transition energy. The system can
then be driven into a state where the transitions between each of the
lower energy states and the higher energy state exactly cancel out, due
to quantum interference, at some specific resonant frequency. In this
way the higher-energy level has null averaged occupation number. This
state is hence called a ``dark state''.  EIT is characterized by a
transparency window, centred around the resonance frequency, where the
medium is \emph{both} almost transparent \emph{and} extremely dispersive
(strong dependence on frequency of the refractive index). This in turn
implies that the group velocity of any light probe would be
characterized by very low real group velocities (with almost vanishing
imaginary part) in proximity to the resonant frequency.

Let us review the most common setup envisaged for this kind of analogue
model. A more detailed analysis can be found in \cite{Leonhardt:2001ye}.
One can start by considering a medium in which an EIT window is opened
via some control laser beam which is oriented perpendicular to the
direction of the flow.  One then illuminates this medium, now along the
flow direction, with some probe light (which is hence perpendicular to
the control beam). This probe beam is usually chosen to be weak with
respect to the control beam, so that it does not modify the optical
properties of the medium. In the case in which the optical properties of
the medium do not vary significantly over several wavelengths of the
probe light, one can neglect the polarization and can hence describe the
propagation of the latter with a simple scalar dispersion
relation~\cite{Leonhardt:2000aa,Fiurasek:2002ay}
%
\begin{equation}
k^2-\frac{\omega^2}{c^2}\left[1+\chi(\omega)\right],
\label{eq:sldr}
\end{equation}
%
where $\chi$ is the susceptibility of the medium, related to the refractive
index $n$ via the simple relation $n=\sqrt{1+\chi}$.

It is easy to see that in this case the group and phase velocities differ
\begin{equation}
v_{\rm g}=\frac{\partial \omega}{\partial k}
=\frac{c}{\sqrt{1+\chi}+\frac{\textstyle \omega}{\textstyle 2n}
\frac{\textstyle \partial\chi}{\textstyle \partial\omega}};
\qquad 
v_{\rm ph}=\frac{\omega}{k}=\frac{c}{\sqrt{1+\chi}}.
\end{equation}
%
So even for small refractive indexes one can get very low group
velocities, due to the large dispersion in the transparency window, and
in spite of the fact that the phase velocity remains very near to $c$.
(The phase velocity is exactly $c$ at the resonance frequency
$\omega_0$). In an ideal EIT regime the probe light experiences a
vanishing susceptibility $\chi$ near the the critical frequency
$\omega_0$, this allows us to express the susceptibility near the
critical frequency via the expansion

\begin{equation}
\chi(\omega)=\frac{2\alpha}{\omega_0}\left(\omega-\omega_0\right)+O\left[\left(\omega-\omega_0\right)^3\right],
\label{eq:grp-ind}
\end{equation}
%
where $\alpha$ is sometimes called the ``group refractive index''.  The
parameter $\alpha$ depends on the dipole moments for the transition from
the metastable states to the high energy one, and most importantly
depends on the ratio between the probe-light energy per photon,
$\hbar\omega_0$, and the control-light energy per
atom~\cite{Leonhardt:2001ye}. This might appear paradoxical because it
seems to suggest that for a dimmer control light the probe light would
be further slowed down. However this is just an artificial feature due
to the extension of the EIT regime beyond its range of applicability. In
particular in order to be effective the EIT requires the control beam
energy to dominate any all processes and hence it cannot be dimmed at
will.

At resonance we have
%
\begin{equation}
v_{\rm g}=\frac{\partial \omega}{\partial k}
\to\frac{c}{1+\alpha} \approx \frac{c}{\alpha};
\qquad 
v_{\rm ph}=\frac{\omega}{k}\to c.
\end{equation}
%
We can now generalize the above discussion to the case in which our
highly dispersive medium flows with a characteristic velocity profile
${\bf u}({\bf x},t)$. In order to find the dispersion relation of the
probe light in this case we just need to transform the dispersion
relation (\ref{eq:sldr}) from the comoving frame of the medium to the
laboratory frame. Let us consider for simplicity a monochromatic probe
light (more realistically a pulse with a very narrow range of
frequencies $\omega$ near $\omega_0$). The motion of the dielectric
medium creates a local Doppler shift of the frequency
%
\begin{equation}
\omega\rightarrow \gamma\left(\omega_0-{\mathbf u}\cdot {\mathbf k}\right),
\label{eq:tilt}
\end{equation}
%
where $\gamma$ is the usual relativistic factor. Given that
$k^2-\omega^2/c^2$ is a Lorentz invariant, it is then easy to see
that this Doppler detuning affects the dispersion relation
(\ref{eq:sldr}) only via the susceptibility dependent term. Given
further that in any realistic case one would deal with non-relativistic
fluid velocities ${\bf u}\ll c$ we can then perform an expansion of the
dispersion relation up to second order in $u/c$. Expressing the
susceptibility via (\ref{eq:grp-ind}) we can then rewrite the dispersion
relation in the form~\cite{Leonhardt:2000aa}
%
\begin{equation}
g^{\mu\nu}k_\mu k_\nu=0,
\end{equation}
%
where 
%
\begin{equation}
k_\nu=\left(\frac{\omega_0}{c},-{\mathbf k}\right),
\end{equation}
and (most of the relevant articles adopt the signature $(+---)$, 
as we also do for this particular section)
%
\begin{equation}
g^{\mu \nu}=
\left[
\begin{array}{c|c}
1+\alpha{u^2}/{c^2}            & \alpha {\mathbf u}^{T} /c^2 \\
\hline
%--- & & ---------- \\
\vphantom{\Big|}
 \alpha {\mathbf u}/c^2  & 
-\mathbf{I}_{3\times 3} +4\alpha {\mathbf u}\otimes {\mathbf u}^T /c^2
\end{array}
\right].
\label{eq:sl-cont-metr}
\end{equation}
%
The inverse of this tensor will be the covariant effective metric
experienced by the probe light, whose rays would then be null geodesics
of the line element $\d s^2=g_{\mu\nu} \d x^\mu \d x^\nu$. In this sense the
probe light will propagate as in a curved background. Explicitly one
finds the covariant metric to be
%
\begin{equation}
g_{\mu \nu}=
\left[
\begin{array}{c|c}
A            & B {\mathbf u}^T \\
\hline
%--- & & ---------- \\
\vphantom{\Big|}
 B {\mathbf u}  & 
-\mathbf{I}_{3\times 3} +C{\mathbf u}\otimes {\mathbf u}^T
\end{array}
\right],
\label{eq:sl-cov-metr}
\end{equation}
where
\begin{eqnarray}
A&=&\frac{1-4\alpha u^2/c^2}{1+(\alpha^2-3\alpha)u^2/c^2-4\alpha^2u^4/c^4};
\\
B&=&\frac{1}{1+(\alpha^2-3\alpha)u^2/c^2-4\alpha^2u^4/c^4};
\\
C&=&\frac{1-(4/\alpha+4 u^2/c^2)}{1+(\alpha^2-3\alpha)u^2/c^2-4\alpha^2u^4/c^4}.
\end{eqnarray}


Several comments are in order concerning the metric
(\ref{eq:sl-cov-metr}). First of all it is clear that although more
complicated than an acoustic metric it will be still possible to cast it
into the  Arnowitt--Deser--Misner-like form~\cite{Visser:2000pk}

\begin{equation}
g_{\mu \nu}=
\left[
\begin{array}{c|c}
-[c_{\rm eff}^2-g_{ab}u_{\rm eff}^a u_{\rm eff}^b]            & [u_{\rm eff}]_i \\
\hline
%--- & & ---------- \\
\vphantom{\Big|}
[u_{\rm eff}]_j & 
[g_{\rm eff}]_{ij}
\end{array}
\right],
\label{eq:adm}
\end{equation}
%
where the effective speed ${\bf u}_{\rm eff}$ is proportional to the
fluid flow speed ${\bf u}$ and the three space effective metric $g_{\rm
eff}$ is (rather differently from the acoustic case) non-trivial.


In any case, the existence of this ADM form already tells us that an
ergoregion will always appear once the norm of the effective velocity
exceeds the effective speed of light (which for slow light is
approximately $c/\alpha$ where $\alpha$ can be extremely large due to
the huge dispersion in the transparency window around the resonance
frequency $\omega_0$). However a trapped surface (and hence an optical
black hole) will form only if the \emph{inward normal component} of the
effective flow velocity exceeds the group velocity of light. In the slow
light setup so far considered such velocity turns out to be
$u=c/(2\sqrt{\alpha})$. 

The realization that ergoregions and event horizons can be simulated via
slow light may lead one to the (erroneous) conclusion that this is an
optimal system for simulating particle creation by gravitational fields.
 However, as pointed out by Unruh in~\cite{Novello:2002qg,Unruh:2003ss},
such conclusion would turn out to be over-enthusiastic. In order to
obtain particle creation an inescapable requirement is to have so-called
``mode mixing", that is, mixing between the positive and negative
frequency modes of the incoming and outgoing states. This is tantamount
to saying that there must be regions where the frequency of the quanta
as seem by a stationary observer at infinity (laboratory frame) becomes
negative beyond the ergosphere at $g_{00}=0$.

In a flowing medium this can in principle occur thanks to the tilting of
the dispersion relation due to the Doppler effect caused by the velocity
of the flow equation~(\ref{eq:tilt}), but this also tell us that the
condition $\omega_0-{\bf u}\cdot{\bf k}<0$ can be satisfied only if the
velocity of the medium exceeds $|\omega_0/k|$ which is the {\em phase}
velocity of the probe light, not its group velocity. Since the phase
velocity in the slow light setup we are considering is very close to
$c$, the physical speed of light in vacuum, not very much hope is left
for realizing analogue particle creation in this particular laboratory
setting.

However it was also noticed by Unruh and Schutzhold
in~\cite{Unruh:2003ss} that a different setup for slow light might deal
with this and other issues (see~\cite{Unruh:2003ss} for a detailed 
summary). In the setup suggested by these authors there are two strong
background counter-propagating control beams illuminating the atoms. The
field describing the beat fluctuations of this electromagnetic
background can be shown to satisfy, once the dielectric medium is in
motion, the same wave equation as that on a curved background. In this
particular situation the phase velocity and the group velocity are
approximately the same, and both can be made small, so that the
previously discussed obstruction to mode mixing is removed. So in this
new setup it \emph{is} concretely possible to simulate classical
particle creation such as, \emph{e.g.}, super-radiance in the presence
of ergoregions.

Nonetheless the same authors showed that this does not open the
possibility for a simulation of quantum particle production
(\emph{e.g.}, Hawking radiation). This is because that effect also
requires the commutation relations of the field to generate the
appropriate zero-point energy fluctuations (the vacuum structure)
according to the Heisenberg uncertainty principle. This is not the case
for the effective field describing the beat fluctuations of the system
we have just described, which is equivalent to saying that it does not
have a proper vacuum state (\emph{i.e.} analogue to any physical field).
Hence one has to conclude that any simulation of quantum particle
production is precluded.


%-------------------------------------
%%% end of section on slow light
%----------------------------------------------


%----------------------------------------------
\subsection{Going further}
%----------------------------------------------
\label{SS:going-further-5}
%----------------------------------------------

We feel that the catalogue we have just presented is reasonably complete
and covers the key items. For additional background on any of these
topics, we would suggest sources such as the books ``Artificial Black
Holes''~\cite{Novello:2002qg} and ``The universe in a Helium
droplet''~\cite{Volovik:2003fe}.  For more specific detail, check this
review's bibliography, and use {\sf Spires} to check for recent
developments.


%--------------------------------
% EOF
%--------------------------------- 

conclusions.tex
%---------------------------------
% conclusions.tex
%---------------------------------



In this review article we have seen the interplay between standard
general relativity and various analogies that can be used to capture
aspects of its behaviour. These analogies have ranged from rather
general but very physical analogue models based on fluid-acoustics,
geometrical optics, and wave optics, to highly specific models based on
BECs, liquid helium, slow light, etc. Additionally, we have seen several
rather abstract mathematical toy models that bring us to such exotic
structures and ideas such as birefringence, bimetricity, Finsler spaces,
and Sakharov's induced gravity.

The primary reason that these analogies were developed within the
general relativity community was to help in the understanding of general
relativity by providing very down-to-earth models of otherwise subtle
behaviour in general relativity. Secondary reasons include the rather
speculative suggestion that there may be more going on than just analogy
--- it is conceivable [though perhaps unlikely] that one or more of
these analogue models could suggest a relatively simple and useful way
of quantizing gravity that side-steps much of the technical machinery
currently employed in such efforts. A tertiary concern [at least as far
as the general relativity community is concerned] is the use of
relativity and differential geometric techniques to improve
understanding of various aspects of condensed matter physics.

The authors expect interest in analogue models to continue unabated, and
suspect that there are several key but unexpected issues whose
resolution would be greatly aided by the analysis of appropriate analogue models
%----------------------------------------------
\subsection{Going further}
%----------------------------------------------
\label{SS:going-further-8}
%----------------------------------------------


Though every practicing scientist already know this, for the sake
of any student reading this we mention the following resources:

\begin{itemize}

\item {\sf http://www.slac.stanford.edu/spires/} --- the bibliographic
database for keeping track of (almost all of) the relevant literature.

\item {\sf http://www.arXiv.org} --- the electronic-preprint (e-print) 
database for accessing the text of (almost all, post 1992) relevant
articles.

\item {\sf http://www.livingreviews.org/} --- the Living Reviews portal.

\end{itemize}
Those three access points should allow you to keep abreast of what is
going on in the field.


%----------------------------
% EOF
%---------------------------- 
future-directions.tex
%-------------------------------------
% future-directions.tex
%-------------------------------------
% BOF
%--------------------------------------

The key question one should ask at this stage is: ``Where can we go
from here?'' Apart from continuing with the analysis of the issues
described in the previous section, there are additional interesting
routes, both theoretical and experimental, worth of future exploration.
In particular, the following topics come to mind:
%
\begin{itemize}
\item
Back reaction,
\item
Equivalence principle,
\item
Emergent gravity,
\item
%Finsler space and beyond,
%\item
Quantum gravity phenomenology,
\item
Quantum gravity,
\item
Experimental analogue gravity.
\end{itemize}
%
There has been already some works dealing with this topics in the contest
of analogue gravity. Let us now expand on these issues a little.

%
%----------------------------------------------
\subsection{Back reaction}
%----------------------------------------------


There are important phenomena on gravitational physics whose
understanding needs of analysis well beyond classical general relativity
and field theory on curved background spacetimes. The black hole
evaporation process can be considered as paradigmatic among these
phenomena. Here, we particularize our discussion to this case. Since we
are currently unable to analyze the entire process of black hole
evaporation within a complete quantum theory of gravity, a way of
proceeding is to analyze the simpler (but still extremely difficult)
problem of semiclassical back reaction (see for
example~\cite{Davies:1976ei,Christensen:1977jc,Birrell,Fulling,
Brout:1995rd,Massar:1999wg}). One takes a background black hole
spacetime, calculates the expectation value of the quantum
energy-momentum tensor of matter fields in the appropriate quantum state
(the Unruh vacuum state for a radiating black hole), and then takes this
expectation value as a source for the perturbed Einstein equations. This
calculation gives us information about the tendency of spacetime to
evolve under vacuum polarization effects.

A nice feature of analogue models of general relativity is that although
the underlying classical equations of motion have nothing to do with
Einstein equations, the tendency of the analogue geometry to evolve due
to quantum effects is formally equivalent (approximately, of course) to
that in semiclassical general relativity. Therefore, the onset of the
back reaction effects (if not their precise details) can be simulated
within the class of analogue models. An example of the type of
back-reaction calculations one can perform are those in
\cite{Balbinot:2004dc,Balbinot:2004da}. These authors started from an
effectively one-dimensional acoustic analogue model, configured to have
an acoustic horizon by using a Laval nozzle to control the flow's speed.
They then considered the effect of quantizing the acoustic waves over
the background flow.  To calculate the appropriate back reaction terms
they took advantage of the classical conformal invariance of the
(1+1)-dimensional reduction of the system. In this case, we know
explicitly the form of the expectation value of the energy-momentum
tensor trace (via the trace anomaly). The other two independent
components of the energy-momentum tensor were approximated by the
Polyakov stress tensor. In this way, what they found is that the
tendency of a left-moving flow with one horizon is for it to evolve in
such a manner as to push the horizon down-stream at the same time that
its surface gravity is decreased. This is a behaviour similar to what is
found for near-extremal Reissner-Nordstr\"om black holes. (However, we
should not conclude that acoustic black holes are in general closely
related to near-extremal Reissner-Nordstr\"om black holes, rather than
to Schwarzschild black holes. This result is quite specific to the
particular one-dimensional configuration analyzed.)

Can we expect to learn something new about gravitational physics by
analysing the problem of back reaction in different analogue models? As
we have repeatedly commented, the analyses based on analogue models
force us to consider the effects of modified high-energy dispersion
relations.  For example, in BECs, they affect the ``classical''
behaviour of the background geometry as much as the behaviour of the
quantum fields living on the background. In seeking a semiclassical
description for the evolution of the geometry, one would have to compare
the effects caused by the modified dispersion relations to those caused
by pure semiclassical back reaction (which incorporates deviations from
standard general relativity as well). In other words, one would have to
understand the differences between the standard back reaction scheme in
general relativity, and that based on equations
(\ref{bec-self-consistent1}) and (\ref{quantum-field}).

To end this subsection we would like to comment that one can go beyond
the semiclassical back-reaction scheme by using the so-called stochastic
semiclassical gravity programme \cite{Hu:1999mm, Hu:2002jm, Hu:2003qn}.
This programme aims to pave the way from semiclassical gravity toward a
complete quantum-gravitational description of gravitational phenomena.
This stochastic gravity approach not only considers the expectation
value of the energy-momentum tensor but also its fluctuations, encoded
in the semiclassical Einstein--Langevin equation. In a very interesting
paper~\cite{Parentani:2000ts}, Parentani showed that the effects of the
fluctuations of the metric (due to the in-going flux of energy at the
horizon) on the out-going radiation led to a description of Hawking
radiation similar to that obtained with analogue models. It would be
interesting to develop the equivalent formalism for quantum analogue
models and to investigate the different emerging approximate regimes.



%----------------------------------------------
\subsection{Equivalence principle}
%----------------------------------------------

Analogue models are of particular interest in that the analogue
spacetimes that emerge often violate, to some extent or another, the
Einstein equivalence principle~\cite{Barcelo:2001cp,Visser:2002sf}.
Since the Einstein equivalence principle (or more precisely the
universality of free fall) is experimentally tested to the accuracy of
about 1 part in $10^{13}$, it is important to build this principle into
realistic models of gravitation --- most likely at a fundamental level.

One way of interpreting the Einstein equivalence principle is as a
``principle of universality'' for the geometrical structure of
spacetime. Whatever the spacetime geometrical structure is, if all excitations
``see" the same geometry one is well on the way to satisfying the
observational and experimental constraints. In a metric theory, this
amounts to the demand of mono-metricity: a \emph{single} universal metric
must govern the propagation of all excitations.

Now it is this feature that is relatively difficult to arrange in
analogue models. If one is dealing with a single degree of freedom then
mono-metricity is no great constraint. But with multiple degrees of
freedom, analogue spacetimes generally lead to refringence --- that is
the occurrence of Fresnel equations that often imply multiple
propagation speeds for distinct normal modes. To even obtain a bi-metric
model (or more generally, a multi-metric model), requires an algebraic
constraint on the Fresnel equation that it completely factorize
into product of quadratics in frequency and wavenumber. Only if this
algebraic constraint is satisfied can one assign a ``metric'' to each of
the quadratic factors. If one further wishes to impose mono-metricity
then the Fresnel equation must be some integer power of some single
quadratic expression, an even stronger algebraic
statement~\cite{Barcelo:2001cp,Visser:2002sf}.

Faced with this situation, there are two ways in which the analogue
gravity community might proceed:

\begin{itemize} 

\item Try to find a broad class of analogue models (either physically
based or mathematically idealized) that naturally lead to
mono-metricity. Little work along these lines has yet been done; at
least partially because it is not clear what features such a model should
have in order to be ``clean'' and ``compelling''.

\item Accept refringence as a common feature of the analogue models and attempt to use refringence to ones benefit in one or more ways:

\begin{itemize}

\item There are real physical phenomena in non-gravitational settings
that definitely do exhibit refringence and sometimes multi-metricity.
Though situations of this type are not directly relevant to the gravity
community, there is significant hope that the mathematical and
geometrical tools used by the general relativity community might in
these situations throw light on other branches of physics.

\item Use the refringence that occurs in many analogue models as a way of ``breaking'' the Einstein equivalence principle, and indeed as a way of ``breaking'' even more fundamental symmetries and features of standard general relativity, with a view to exploring possible extensions of general relativity.
While the analogue models are not themselves primary physics, they can nevertheless be used as a way of providing \emph{hints} as to how more fundamental physics might work.

\end{itemize}

\end{itemize}

%----------------------------------------------
\subsection{Emergent gravity}
%----------------------------------------------

One of the more fascinating approaches to ``quantum gravity'' is the
suggestion, typically attributed to
Sakharov~\cite{Sakharov:1967pk,Visser:2002ew}, that gravity itself may
not be ``fundamental physics''. Indeed it is now a relatively common
opinion, maybe not mainstream but definitely a strong minority opinion,
that gravity (and in particular the whole notion of spacetime and
spacetime geometry) might be no more ``fundamental'' than is fluid
dynamics. The word ``fundamental'' is here used in a rather technical
sense --- fluid mechanics is not fundamental because there is a known
underlying microphysics; that of molecular dynamics, of which fluid
mechanics is only the low-energy low-momentum limit. Indeed the very
concepts of density and velocity field, which are so central to the
Euler and continuity equations, make no sense at the microphysical level
and emerge only as one averages over timescales and distance-scales
larger than the mean free time and mean free path.

In the same way, it is plausible (even though no specific and compelling
model of the relevant microphysics has yet emerged) that the spacetime
manifold and spacetime metric might arise only once one averages over
suitable microphysical degrees of freedom. Sakharov had in mind a
specific model in which gravity could be viewed as an ``elasticity'' of
the spacetime medium, and was ``induced'' via one-loop physics in the
matter sector~\cite{Sakharov:1967pk,Visser:2002ew}. In this way Sakharov
had hoped to relate the observed value of Newton's constant (and the
cosmological constant) to the spectrum of particle masses.

More generally the phrase ``emergent gravity'' is now used to describe
the whole class of theories in which the spacetime metric arises as a
low-energy approximation, and in which the microphysical degrees of
freedom might be radically different. Analogue models, and in particular
analogue models based on fluid mechanics or the fluid dynamic
approximation to BECs, are specific examples of ``emergent physics'' in
which the microphysics is well understood. As such they are useful for
providing \emph{hints} as to how such a procedure might work in a more
fundamental theory of quantum gravity.



%----------------------------------------------
\subsection{Quantum gravity --- phenomenology}
%----------------------------------------------


Over the last few years a widespread consensus has emerged that
observational tests of quantum gravity are for the foreseeable future
likely to be limited to precision tests of dispersion relations and
their possible deviations from Lorentz
invariance~\cite{Mattingly:2005re}. The key point is that at low
energies (well below the Planck energy) one expects the locally
Minkowskian structure of the spacetime manifold to guarantee that one sees
only special relativistic effects; general relativistic effects are
negligible at short distances. However as ultra high energies are
approached (although still below Planck scale energies) several quantum
gravity models seem to predict that the locally Euclidean geometry of
the spacetime manifold will break down. There are several scenarios for
the origin of this breakdown ranging from string
theory~\cite{Kostelecky:1988zi, Ellis:1999sf} to brane
worlds~\cite{Burgess:2002tb} and loop quantum
gravity~\cite{Gambini:1998it}. Common to all such scenarios is that the
microscopic structure of spacetime is likely to show up in the form of a
violation of Lorentz invariance leading to modified dispersion relations
for elementary particles. Such dispersion relations are characterized by
extra terms (with respect to the standard relativistic form) which are
generally expected to be suppressed by powers of the Planck energy.
Remarkably the last years have seen a large wealth of work in testing
the effects of such dispersion relations and in particular strong
constraints have been cast by making use of high energy astrophysics
observations (see \eg~\cite{Amelino-Camelia:1997gz, Coleman:1998ti,
Jacobson:2002hd, Jacobson:2002ye, Jacobson:2004rj, Jacobson:2004qt,
Mattingly:2005re} and references therein).

Several of the analogue models are known to exhibit similar behaviour,
with a low-momentum effective Lorentz invariance eventually breaking
down at high momentum once the microphysics is explored.~\footnote{It is
however important to keep in mind that not all the above cited quantum
gravity models violate the Lorentz symmetry in the same manner. The
discreteness of spacetime at short scales is not the only way of
breaking Lorentz invariance.} Thus some of the analogue models provide
controlled theoretical laboratories in which at least some forms of
subtle high-momentum breakdown of Lorentz invariance can be explored. As
such the analogue models provide us with \emph{hints} as to what sort of
modified dispersion relation might be natural to expect given some
general characteristics of the microscopic physics. Hopefully
investigation of appropriate analogue models might be able to illuminate
possible mechanisms leading to this kind of quantum gravity
phenomenology,  and so might be able to provide us new ideas about other
effects of physical quantum gravity that might be observable at
sub-Planckian energies.




%----------------------------------------------
\subsection{Quantum gravity --- fundamental models}
%----------------------------------------------

When it comes to dealing with ``fundamental'' theories of quantum
gravity, the analogue models play an interesting role that is
complementary to the more standard approaches.  The search for a quantum
theory of gravity is fundamentally a search for an appropriate
mathematical structure in which to simultaneously phrase both quantum
questions and gravitational questions. More precisely, one is searching
for a mathematical framework in which to develop an abstract  quantum
theory that then itself encompasses classical Einstein gravity (the
general relativity), and reduces to it in an appropriate
limit~\cite{Carlip:2001wq, Smolin:2003rk, Friedan:2002aa}.

The three main approaches to quantum gravity currently in vogue,
``string models'' [also known as ``M-models''], ``loop space'' [and the
related ``spin foams''], and ``lattice models'' [Euclidean or
Lorentzian] all share one feature: They attempt to develop a
``pre-geometry'' as a replacement for classical differential geometry
(which is the natural and very successful mathematical language used to
describe Einstein gravity)~\cite{Carlip:2001wq, Smolin:2003rk,
Friedan:2002aa, Rovelli, Rovelli:2003cu, Bousso:2004fc}.  The basic idea
is that the mooted replacement for differential geometry would be
relevant at extremely small distances (where the quantum aspects of the
theory would be expected to dominate), while at larger distances (where
the classical aspects dominate) one would hope to recover both ordinary
differential geometry and specifically Einstein gravity or possibly some generalization of it.  The
``string'', ``loop'', and ``lattice'' approaches to quantum gravity
differ in detail in that they emphasise different features of the
long-distance model, and so obtain rather different short-distance
replacements for classical differential geometry.  Because the relevant
mathematics is extremely difficult, and by and large not particularly
well understood, it is far from clear which if any of these three
standard approaches will be preferable in the long
run~\cite{Smolin:2003rk}.


We feel it likely that analogue models can shed new light on this very
confusing field by providing a concrete specific situation in which the
transition from the short-distance ``discrete'' or ``quantum'' theory to
the long-distance ``continuum'' theory is both well understood and
non-controversial.  Here we are specifically referring to fluid
mechanics, where at short distances the system must be treated using
discrete atoms or molecules as the basic building blocks, while at large
distances there is a well-defined continuum limit that leads to the
Euler and continuity equations. Furthermore once one is in the continuum
limit, there is a well-defined manner in which a notion of ``Lorentzian
differential geometry'', and in particular a ``Lorentzian effective
spacetime metric'' can be assigned to any particular fluid
flow~\cite{Unruh:1981cg, Visser:1997ux, Novello:2002qg}.  Indeed, the
``analogue gravity programme'' is extremely successful in this regard,
providing a specific and explicit example of a  ``discrete'' $\to$
``continuum" $\to$ ``differential geometry" chain of development. What
the ``analogue gravity programme'' does not seem to do as easily is to
provide a natural direct route to the Einstein equations of general
relativity, but that merely indicates that current analogies have their limits and therefore, one should not take them too literally [378,283].
Fluid mechanics is a guide to the mathematical possibilities, not an end in itself.  The
parts of the analogy that do work well are precisely the steps where the
standard approaches to quantum gravity have the most difficulty, and so
it would seem useful to develop an abstract mathematical theory of the
``discrete'' $\to$ ``continuum" $\to$ ``differential geometry" chain
using this fluid mechanical analogy (and related analogies) as
inspiration.



%----------------------------------------------
\subsection{Going further}
%----------------------------------------------
\label{SS:going-further-7}
%----------------------------------------------


Beyond the various theoretical issues we have discussed above there is
the important question of ``experimental analogue gravity'' --- to what
extent can all these ideas be tested in direct laboratory experiments?
Currently several experimental groups are interested, but to the best of
our knowledge no actual experiments are currently underway. Broadly
speaking, for any experimental group interested in analogue spacetimes
the two key issues to address are:

\begin{itemize}

\item Identify a particular analogue model easily amenable to laboratory
investigation, and double check the extent to which the model provides a
theoretically robust and clean analogue to general relativistic curved
spacetime.

\item Identify the technical issues involved in actually setting up a
laboratory experiment.

\end{itemize}

While the consensus in the theoretical community is that Bose--Einstein condensates are likely to provide the best working model for analogue gravity, it is possible that we might still be surprised by experimental developments.
We leave this as an open  challenge to the experimental community.

%------------------------------------
% EOF
%------------------------------------ 
history.tex
%------------------------
% history.tex
%------------------------
% BOF
%------------------------

From the point of view of the general relativity community the history
of analogue models can reasonably neatly (but superficially) be divided
into a ``historical'' period (essentially pre-1981) and a ``modern''
period (essentially post-1981).

%----------------------------------------------
\subsection{Modern period}
%----------------------------------------------
%\noindent
%Recent history:
%\begin{itemize}
%\item
%Unruh 81
%\item
%Jacobson 91
%\item
%Jacobson 9303
%\item
%Visser 9311
%\item
%Unruh 9409
%\item
%Plus hundreds more....
%\end{itemize}
%----------------------------------------------
\subsubsection{1981-1999:}
%----------------------------------------------


The key event in the ``modern'' period  (though largely unrecognized at
the time) was the 1981 publication of Unruh's paper ``Experimental black
hole evaporation''~\cite{Unruh:1981cg},  which  implemented an analogue
model based on fluid flow, and then used the power of that analogy to
probe fundamental issues regarding Hawking radiation from ``real''
general relativity black holes.

We believe that Unruh's 1981  article represents the first observation
of the now widely established fact that Hawking radiation has nothing to
do with general relativity \emph{per se}, but that Hawking radiation is
instead a fundamental curved-space quantum field theory phenomenon that
occurs whenever a horizon is present  in an effective
geometry.\footnote{We emphasise: To get Hawking radiation you need an
effective geometry, a horizon, and a suitable quantum field theory on
that geometry.} Though Unruh's 1981 paper was seminal in this regard, it
lay largely unnoticed for many years.

Some 10 years later Jacobson's article ``Black-hole evaporation and
ultrashort distances''~\cite{Jacobson:1991gr} used Unruh's analogy to
build a physical model for the ``trans--Planckian modes'' believed to be
relevant to the Hawking radiation process. Progress then sped up with
the relatively rapid appearance of~\cite{Jacobson:1993hn}
and~\cite{Unruh:1994zw,Unruh:1995je}. (This period also saw the
independent rediscovery of the fluid analogue model by one of the
present authors~\cite{Visser:1993ub}, and the first explicit
consideration of superfluids in this regard~\cite{Comer}.) 

The later 1990's then saw continued work by Jacobson and his
group~\cite{Jacobson:1995ak, Jacobson:1996zs, Corley:1996ar,
Corley:1998rk, Jacobson:1999ay}, with new and rather different
contributions coming in the form of the solid state models considered by
Reznik~\cite{Reznik:1997ag,
Reznik:1997xt}.~\footnote{Reference~\cite{Hochberg} is an attempt at
connecting Hawking evaporation with the physics of collapsing bubbles.
This was part of a more general programme aimed at connecting black hole
thermodynamics with perfect fluid
thermodynamics~\cite{Hochberg:1996mb}.}  This period also saw the
introduction of the more general class of superfluid  models considered
by Volovik and his collaborators~\cite{Volovik:1995ja, Volovik:1996he,
Kopnin:1997jy, Eltsov:1998fv, Volovik:1998de, Volovik:1998pf2,
Volovik:1998pf,  Jacobson:1998he, Volovik:1999fc, Volovik:1999cn}, more
precise formulations of the notions of horizon, ergosphere, and surface
gravity in analogue models~\cite{Visser:1997ux, Visser:1998qn}, and
discussions of the implications of analogue models regarding
Bekenstein--Hawking entropy~\cite{Visser:1997yu, Visser:1998qn}.
Finally, analogue spacetimes based on \emph{special} relativistic
acoustics were considered in~\cite{Bilic:1999sq}.

By the year 2000, articles on one or another aspect of analogue gravity
were appearing at the rate of over 20 per year, and it becomes
impractical to summarize more than a selection of them.


%{\bf *** Will need to check and re-check this to see who we have offended ***}


%----------------------------------------------
\subsubsection{2000:}
%----------------------------------------------

Key developments in 2000 were the introduction, by Garay and
collaborators, of the use of Bose--Einstein condensates as a working
fluid~\cite{Garay:1999sk,Garay:2000jj}, and the extension of those ideas
by the present authors~\cite{Barcelo:2000tg}. Further afield, the
trans-Planckian problem also reared its head in the context of
cosmological inflation, and analogue model ideas previously applied to
Hawking radiation were reused in that context~\cite{Kempf:2000ac,
Niemeyer:2000eh}.

That year also marked the appearance of a review article on superfluid
analogues~\cite{Volovik:2000ua}, more work on ``near-horizon''
physics~\cite{Fischer:2000hg}, and the transference of the idea of
analogue-inspired ``multiple metric'' theories into cosmology where they
can be used as the basis for a precise definition of what is meant by a
VSL (``variable speed of light'') cosmology~\cite{Bassett:2000wj}.
Models based on nonlinear electrodynamics were investigated
in~\cite{Baldovin:2000xy}, ${}^3$He--A based models were reconsidered
in~\cite{Jacobson:2000gs,Volovik:1999mj}, and ``slow light'' models in
quantum dielectrics were considered in~\cite{Leonhardt:2000aa,
Leonhardt:2000hf, Leonhardt:2000zp}.



The most radical proposal to appear in 2000 was that of Laughlin
\emph{et al.}~\cite{Chapline:2000en}. Based on taking a superfluid
analogy rather literally they mooted an actual physical breakdown of
general relativity at the horizon of a black
hole~\cite{Chapline:2000en}.

Additionally, the workshop on ``Analogue models of general relativity'',
held at CBPF (Rio de Janeiro) gathered some 20 international
participants and greatly stimulated the field, leading ultimately to the
publication of the book~\cite{Novello:2002qg} in 2002.

%----------------------------------------------
\subsubsection{2001:}
%----------------------------------------------

This year saw more applications of analogue-inspired ideas to
cosmological inflation~\cite{Easther:2001fi, Mersini:2001su,
Mersini:2001jt, Kempf:2001fa, Niemeyer:2001qe}, to neutron star
cores~\cite{Carter:2001tg}, and to the cosmological
constant~\cite{Volovik:2001fm, Volovik:2001jw}.

Closer to the heart of the analogue programme were the development of a
``normal mode'' analysis in~\cite{Barcelo:2001ah, Barcelo:2001cp,
Visser:2001fe}, the development of dielectric analogues
in~\cite{Schutzhold:2001fw}, speculations regarding the possibly
emergent nature of Einstein gravity~\cite{Barcelo:2001tb,
Visser:2001fe}, and further developments regarding the use of
${}^3$He--A~\cite{Dziarmaga:2001st} as an analogue for electromagnetism.
Experimental proposals were considered in~\cite{Barcelo:2001ca,
Visser:2001fe, Sakagami:2001ph}.

Vorticity was discussed in~\cite{PerezBergliaffa:2001nd}, and the use of
BECs as a model for the breakdown of Lorentz invariance
in~\cite{Visser:2001jd}.  Analogue models based on nonlinear
electrodynamics were discussed in~\cite{DeLorenci:2001ch}. Acoustics in
an irrotational vortex were investigated in~\cite{Fischer:2001jz}.

The excitation spectrum in superfluids, specifically the fermion zero
modes, were investigated in~\cite{Volovik:2001rx,Huhtala:2001cj}, while
the relationship between rotational friction in superfluids and
super-radiance in rotating spacetimes was discussed
in~\cite{Calogeracos:2001pw}. More work on ``slow light'' appeared
in~\cite{Brevik:2001nf}. The possible role of Lorentz violations at
ultra-high energy was emphasized in~\cite{Jacobson:2001kz}.




%----------------------------------------------
\subsubsection{2002:}
%----------------------------------------------

``What did we learn from studying acoustic black holes?'' was the title
and theme of Parentani's article in 2002~\cite{Parentani:2002bd}, while
Schutzhold and Unruh developed a rather different fluid-based analogy
based on gravity waves in shallow water~\cite{Schutzhold:2002rf,
Schutzhold:2002xm}. Super-radiance was investigated
in~\cite{Basak:2002aw}, while the propagation of phonons and
quasiparticles was discussed in~\cite{Fischer:2002jn, Fischer:2002ma}.
More work on ``slow light'' appeared in~\cite{Fiurasek:2002ay,
Piwnicki:2002xw}.

The stability of an acoustic white hole was investigated
in~\cite{Leonhardt:2002yu}, while further developments regarding
analogue models based on nonlinear electrodynamics were presented by
Novello and collaborators in~\cite{DeLorenci:2002mi, DeLorenci:2002ws,
Novello:2002ed, Novello:2002iq, Fonseca-Barbatti:2002hq}. Analogue
spacetimes relevant to braneworld cosmologies were considered
in~\cite{Barcelo:2002xw}.

Though analogue models lead naturally to the idea of high-energy
violations of Lorentz invariance, it must be stressed that definite
observational evidence for violations of Lorenz invariance is lacking
--- in fact there are rather strong constraints on how strong any
possible Lorentz violating effect might be~\cite{Jacobson:2002hd,
Jacobson:2002ye}.


%----------------------------------------------
\subsubsection{2003:}
%----------------------------------------------

That year saw further discussion of analogue-inspired models for black
hole entropy and the cosmological constant~\cite{Volovik:2003ga,
Volovik:2003kt}, and the development of analogue models for FRW
geometries~\cite{Fedichev:2003bv, Fedichev:2003id, Barcelo:2003et,
Dumin:2003gs, Lidsey:2003ze}. There were several further developments
regarding the foundations of BEC-based models in~\cite{Barcelo:2003wu,
Fedichev:2003dj}, while analogue spacetimes in superfluid neutron stars
were further investigated in~\cite{Carter:2003im}.

Effective geometry was the theme in~\cite{Novello:2003je}, while
applications of nonlinear electrodynamics (and its effective metric) to
cosmology were presented in~\cite{Novello:2003kh}. Super-radiance was
further investigated in~\cite{Basak:2003uj, Basak:2003yt},  while the
limitations of the ``slow light'' analogue were explained
in~\cite{Unruh:2003ss}. Vachaspati argued for an analogy between phase
boundaries and acoustic horizons in~\cite{Vachaspati:2003rb}. Emergent
relativity was again addressed in~\cite{Laughlin:2003yh}.


The review article by Burgess~\cite{Burgess:2003jk}, emphasised the role
of general relativity as an effective field theory --- the \emph{sine
qua non} for any attempt at interpreting general relativity as an
emergent theory. The lecture notes by Jacobson~\cite{Jacobson:2003vx}
give a nice introduction to Hawking radiation and its connection to
analogue spacetimes.



%----------------------------------------------
\subsubsection{2004:}
%----------------------------------------------

The year 2004 saw the appearance of some 30 articles on (or closely
related to) analogue models. Effective geometries in astrophysics were
discussed by Perez--Bergliaffa~\cite{PerezBergliaffa:2004xg}, while the
physical realizability of acoustic Hawking radiation was addressed
in~\cite{Das:2004cs,Vachaspati:2004wn}. More cosmological issues were
raised in~\cite{Vachaspati:2004wn,Weinfurtner:2004mu}, while a
specifically astrophysical use of the acoustic analogy was invoked
in~\cite{Das:2004gn,Das:2004zm,Das:2004wf}.

BEC-based horizons were again considered in~\cite{Giovanazzi:2004pm, 
Giovanazzi:2004zv}, while backreaction effects were the focus of
attention in~\cite{Balbinot:2004da,Balbinot:2004dc,Kim:2004sf}. More issues
relating to the simulation of FRW cosmologies were raised
in~\cite{Fischer:2004hr,Fischer:2004bf}.

Unruh and Schutzhold discussed the universality of the Hawking
effect~\cite{Unruh:2004zk}, and a new proposal for possibly detecting
Hawking radiation in a electromagnetic wave
guide~\cite{Schutzhold:2004tv}. The causal structure of analogue
spacetimes was considered in~\cite{Barcelo:2004wz}, while quasinormal
modes attracted attention in~\cite{Berti:2004ju, Lepe:2004kv,
Cardoso:2004fi, Nakano:2004ha}. Two dimensional analogue models were
considered in~\cite{Cadoni:2004my}.

There were attempts at modelling the Kerr geometry~\cite{Visser:2004zs},
a proposal for using analogue models to generate massive phonon modes in
BECs~\cite{Visser:2004qp}, and an extension of the usual formalism for
representing weak-field gravitational lensing in terms of an analogue
refractive index~\cite{Boonserm:2004wp}.

Finally we mention the development of yet more strong observational
bounds on possible ultra high energy Lorentz
violation~\cite{Jacobson:2004rj, Jacobson:2004qt}.


%----------------------------------------------
\subsubsection{2005:}
%----------------------------------------------

The first few months of 2005 have seen continued and vigourous activity
on the analogue model front.

More studies of the super-resonance phenomenon
have appeared~\cite{Basak:2005fv, Federici:2005ty, Kim:2005pb,Savage}, and a
mini-survey was presented in~\cite{Cardoso:2005ij}. Quasinormal modes
have again received attention in ~\cite{Cherubini:2005dh}, while the
Magnus force is reanalyzed in terms of the acoustic geometry
in~\cite{Zhang:2005va}. Singularities in the acoustic geometry are
considered in~\cite{Cadoni:2005jg}, while back-reaction has received
more attention in~\cite{Schutzhold:2005ex}.

Interest in analogue models is intense and shows no signs of abating.


%----------------------------------------------
\subsubsection{Analogue spacetimes as background gelstat}
%----------------------------------------------

To further complicate matters, there is large body of work for which
analogue spacetime ideas provide part of the background gelstat, even if
the specific connection may sometimes be somewhat tenuous. Among such
articles we mention~\cite{Amati:1996pu, Anglin:1992ur, Antunes:2003jr,
Arbona:2003re, Arteaga:2003we, 
Barrabes:1998iw, Barrabes:2000fr, Bastero-Gil:2001js, Belgiorno:2002iv,
Bhattacharyya:2004dy, Blaut:2001fy, Bombelli:1994rh, Brout:1998ei,
Calzetta:1999zr, Canfora:2003zx, Canfora:2003cc, M.deCarvalho:2004ic,
Casadio:2001hp, Casadio:2003vk, Casher:1996ct, Chang:2003he,
Chang:2003sa, Chapline:2004mu, Chrusciel:2002mi, Chu:2000ww,
Collins:2003ze, Consoli:2003up, Czerniawski:2002sv, Damour:2004kw,
Englert:1994sm, Englert:1994qe, Fischer:1999cp, Ford:1997hb,
Ford:1997zb, Frolov:1995pt, Frolov:1995qp, Furtado:2004ei, Garay:1994en,
Ghafarnejad:1997ps, Glass:1999yt, Hambli:1995pp, Hamilton:2004au,
Helfer:2000wh, Helfer:2003va, Helfer:2004jx, Ho:2003zm, Horwitz:2004fm,
Hossenfelder:2004gj, Hossenfelder:2004up, Hu:1987rr, Hu:1994dx,
Jevicki:2002fq, Kempf:2003qu, Kiss:2003kp, Kokkotas:1999bd,
Kowalski-Glikman:2000dz,  Kowalski-Glikman:2002ft, Kowalski-Glikman:2002be,
 Larsen:1996ah, Laschkarew, Leonhardt:2003mp,
Liberati:2000ag, Liberati:2000pt,  Liberati:2000mp, Liberati:2001sd,
Lubo:1999xg,  Lubo:2003mr, Lubo:2003rs, Martin:2004um, Massar:1994iy,
Massar:1994xz,  Massar:1996tx, Massar:1995im, Massar:1999wg,
Mersini:2001jt, Nandi:2004rr, Nandi:2002fe,  Nandi:2004nw,
Nikolic:2004wu, Nikolic:2003xs, Obadia:2001hj, Obadia:2002qe,
Obukhov:2003vv, Oppenheim:2002kx, Oron:2002xk, Padmanabhan:2003gd, Paddy3,
 Parentani:1995iw, Parentani:1996sz, Parentani:1995ts,  Parentani:1997ua,
Parentani:1999qv, Renaud:2004ta, Pashaev:1998sk,  Radu:1998sk,
Raval:1996vt, Ray:2002gd, Ray:2003gb, Rosquist:2003qs, Rosu:1989cb, Rosu:1996pw,
Rosu:1994je, Rosu:1994ky,  Rosu:1999sa, Rovelli:1994ge,
Russo:1996qa, Salehi:1994em, Salehi:2003pj, Saul:2004ch,
Schmelzer:2000da, Schmelzer:2000db, Schmelzer:2001kx, Schmelzer:2002es,
Schutzhold:2000ju, Schutzhold:2000jn, Schutzhold:2003yp, Paddy2,
Siemienic-Ozieblo:2001ef, Smolin:1995ai, Smolyaninov:2003wb, Smolyaninov:2003qt, Sorkin:2003bx, Paddy1, Starobinsky:2001kn,
Starobinsky:2002rp, Stephens:2001sd, Stone:1997nj, Stone:1999gi,
Stone:2000vortex, Stone:2000gb,  Tamaki:2002iz, Tanaka:2000jw, Trautman,
Visser:2003tt, Volovik:1998pc, Winicour:2000zj, Zaslavsky:1998in,
Zloshchastiev:1998uz}.



%----------------------------------------------
\subsection{Historical Period}
%----------------------------------------------
%\noindent
%Prehistory:
%\begin{itemize}
%\item
%Gordon
%\item
%Landau Lifshitz
%\item
%Matarrese
%\item
%de Felice
%\item
%Many others... 
%\item
%Use ``analog models'' website as basis for first sweep.
%\end{itemize}
%--------------------------------------------------------

Of course the division into pre-1981 and post-1981 articles is at a
deeper level somewhat deceptive. There have been several analogue models
investigated over the years, with different aims, different levels of
sophistication, and ultimately different levels of development. Armed
with a good library and some hindsight it is possible to find
interesting analogues in  a number of places.\footnote{Indeed historically, though not of direct relevance to general relativity, analogue models played a key role in the development of electromagnetism --- Maxwell's derivation of his equations for the electromagnetic field was guided by a rather complicated analogue model in terms of spinning vortices of aether. Of course, once you have the equations in hand you can treat them in their own right and forget the model that guided you --- which is exactly what happened in this particular case.}

%----------------------------------------------
\subsubsection{Optics}
%----------------------------------------------


Perhaps the first paper to seriously discuss analogue models and
effective metric techniques was that of Gordon (yes, he of the
Klein--Gordon equation)~\cite{Gordon} Note that Gordon seemed largely
interested in trying to describe dielectric media by an ``effective
metric". That is: Gordon wanted to use a gravitational field to mimic a
dielectric medium. What is now often referred to as the Gordon metric is
the expression %

\begin{equation}
[g_\mathrm{effective}]_{\mu\nu}=\eta_{\mu \nu} +\left[1- n^{-2}\right] V_{\mu} V_{\nu}, 
\end{equation}
%
where $\eta_{\mu\nu}$ is the flat Minkowski metric, $n$ is the refractive index, and $V_\mu$ is the 4-velocity of the medium.


After that, there was sporadic interest in effective metric techniques.
One historically important contribution was one of the \emph{problems}
in the well-known book ``The classical theory of fields" by Landau and
Lifshitz~\cite{Landau:Lifshitz}. See the end of chapter 10, paragraph
90, and the problem immediately thereafter:  ``Equations of
electrodynamics in the presence of a gravitational field". Note that in
contrast to Gordon, here the interest is in using dielectric media to
mimic a gravitational field.

In France the idea was taken up by Pham Mau Quan~\cite{Pham-Mau-Quan},
who showed that (under certain conditions) Maxwell's equations can be
expressed directly in terms of the effective metric specified by the
coefficients

%
\begin{equation}
[g_\mathrm{effective}]_{\mu\nu}=g_{\mu \nu} +\left[1- {1\over\epsilon\mu}\right]
V_{\mu} V_{\nu}, 
\end{equation}
%
where $g_{\mu\nu}$ is the ordinary spacetime metric, $\epsilon$ and $\mu$ are 
the permeability and permittivity, and $V_\mu$ is the 4-velocity of the medium.
The trajectories of the electromagnetic rays are interpreted in this case as 
geodesics of null length of this new effective metric.


Three articles that directly used the dielectric analogy to analyze
specific physics problems are those of Skrotskii~\cite{Skrotskii},
Balazs~\cite{Balazs}, and Winterberg~\cite{Winterberg}. The general
formalism was more fully developed in articles such as those by
Peblanski~\cite{Peblanski,Peblanski2}, and  good summary of this
classical period can be found in the article by de
Felice~\cite{deFelice}.

In summary and with the benefit of hindsight:
An arbitrary gravitational field can always be represented as an equivalent
optical medium, but subject to the somewhat unphysical restriction that 
%
\begin{equation}
\hbox{ [magnetic permitivity]} \propto \hbox{[electric permeability].}
\end{equation}
%
If an optical medium does not satisfy this constraint (with a position
independent proportionality constant) then it is not completely equivalent
to a gravitational field. 
For a position dependent proportionality constant complete equivalence can be
established in the geometric optics limit, but for wave optics the equivalence
is not complete.


%----------------------------------------------
\subsubsection{Acoustics}
%----------------------------------------------

There were several papers in the 1980's using an acoustic analogy to
investigate the propagation of shockwaves in astrophysical situations,
most notably those of Moncrief~\cite{Moncrief} and 
Matarrese~\cite{Matarrese,Matarrese2,Matarrese3}.  (Logically they are
orthogonal to modern developments and belong to the historical period.
Temporally several of these articles post-date Unruh's 1981 paper by a
few years, but there seems to have not been any cross-connection.)

%----------------------------------------------
\subsubsection{Electro-mechanical analogy}
%----------------------------------------------

The so-called ``electro-mechanical analogy'' has also had a long history
within the engineering community. It is sometimes extended to obtain an
``electro-mechanical-acoustic'' analogy, or even an ``electro-thermal''
analogy. Unfortunately the issues of interest to the engineering
community rarely resonate within the relativity community, and these
engineering analogies (though powerful in their own right) have no
immediate impact for our purposes.~\footnote{A recent attempt at
connecting the electro-mechanical analogy back to relativity can be
found in~\cite{Zloshchastiev:1999rr}.}



%----------------------------------------------
\subsection{Motivation}
%----------------------------------------------


The motivation for these investigations (both historical and current) is
rather mixed. In modern language the reasons to investigate analogue
models are:

\begin{itemize}

\item Partly to use condensed matter to gain insight into classical
general relativity.

\item Partly to use condensed matter to gain insight into curved-space
quantum field theory.

\item Partly to develop an observational window on curved-space quantum
field theory.

\item Partly to use classical general relativity to gain insight into
condensed matter physics.

\item Partly (much more tenuous) to gain insight into new and radically
different ways of dealing with ``quantum gravity''.

\end{itemize}



%----------------------------------------------
\subsection{Going further}
%----------------------------------------------
\label{SS:going-further-3}
%----------------------------------------------

There is not much more that we can usefully say here. We have doubtless
missed some articles of historical importance, but with a good library
or a fast internet connection the reader will be in as good a position
as we are to find any additional historical articles.


%------------------------
% EOF
%------------------------ 
introduction.tex
%---------------------------
% introduction.tex
%---------------------------
 
 
\begin{quote}

And I cherish more than anything else the Analogies, my most trustworthy
masters. They know all the secrets of Nature, and they ought least to be
neglected in Geometry.

\null\hfill --- Johannes Kepler
\end{quote}
% See: G Jona-Lasinio; Highlights of Mathematical Physics; 
% AMS 2002; p 143.
% See also: G Polya, Induction and Analogy in Mathematics.


Analogies have played a very important role in physics and mathematics
--- they provide new ways of looking at problems that permit
cross-fertilization of ideas among different branches of science. A
carefully chosen analogy can be extremely useful in focussing
attention on a specific problem, and in suggesting unexpected routes
to a possible solution.  In this review article we will focus on
``analogue gravity'', the development of analogies (typically but not
always based on condensed matter physics) to probe aspects of the
physics of curved spacetime --- and in particular to probe aspects of
curved space quantum field theory.

The most well-known of these analogies is the use of sound waves in a
moving fluid as an analogue for light waves in a curved
spacetime. Supersonic fluid flow can then generate a ``dumb hole'',
the acoustic analogue of a ``black hole'', and the analogy can be
extended all the way to mathematically demonstrating the presence of
phononic Hawking radiation from the acoustic horizon.  This particular
provides (at least in principle) a concrete laboratory model for
curved-space quantum field theory in a realm that is technologically
accessible to experiment.

There are many other ``analogue models'' that may be useful for this
or other reasons --- some of the analogue models are interesting for
experimental reasons, others are useful for they way they provide new
light on perplexing theoretical questions. The information flow is in
principle bi-directional and sometimes insights developed within the
context of general relativity can be used to understand aspects of the
analogue model.

Of course analogy is not identity, and we are in no way claiming that
the analogue models we consider are completely equivalent to general
relativity --- merely that the analogue model (in order to be
interesting) should capture and accurately reflect a sufficient number
of important features of general relativity (or sometimes special
relativity). The list of analogue models is extensive, and in this
review we will seek to do justice both to the key models, and to the key
features of those models.


\noindent
The plan for this review is as follows. In the following chapters we shall:
\begin{itemize}
\item Discuss the flowing fluid analogy in some detail.
\item Summarize the history and motivation for various analogue models.
\item Discuss the many physics issues various researchers have addressed.
\item Provide a catalogue (hopefully complete) of extant models.
\item Discuss the main physics results obtained to date.
\item Outline the many possible directions for future research.
\item Summarize the current state of affairs.
\end{itemize}

By that stage the interested reader will have had a quite thorough
introduction to the ideas, techniques, and hopes of the analogue gravity
programme.

%----------------------------------------------
\subsection{Going further}
%----------------------------------------------
\label{SS:going-further-1}
%----------------------------------------------

Apart from this present review article, and the references contained herein, there are 
several key items that stand out as starting points for any deeper investigation:
\begin{itemize}
\item The book ``Artificial Black Holes'',  
edited by Mario Novello, Matt Visser, and Grigori Volovik~\cite{Novello:2002qg}.
\item The websites for the ``Analogue models'' workshop:
\begin{itemize}
\item {\sf http://www.cbpf.br/\verb#~#bscg/analog/}
\item {\sf http://www.mcs.vuw.ac.nz/\verb#~#visser/Analog/ }
\item {\sf http://www.physics.wustl.edu/\verb#~#visser/Analog/ }
\end{itemize}
\item The book ``The Universe in a Helium droplet'', by Grigori Volovik~\cite{Volovik:2003fe}.
\item The Physics Reports article, ``Superfluid analogies of cosmological phenomena", by Grigori Volovik~\cite{Volovik:2000ua}.
\end{itemize}

%-------------------
% EOF 
%-------------------


lessons-from-analogue.tex
%--------------------------------------- 
%\section{Lessons from analogue gravity} 
%--------------------------------------- 
% BOF
%---------------------------------------
\label{S-lessons}
%---------------------------------------



Of course the entire motivation for looking at analogue models is to be
able to learn more physics. One could start studying analogue models
with the idea of seeing whether it is possible (either theoretically or
in practice) to reproduce in the laboratory various gravitational
phenomena whose real existence in nature cannot be currently checked.
These are phenomena that surpass our present (and foreseeable)
observational capabilities, but yet, we believe in their existence
because it follows from seemingly strong theoretical arguments within
the standard frameworks of general relativity and field theory in curved
space. However, the interest of this approach is not merely to reproduce
these gravitational phenomena  in some formal analogue model, but to
see which departures from the ideal case show up in real analogue
models, and to analyse whether similar deviation are likely to appear in
real gravitational systems.

When one thinks about emergent gravitational features in condensed
matter systems, one immediately realizes that these features only appear
in the low-energy regime of the analogue systems. When the systems are
probed at high energies (short length scales) the effective geometrical
description of the analogue models break down, as one starts to be aware
that the systems are actually composed of discrete pieces (atoms and
molecules). This scenario is quite similar to what one expects to happen
with our geometrical description of the Universe, when explored with
microscopic detail at the Planck scale.

That is, the study of analogue models of general relativity is giving us
insights as to how the standard theoretical picture of different
gravitational phenomena could change when taking into account additional
physical knowledge coming from the existence of an underlying
microphysical structure. Quite robustly, these studies are telling us
already that the first deviations from the standard general relativity
picture can be encoded in the form of high-energy violations of Lorentz
invariance in particle dispersion relations. Beyond these first
deviations, the analogue models of general relativity provide well
understood examples (the underlying physics is well known) in which a
description in terms of fields in curved spacetimes shows up as a
low-energy-regime emergent phenomena.

The analogue models are being used to throw light upon these general
questions through a number of specific routes. Let us now turn to
discussing several specific physics issues that are being analyzed from
this perspective.



%---------------------------------------------- 
\subsection{Hawking radiation} 
%----------------------------------------------


%----------------------------------------------
\subsubsection{Basics}
%----------------------------------------------

As is well known, in 1974 Stephen Hawking announced that quantum
mechanically even a spherical distribution of matter collapsing to form
a black hole should emit radiation; with a spectrum approximately that
of a black body \cite{Hawking:1974rv,Hawking:1974sw}. A black hole will
tend to evaporate by emitting particles from its horizon toward
infinity. Hawking radiation is a quantum-field-in-curved-space effect.
The existence of radiation emission is a kinematic effect that does not
rely on Einstein equations.  Therefore, one can aim to reproduce it in a
condensed matter system.  Within standard field theory, a minimal
requirement for having Hawking radiation is the existence in the
background configuration of an apparent horizon \cite{Visser:2001kq}.
So, in principle, to be able to reproduce Hawking radiation in a
laboratory one would need at least two requirements:

\begin{itemize}

\item To choose an adequate analogue system; it has to be a quantum
analogue model (see Section \ref{S:catalogue}) such that its description
could be separated into a classical effective background spacetime plus
some standard relativistic quantum fields leaving on it (it can happen
that the quantum fields do not satisfy the appropriate commutation or
anti-commutation relations \cite{Unruh:2003ss}).

\item To configure the analogue geometry such that it includes an
apparent horizon. That is, within an appropriate quantum analogue model,
the formation of an apparent horizon for the propagation of the quantum
fields should excite the fields as to result in the emission of a
thermal distribution of field particles.

\end{itemize}

This is a straight and quite naive translation of the standard Hawking
effect derivation to the condensed matter realm. However, in reality,
this translation process has to take into account additional issues
that, as we are trying to convey, instead of problems, are where the
interesting physics lies.

--- (1) The effective description of the quantum analogue systems as
fields over a background geometry breaks down when probed at
sufficiently short length scales. This could badly influence the main
features of Hawking radiation. In fact, immediately after the inception
of the idea that black holes radiate, it was realized that there was a
potential problem with the calculation \cite{Unruh:1976db}. It strongly
relies on the validity of quantum field theory on curved backgrounds up
to arbitrary high energies. Following a wave packet with a certain
frequency at future infinity backwards in time, we can see that it had
to contain arbitrarily large frequency components with respect to a
local free fall observer (well beyond the Planck scale) when it was
close to the horizon. In principle any unknown physics at the Planck
scale could strongly influence the Hawking process so that one should
view it with suspicion. This is the so-called trans-Planckian problem of
Hawking radiation. To create an analogue model exhibiting Hawking
radiation will be, therefore, equivalent to give a solution to the
trans-Planckian problem.

--- (2) In order to clearly observe Hawking radiation, one should first
be sure that there is no other source of instabilities in the system
that could mask the effect. In analogue models such as liquid Helium or
BECs the interaction of a radial flow (with speed of the order of the
critical Landau speed, that in these cases coincides with the sound
speed \cite{Kopnin:1997jy}) with the surface of the container (an
electromagnetic potential in the BECs case) might cause the production
of rotons and quantized vortices, respectively. Thus, in order to
produce an analogue model of Hawking radiation, one has to be somewhat
ingenious.  For example in the liquid Helium case, instead of taking
acoustic waves in a supersonic flow stream as the analogue model, it is
preferable to use as analogue model ripplons in the interface between
two different phases, A and B phases, of Helium-3 \cite{Volovik:2002ci}.
Another option is to start from a moving domain wall configuration.
Then, the topological stability of the configuration prevents its
destruction when creating an horizon \cite{Jacobson:1998he,
Jacobson:1998ms}. In the case of BECs a way to suppress the formation of
quantized vortices is to take effectively one-dimensional
configurations. If the transverse dimension of the flow is smaller than
the healing length then there is no space for the existence of a vortex
\cite{Barcelo:2001ca}. In either liquid Helium or BECs, there is also
the possibility of creating an apparent horizon by rapidly approaching a
critical velocity profile (see figure
\ref{F:critical-dynamical-configuration}), but without actually crossing
into the supersonic regime \cite{Barcelo:2004wz}, softening in this way
the appearance of dynamical instabilities.


%====================================================
\begin{figure}[htb] \vbox{

\centerline{{\includegraphics[width=5in]{F-v-spike-dyn}}}

\caption[Critical collapse velocity profile]{Velocity profile for a
left going flow; the profile is dynamically modified with time so that
it reaches the profile with a sonic point at the asymptotic future.
\smallskip \label{F:critical-dynamical-configuration}}

} \end{figure} 
%====================================================





-- (3) Real analogue models cannot, strictly speaking, reproduce eternal
black-hole configurations. An analogue model of a black hole has always
to be created at some finite laboratory time. Therefore, one is forced
to carefully analyze the creation process, as it can greatly influence
Hawking effect. Depending on the procedure of creation, one could end up
in quite different quantum states for the field and only some of them
might exhibit Hawking radiation. This becomes more important when
considering that the analogue models incorporate modified dispersion
relations. An inappropriate preparation, together with modified
dispersion relation effects, could completely eliminate Hawking
radiation \cite{Unruh:2004zk}.

--- (4) Another important issue is the need to characterize ``how
quantum'' a specific analogue model is. Even though, strictly speaking,
one could say that any system undergoes quantum fluctuations, the point
is how important they are in its description. In trying to build an
analogue model of Hawking's quantum effect, the relative value of
Hawking temperature with respect to the environment is going to tell us
whether the system can be really thought as a quantum analogue model or
as effectively classical. For example, in our standard cosmological
scenario, for a black hole to radiate at temperatures higher than that
of the Cosmic Microwave Background, $\approx 3$ K, the black hole should
have a diameter of the order of micrometers or less. We would have to
say that such black holes are no longer classical, but semiclassical.
The black holes for which we have some observational evidence are of
much higher mass and size, so their behaviour can be thought of as
completely classical. Estimates of the Hawking temperature reachable in
BECs yield $T \sim 100$nK \cite{Barcelo:2001ca}. This has the same order
of magnitude of the temperature of the BECs themselves. This is telling
us that regarding the Hawking process, BECs can be considered to be
highly-quantum analogue models.

Because of its importance, let us now review what we know about the
effects of high-energy dispersion relations on the Hawking process.

%----------------------------------------------
\subsubsection{Trans-Planckian problem}
%----------------------------------------------

We saw in the introduction to this section that the trans-Planckian
problem of Hawking radiation was one of the strongest motivations for
the modern research into analogue models of gravity.  In fact it was
soon realized that such models could provide a physical framework within
which a viable solution of the problem could be found. Let us explain
why and how.

As we have said, the requirement of a reservoir of ultra-high frequency
modes nearby the horizon seems to indicate a possible (and worrisome)
sensitivity of the black hole radiation to the microscopic structure of
spacetime.  Actually by assuming exact Lorentz invariance one could in
principle always locally transform the problematic ultra high frequency
modes to low energy ones via some appropriate Lorentz
transformation~\cite{Jacobson:1991gr}. However in doing so it would have
to rely on the physics of reference frames moving ultra fast with
respect to us, as the reference frame needed would move arbitrarily
close to the speed of light. Hence we would have to apply Lorentz
invariance in a regime of arbitrary large boots yet untested and in
principle never completely testable given the non-compactness of the
boost subgroup. The assumption of an exact boost symmetry is linked to a
scale-free nature of spacetime given that unbounded boosts expose
ultra-short distances. Hence the assumption of \emph{exact} Lorentz
invariance needs, in the end, to rely on some idea on the nature of
spacetime at ultra-short distances.

It was this type of reasoning that led in the nineties to a careful
reconsideration of the crucial ingredients required for the derivation
of Hawking radiation~\cite{Jacobson:1991gr, Jacobson:1993hn,
Unruh:1994zw}. In particular investigators explored the possibility that
spacetime microphysics could provide a short distance, Lorentz-breaking
cutoff, but at the same time leave Hawking's results unaffected at energy
scales well below that set by the cutoff.

Of course ideas about a possible cutoff imposed by the discreteness of
spacetime at the Planck scale had already been discussed in the
literature well before Unruh's seminal paper~\cite{Unruh:1981cg}.
However such ideas were running into serious difficulties given that a
naive short distance cutoff posed on the available modes of a free field
theory results in a complete removal of the evaporation process (see
\emph{e.g.},~\cite{Jacobson:1991gr} and references therein). Indeed
there are alternative ways through which the effect of the short scales
physics could be taken into account, and analogue models provide a
physical framework where these ideas could be put to the test.  In fact
analogue models provide explicit examples of emergent spacetime
symmetries, they can be used to simulate black hole backgrounds, they
may be endowed with quantizable perturbations and, in most of the cases,
they have a well known microscopic structure. Given that  Hawking
radiation can be, at least in principle, simulated in such systems one
might ask how and if the trans-Planckian problem is resolved in these
cases.


%-----------------------------------------------
\paragraph{Modified dispersion relations:}
%-----------------------------------------------

The general feature that most of the work on this subject has focussed
on is that in analogue models the quasi-particles propagating on the
effective geometry are actually collective excitations of atoms. This
generically implies that their dispersion relation will be a
relativistic one only at low energies (large
scales),\footnote{Actually, even relativistic behaviour at low energy
can be non-generic, but we assume in this discussion that an analogue
model by definition is a system for which all the linearized
perturbations do propagate on the same Lorentzian background at low
energies.} and in each case there will be some short length scale
(\emph{e.g.}, intermolecular distance for a fluid, coherence length
for a superfluid, healing length for a BEC) beyond which deviations
will be non-negligible. In general such microphysics induced
corrections to the dispersion relation take the form

\begin{equation} 
E^2=c^2\left(m^2c^2+k^2+\Delta(k,K)\right)
\end{equation} 
%
where $K$ is the scale that describes the transition to the full
microscopic system (what we might call the ``analogue Planck scale'').

In general the best one can do is to expand $\Delta(k,K)$ around $k=0$,
so obtaining an infinite power series (of which it will be safe to
retain only the lowest order terms), although in some special models
(like BEC) the series is automatically finite due to intrinsic
properties of the system. [In any case one can see that most of the
analogue models so far considered lead to modifications of the form $\pm
k^3/K^2$ or $\pm k^4/K^2$.] Depending on the sign in front of the
modification the group velocity at high energy can be larger ($+$) or
smaller ($-$) than the low energy speed of light $c$. These cases are
usually referred in the literature as ``superluminal'' and
``subluminal'' dispersion relations.

Most of the work on the trans-Planckian problem in the nineties focussed
on studying the effect on Hawking radiation due to such modifications of
the dispersion relations at high energies in the case of acoustic
analogues \cite{Jacobson:1991gr, Jacobson:1993hn, Unruh:1994zw,
Unruh:1995je, Corley:1996ar}, and the question of whether such
phenomenology could be applied to the case of real black holes (see
\emph{e.g.},~\cite{Brout:1995wp, Jacobson:1996zs, Corley:1996ar,
Parentani:2000ts}).\footnote{However see
also~\cite{Reznik:1997xt, Rosu:1996pw} for a radically different
alternative approach based on the idea of ``superoscillations'' where
ultrahigh frequency modes near the horizon can be mimicked (to
arbitrary accuracy) by the exponential tail of an exponentially large
amplitude mostly hidden behind the horizon.} In all the aforementioned
works Hawking radiation can be recovered under some suitable
assumptions, as long as neither the black hole temperature nor the
frequency at which the spectrum is considered are too close to the scale
of microphysics $K$. However, the applicability of these assumptions to
the real case of black hole evaporation is an open question. Also, in
the case of the analogue models the mechanism by which the Hawking
radiation is recovered is not always the same. We concisely summarize
here the main results (but see \emph{e.g.},~\cite{Unruh:2004zk} for
further details).

%---------------------------------------------- 
\paragraph{Subluminal dispersion relations:} 
%----------------------------------------------

This was the case originally considered by Unruh~\cite{Unruh:1994zw,
Unruh:1995je},
\begin{equation} 
\omega=K\left(\tanh(k/K)^n\right)^{1/n}, 
\label{eq:Un}
\end{equation} 
and by Corley and Jacobson~\cite{Corley:1996ar}
\begin{equation} 
\omega^2=k^2-k^4/K^2,
\label{eq:CJ} 
\end{equation} %
where both dispersion relations are given in the co-moving frame.

The key feature is that in the presence of a subluminal modification
the group velocity of the modes increases with $k$ only up to some
turning point (which is equivalent to saying that the group velocity
does not asymptote to $c$, which could be the speed of sound, but
instead is upper bounded). For values of $k$ beyond the turning point
the group velocity decreases to zero (for (\ref{eq:Un})) or becomes
imaginary (for (\ref{eq:CJ})). In the latter case this can be
interpreted as signifying the breakdown of the regime where the
dispersion relation (\ref{eq:CJ}) can be trusted. The picture that
emerged from these analyses concerning the origin of the outgoing
Hawking modes at infinity is quite surprising. In fact, if one
traces back in time an outgoing mode, as it approaches the horizon it
decreases its group velocity below the speed of sound. At some point
before reaching the horizon, the outgoing mode will appear as an
ingoing one dragged into the black hole by the flow.  
%{\it In fact if
%one traces back in time an outgoing mode, it will approach the horizon
%and so increase its group velocity up to a maximum equal to the speed
%of sound. From that point on the group velocity drop below the
%speed of sound and the outgoing mode will appear as an ingoing one
%dragged into the black hole by the flow.} 
Stepping further back in time it is seen that such a mode was located at
larger and larger distances from the horizon, and tends to a very high
energy mode far away at early times. We have then that in this case one
finds what might be called a ``mode conversion", where the origin of the
outgoing Hawking quanta seems to originate from ingoing modes which have
``bounced off'' the horizon without reaching trans-Planckian frequencies
in its vicinities. Several detailed analytical and numerical
calculations have shown that such a conversion indeed
happens~\cite{Unruh:1995je, Brout:1995wp, Corley:1996ar, Corley:1997pr,
Himemoto:1999kd, Saida:1999ap, Unruh:2004zk} and that the Hawking result
can be recovered for $\kappa\ll K$ where $\kappa$ is the black hole
surface gravity.

%---------------------------------------------- 
\paragraph{Superluminal dispersion relations:} 
%----------------------------------------------

The case of a superluminal dispersion relation is quite different and, as
we have seen, it has some experimental interest given that these are the
kind of dispersion relations that arise in some promising analogue
models (\emph{e.g.}, BECs). In this situation, the outgoing modes are
actually seen as originating from behind the horizon. This implies that
these modes somehow originate from the singularity (which can be a
region of high turbulence in acoustic black hole analogues), and hence
it would seem that not much can be said in this case.  However it is
possible to show that if one imposes vacuum boundary conditions on these
modes near the singularity, then it is still possible to recover the
Hawking result, \emph{i.e.}, thermal radiation outside the
hole~\cite{Corley:1997pr}.  It is particularly interesting to note that
this recovering of the standard result is not always guaranteed in the
presence of superluminal dispersion relations. Corley and
Jacobson~\cite{Corley:1998rk} in fact discovered a very peculiar type of
instability due to such superluminal dispersion in the presence of black
holes with inner horizons. The net result of the investigation carried
out in~\cite{Corley:1998rk} is that the compact ergo-region
characterizing such configurations is unstable to self-amplifying
Hawking radiation. The presence of such an instability seems to be
confirmed by the analysis carried on in~\cite{Garay:1999sk,Garay:2000jj}
where a Bose--Einstein condensate analogue black hole was considered.

%---------------------------------------------- 
\paragraph{General conditions for Hawking radiation}
%----------------------------------------------

Is it possible to reduce the just described rather complex
phenomenology to a few basic assumptions that must be satisfied in order
to recover Hawking radiation in the presence of Lorentz violating
dispersion relations? A tentative answer is given
in~\cite{Unruh:2004zk}, where the robustness of the Hawking result is
considered for general modified (subluminal as well as superluminal)
dispersion relations. The authors of~\cite{Unruh:2004zk} assume that the
geometrical optics approximation breaks down only in the proximity of
the event horizon (which is equivalent to saying that the particle
production happens only in such a region). Here, the would be
trans-Planckian modes are converted in sub-Planckian ones. Then, they
try to identify the minimal set of assumptions that guarantees that such
``converted modes'' are generated in their ground states (with respect
to a freely falling observer), as this is a well known condition in
order to recover Hawking's result. They end up identifying three basic
assumptions that guarantee such emergence of modes in the ground state
at the horizon. First, the preferred frame selected by the breakdown of
Lorentz invariance must be the freely falling one instead of the rest
frame of the static observer at infinity (which coincides in this limit
with the laboratory observer). Second, the Planckian excitations must
start off in the ground state with respect to freely falling observers.
Finally, they must evolve in an adiabatic way (\emph{i.e.}, the Planck
dynamics must be much faster than the external sub-Planckian dynamics).
Of course, although several systems can be found in which such
conditions hold, it is also possible to show~\cite{Unruh:2004zk} that
realistic situations in which at least one of these assumptions is
violated can be imagined. It is hence still an open question whether
real black hole physics does indeed satisfy such conditions, and it is
hence robust against modifications induced by the violation of Lorentz
invariance.

%---------------------------------------------- 
\paragraph{Open issues}
%----------------------------------------------

In spite of the remarkable insight given by the above discussed models
(based on modified dispersion relations) it is not possible to consider
them fully satisfactory in addressing the trans-Planckian problem.  In
particular it was soon recognized~\cite{Corley:1998ef,Jacobson:1999zk}
that in this framework it is not possible to explain the origin of the
short wavelength incoming modes which are ``progenitors'' of the
outgoing modes after bouncing off in the proximity of the horizon. For
example, in the Unruh model (\ref{eq:Un}), one can see that if one keeps
tracking an incoming mode back in time, then its group velocity (in the
co-moving frame) drops to zero as its frequency gets more and more blue
shifted. This is tantamount to saying that the trans-Planckian problem
has just been moved from the region near the horizon out to the region
near infinity. In the Corley--Jacobson model (\ref{eq:CJ}) this
unphysical behaviour is removed thanks to the presence of the physical
cutoff $K$. However it is still true that in tracking the incoming modes
back in time one finally sees a wave packet so blue shifted that
$|k|=K$. At this point one can no longer trust the dispersion relation
(\ref{eq:CJ}) (which anyway in realistic analogue models is emergent and
not fundamental), and hence the model has no predictive power regarding
the ultimate origin of the relevant incoming modes.

These conclusions regarding the impossibility of clearly predicting the
origin at early times of the modes ultimately to be converted into
Hawking radiation is not specific to the particular dispersion relations
(\ref{eq:Un}) or (\ref{eq:CJ}) one is using. The Killing frequency is in
fact conserved on a static background, thus the incoming modes must have
the same frequency as the outgoing ones, hence there can be no
mode-mixing and particle creation. This is why one has actually to
assume that the WKB approximation fails in the proximity of the horizon
and that the modes are there in the vacuum state for the co-moving
observer. In this sense the need for these assumptions can be
interpreted as evidence that these models are not fully capable of
solving the trans-Planckian problem.


%---------------------------------------------- 
\paragraph{Solid state {\bf and lattice} models} 
%----------------------------------------------

It was to overcome this type of issue that alternative ways of
introducing an ultra-violet cutoff due to the microphysics were
considered~\cite{Reznik:1997xt,Reznik:1997ag,Corley:1998ef}. In
particular in~\cite{Reznik:1997ag} the transparency of the refractive
medium at high frequencies has been used to introduce an effective
cutoff for the modes involved in Hawking radiation in a classical
refractive index analogue model (see section~\ref{sec:cri} of this
review). In this model an event horizon for the electromagnetic field
modes can be simulated by a surface of singular electric and magnetic
permeabilities. This would be enough to recover Hawking radiation but it
would imply the unphysical assumption of a refractive index which is
valid at any frequency. However it was shown in~\cite{Reznik:1997ag}
that the Hawking result can be recovered even in the case of a
dispersive medium which becomes transparent above some fixed frequency
$K$ (which we can imagine as the plasma frequency of the medium), the
only (crucial) assumption being again that the ``trans-Planckian" modes
with $k>K$ are in their ground state near the horizon.

An alternative avenue was instead considered in~\cite{Corley:1998ef}.
There a lattice description of the background was used for imposing a
cutoff in a more physical way with respect to the continuum dispersive
models previously considered. In such a discretized spacetime the field
takes values only at the lattice points, and wavevectors are identified
modulo $2\pi/\ell$ where $\ell$ is the lattice characteristic spacing,
correspondingly one obtains a sinusoidal dispersion relation for the
propagating modes. Hence the problem of recovering a smooth evolution of
incoming modes to outgoing one is resolved by the intrinsically
regularized behaviour of the wave vectors field. In~\cite{Corley:1998ef}
the authors explicitly considered the Hawking process for a discretized
version of a scalar field, where the lattice is associated to the
free-fall coordinate system (taken as the preferred system). With such a
choice it possible to preserve a discrete lattice spacing. Furthermore
the requirement of a  fixed short distance cutoff leads to the choice of
a lattice spacing constant at infinity, and that the lattice points are
at rest at infinity and fall freely into the black
hole.~\footnote{Reference~\cite{Corley:1998ef} also considered the case
of a lattice with proper distance spacing constant in time but this has
the problem that the proper spacing of the lattice goes to zero at
spatial infinity, and hence there is no fixed short distance cutoff.} In
this case the lattice spacing grows in time and the lattice points
spread in space as they fall toward the horizon. However this time
dependence of the lattice points is found to be of order $1/\kappa$, and
hence unnoticeable to long wavelength modes and relevant only for those
with wavelengths of the order of the lattice spacing. The net result is
that on such a lattice long wavelength outgoing modes are seen to 
originate from short wavelength incoming modes via a process analogous
to the Bloch oscillations of accelerated electrons in
crystals~\cite{Corley:1998ef,Jacobson:1999zk}.

%---------------------------------------------- 
\subsubsection{Horizon stability} 
%----------------------------------------------
\label{S:Horizon-stability}
%----------------------------------------------

Although closely related, as we will soon see, we have to distinguish
carefully between the mode analysis of a linear field theory (with or
without modified dispersion relations --- MDR) over a fixed background
and the stability analysis of the background itself.

Let us consider a three-dimensional irrotational and inviscid fluid
system with a stationary sink-type of flow (see figures~\ref{F:cascade}
and~\ref{F:ondas}).  The details of the configuration are not important
for the following discussion, only the fact that there is a spherically
symmetric fluid flow accelerating towards a central sink,  that sink
being surrounded by a sphere acting as a sonic horizon). Then, as we
have discussed in section \ref{S:simple}, linearizing the Euler and continuity
equations leads to a massless scalar field theory over a black-hole-like
spacetime. (We are assuming that the hydrodynamic regime remains valid
up to arbitrarily short length scales; for instance, we are neglecting
the existence of MDR). To be specific, let us choose the geometry of the
canonical acoustic black-hole spacetime described
in~\cite{Visser:1997ux}: %

\begin{eqnarray} 
\d s^2=-c^2\left(1-{r_0^4 \over r^4}\right)\d\tau^2
+\left(1-{r_0^4 \over r^4}\right)^{-1}\d r^2
+r^2\left(\d\theta^2+\sin^2\theta \;\d\varphi^2 \right). 
\end{eqnarray} %
%
In this expression we have used the Schwarzschild time coordinate $\tau$
instead of the lab time $t$; $c$ is constant. If we expand the field in
spherical harmonics, %

\begin{eqnarray} 
\phi_{lm}(\tau,r,\theta,\varphi) \equiv e^{-i\omega\tau} \;
{\chi_{lm}(r) \over r} \; Y_{lm}(\theta, \varphi), 
\end{eqnarray} % 
%
we obtain the following equation for the radial part of the field: 
%
\begin{eqnarray} 
{\omega^2 \over c^2} \; \chi = \left(-{\d^2 \over
\d{r^*}^2} + V_l(r) \right)\chi;
\end{eqnarray} %
%
where
% 
\begin{eqnarray} 
V_l(r)=\left(1- {r^4_0 \over r^4}\right)
\left[{l(l+1) \over r^2}+ {4r_0^4 \over r^6}\right]. 
\end{eqnarray} %
%
Here 
%
\begin{equation} 
r^* \equiv r-(r_0/4) \{\ln (r+r_0)/(r-r_0)+ 2\arctan r/r_0 \} 
\end{equation} 
%
is a ``tortoise'' coordinate. 

In a normal mode analysis one requires boundary conditions such that the
field is regular everywhere, even at infinity. However, if one is
analyzing the solutions of the linear field theory as a way of probing
the stability of the background configuration, one can consider less
restrictive boundary conditions. For instance, one can consider the
typical boundary conditions that lead to quasinormal modes: These modes
have to be purely out-going at infinity and purely in-going at the
horizon, but one does not require, for example, the modes to be
normalizable. The quasinormal modes associated to this sink
configuration have been analyzed in \cite{Berti:2004ju}. The results
found are qualitatively similar to those in the classical linear
stability analysis of the Schwarzschild black hole in general relativity
\cite{Vishveshwara:1970, Vishveshwara:1970cc, Regge:1957,
Zerilli:1970se, Moncrief:1974}. Of course, the gravitational field in
general relativity has two dynamical degrees of freedom ---those
associated with gravitational waves--- that have to be analyzed
separately; these are the ``axial'' and ``polar'' perturbations. In
contrast, in the present situation we only have scalar perturbations.
Nevertheless, the potentials associated with ``axial'' and ``polar''
perturbations of Schwarzschild spacetime, and that associated with
scalar perturbations of the canonical acoustic black hole, produce
qualitatively the same behaviour: There is a series of \emph{damped}
quasinormal modes ---proving the linear stability of the system--- with
higher and higher damping rates.

An important point we have to highlight in here is that although in the
linear regime the dynamical behaviour of the acoustic system is similar
to general relativity, this is no longer true once one enters the
non-linear regime. The underlying nonlinear equations in the two cases
are very different.  The differences are so profound, that in the
general case of acoustic geometries constructed from compressible
fluids, there exist sets of perturbations that, independently of how
small they are initially, can lead to the development of shocks, a
situation completely absent in vacuum general relativity.

Now, given an approximately stationary, and at the very least
metastable, classical black-hole-like configuration, a standard quantum
mode analysis leads to the existence of Hawking radiation in the form of
phonon emission. This shows, among other things, that quantum
corrections to the classical behaviour of the system must make the
configuration with a sonic horizon dynamically unstable against Hawking
emission. Moreover, in an analogue system with quantum fluctuations that
maintain strict adherence to the equivalence principle (no MDR) it must
then be impossible to create an isolated truly stationary horizon by
external means --- any truly stationary horizon must be provided with an
external power source to stabilize it against Hawking emission. That is,
in an analogue system one could in principle, by manipulating external
forces, compensate for the backreaction effects that in a physical
general relativity scenario cause the horizon to shrink (or evaporate)
and thus becomes non-stationary.




%However we know that in the absence of backreaction a stationary black
%hole horizon would produce and infinite amount of particles and
%therefore, to really create a stationary horizon in such an (imaginary)
%fluid system with no MDR whatsoever would require the injection into the
%system of an infinite amount of energy during a finite amount of time.
%Obviously, no one can do that. In reality, a sonic stationary horizon
%can be dynamically created in a laboratory, but just because in real
%analogue systems the equivalence principle gets broken at high energies
%(short wavelengths) so that some MDR finally show up.

Let us describe what happens when one takes into account the existence
of MDR. A wonderful physical system that has MDR explicitly incorporated
in its description is the Bose-Einstein condensate. The macroscopic wave
function of the BEC behaves as a classical irrotational fluid but with
some deviations when short length scales become involved. (For length
scales of the order of or shorter than the healing length.) What are the
effects of the MDR on the dynamical stability of a black-hole like
configuration in a BEC?  The stability of a sink configuration in a BEC
has been analyzed in \cite{Garay:1999sk,Garay:2000jj} but taking the
flow to be effectively one-dimensional. What they found is that these
configurations are dynamically unstable: There are modes satisfying the
appropriate boundary conditions such that the imaginary parts of their
associated frequencies are positive. These instabilities are associated
basically with the bound states inside the black hole.  The dynamical
tendency of the system to evolve is suggestively similar to that in the
standard evaporation process of a black hole in semiclassical general
relativity. This observation alone could make us question whether the
first signatures of a quantum theory underlying general relativity might
show up in precisely this manner. Interest in this question is
reinforced by a specific analysis in the ``loop quantum gravity''
approach to quantizing gravity that points towards the the existence of
fundamental MDR at high energies \cite{Gambini:1998it}. The formulation
of effective gravitational theories that incorporate some sort of MDR at
high energies is currently under investigation (see for example
\cite{Magueijo:2001cr,Amelino-Camelia:2002vy}); these exciting
developments are however beyond the scope of this review.

Before continuing with the discussion on the stability of configurations
with horizons, and in order not to cause confusion between the different
wording used when talking about the physics of BECs and the emergent
gravitational notions on them, let us write down a quite loose but
useful translation dictionary:

\begin{itemize}

\item The ``classical'' or macroscopic wave function of the BEC
represents the classical spacetime of GR, but only when probed at long
enough wavelengths.

\item The ``classical'' long-wavelength perturbations to a background
solution of the Gross--Pitaevskii equation correspond to classical
gravitational waves in GR. Of course this analogy does not imply that
these are spin 2 waves, it only points out that the perturbations
are made from the same ``substance'' as the background configuration
itself.


\item The macroscopic wave function of the BEC, without the restriction
of being probed only at long wavelengths, corresponds to some sort of
semiclassical vacuum gravity. Its ``classical'' behaviour (in the sense
that does not involve any probability notion) is already taking into
account its underlying quantum origin.

\item The Bogoliubov quantum quasiparticles over the ``classical'' wave
function correspond to a further step away from semiclassical gravity in
that they are analogous to the existence of quantum gravitons over a
(semiclassical) background spacetime.

\end{itemize} 

At this point we would like to remark, once again, that the analysis
based on the evolution of a BEC has to be used with care. For example,
they cannot directly serve to shed light on what happens in the final
stages of the evaporation of a black hole, as the BEC does not fulfil,
at any regime, the Einstein equations.

Now continuing the discussion, what happens when treating the
perturbations to the background BEC flow as quantum excitations
(Bogoliubov quasiparticles)?. What we certainly know is that the
analysis of modes in a collapsing-to-form-a-black-hole background
spacetime leads to the existence of radiation emission very much like
Hawking emission \cite{Brout:1995wp, Corley:1997pr, Corley:1997bx},
(this is why it is said that Hawking's process is robust against
modifications of the physics at high energies). The comparison of these
calculations with that of Hawking, (without MDR), tells us that the main
modification to Hawking's result is that now the Hawking flux of
particles would not last forever but would vanish after a long enough
time (this is why, in principle, we can dynamically create a
configuration with a sonic horizon in a BEC). The emission of quantum
particles reinforces the idea that the supersonic sink configurations
are unstable. 

Summarizing: 

\begin{itemize}

\item If the perturbation to the BEC background configuration have
``classical seeds'' (that is, are describable by the linearized
Gross--Pitaevskii equation alone), then, one will have ``classical''
instabilities;

\item If the perturbations have ``quantum seeds'' (that is, are
described by the Bogoliubov equations), then, one will have ``quantum''
instabilities.

\end{itemize}

%====================================================
\begin{figure}[htb] \vbox{

\centerline{{\includegraphics[width=5in]{F-bh-wh-flow}}}

\caption[Black hole and white hole velocity profile]{One-dimensional
velocity profile with a black hole horizon and a white hole horizon.
\smallskip \label{F:bh-wh-flow}}

} 
\end{figure} 
%====================================================

In the light of the acoustic analogies it is natural to ask whether
there are other geometric configurations with horizons of interest,
besides the sink type of configurations (these are the most similar to
the standard description of black holes in general relativity, but
probably not the simplest in terms of realizability in a real
laboratory; for an entire catalogue of them see \cite{Barcelo:2004wz}).
Here, we are going to specifically mention two effectively one-dimensional black
hole--white hole configurations, one in a straight line and one in a ring
(see figures \ref{F:bh-wh-flow}, \ref{F:bh-wh-ring-flow}, respectively).

A quantum mode analysis of the black hole--white hole configuration in a
straight line, taking into account the existence of superluminal
dispersion relations (similar to those in a BEC), led to the conclusion
that the emission of particles in this configuration proceeds in a
self-amplified (or runaway) manner \cite{Corley:1998rk}. We can
understand this effect as follows: At the black horizon a virtual pair
of phonons are converted into real phonons, the positive energy phonon
goes towards infinity while the negative energy pair falls beyond the
black horizon. However, the white horizon makes this negative energy
phonon bounce back towards the black horizon (thanks to superluminal
motion) stimulating the emission of additional phonons. Although
related to Hawking's process this phenomenon has a quite different
nature: For example, there is no temperature associated with it. A
stability analysis of a configuration like this in a BEC would lead to
strong instabilities. This same configuration, but compactified into a
ring configuration, has been dynamically analyzed in
\cite{Garay:1999sk,Garay:2000jj}. What they found is that there are
regions of stability and instability depending on the parameters
characterizing the configuration. We suspect that the stability regions
appear because of specific periodic arrangements of the modes around the
ring.  Among other reasons, these arrangements are interesting because
they could be easier to create in laboratory with current technology and
their instabilities easier to detect than Hawking radiation itself.

%====================================================
\begin{figure}[htb] \vbox{

\centerline{{\includegraphics[width=5in]{F-bh-wh-ring-flow}}}

\caption[Black hole and white hole horizons in a ring]{One-dimensional
velocity profile in a ring; the fluid flow exhibits two sonic horizons,
one of black hole type and the other of white hole type. \smallskip
\label{F:bh-wh-ring-flow}}

} 
\end{figure} 
%====================================================



To conclude this subsection, we would like to highlight that there is
still much to be learned by studying the different levels of description
of an analogue system, and how they influence the stability or
instability of configurations with horizons.


%----------------------------------------------
\subsection{Super-radiance}
%----------------------------------------------

Another phenomena that has been (and is being) analyzed from the
analogue gravity perspective is super-radiance. The rotational kinetic
energy accumulated in a rotating black hole can be extracted from it by
scattering into it waves of sufficiently low frequency and high angular
momentum. In general, in order that the wave can extract rotational
energy from the system it has to satisfy the condition

\begin{eqnarray} 
\omega < m\;\Omega 
\end{eqnarray} 
%
where $\Omega$ is the angular speed of the black hole at the event
horizon and $m$ is the harmonic azimuthal number of the wave.  This is a
purely classical process first considered by Penrose
\cite{Penrose:1969pc}. When dealing with quantum fields, as opposed to
classical fields, this process can proceed spontaneously: Quantum
mechanically, a rotating black hole will tend to radiate away all of its
angular momentum, eventually approaching a non-rotating Schwarzschild
black hole \cite{Zeldovich:1971,Zeldovich:1971t}. This process is known
as super-radiance. (The term super-radiance was already used in
condensed matter to describe processes in which there was some coherent
emission of radiation from an otherwise disordered system).

Again, these processes have a purely kinematical origin so they are
perfectly suitable for being reproduced in an analogue
model. Regarding these processes, the simplest geometry that one can
reproduce, thinking of analogue models based on fluid flows, is that
of the draining bathtub of Section \ref{S:simple}. Of course, this
metric does not exactly correspond to Kerr geometry, nor even to a
section of it \cite{Visser:2004zs}. However, it is
\emph{qualitatively} similar. It can be used to simulate both Penrose's
classical process and quantum super-radiance as these effects do not
depend on the specific multipole decomposition of Kerr's geometry, but
only on its rotating character. A specific experimental setup has been
put forward by Schutzhold and Unruh using gravity waves in a shallow
basin mimicking an ideal draining bathtub \cite{Schutzhold:2002xm}.
Equivalently to what happens with Kerr black holes, this configuration
is classically stable (in the linear regime) \cite{Berti:2004ju}. A word
of caution is in order here: Interactions of the gravity surface waves
with bulk waves (neglected in the analysis) could cause the system to
became unstable \cite{Volovik:2002ci}. This instability has no
counterpart in standard general relativity (though it might have one in
braneworld theories). Super-resonant scattering of waves in this
rotating sink configuration, or in a simple purely rotating vortex,
could in principle be observed in this and other analogue models.  There
are already several articles dealing with this problem \cite{Basak:2005fv,
Basak:2002aw, Basak:2003uj, Cardoso:2004fi, Federici:2005ty,
Lepe:2004kv}.

A related phenomenon one can consider is the black-hole bomb mechanism
\cite{Press:1972}. One would only have to surround the rotating
configuration by a mirror for it to become grossly unstable. What causes
the instability is that those in-going waves that are amplified when
reflected in the ergosphere would then in turn be reflected back toward
the ergoregion, due to the exterior mirror, thus being amplified again,
and so on.

An interesting phenomenon that appears in many condensed matter systems
is the existence of quantized vortices. The angular momentum of these
vortices come in multiples of some fundamental unit (typically $\hbar$
or something proportional to $\hbar$). The extraction of rotational
energy by Penrose process in these cases could only proceed via
finite-energy transitions. This would supply an additional specific
signature to the process. In such a highly quantum configuration it is
also important to look for the effect of having high-energy dispersion
relations. For example, in BECs, the radius of the ergoregion of a
single quantized vortex is of the order of the healing length, so one
cannot directly associate an effective Lorentzian geometry to this
portion of the configuration. Any analysis that neglects the high energy
terms is not going to give any sensible result in these cases.



%---------------------------------------------- 
\subsection{Cosmological geometries} 
%----------------------------------------------

Analogue model techniques have also been applied to cosmology. In a
cosmological framework the key items of interest are the
Friedmann--Robertson--Walker [FRW] geometries, more properly called the
Friedmann--\Lemaitre--Robertson--Walker [FLRW] geometries. The
simulation of such geometries has been considered in various works such
as~\cite{Barcelo:2003et, Barcelo:2003wu, Fedichev:2003dj,
Fedichev:2003bv, Fedichev:2003id, Lidsey:2003ze, Calzetta:2002jz,
Calzetta:2002zz, Weinfurtner:2004mu} with a specific view to enhancing
our understanding of ``cosmological particle production'' driven by the
expansion of the universe. 

The acoustic metric can be written as 
%
\begin{eqnarray}
\d s^2={\rho \over c_s}
\left[-(c_s^2- v^2)\,
\d t^2-2\v \cdot \mathbf{dx} \, \d t +{\mathbf{dx}}^2 \right].
\label{fluid-geometry}
\end{eqnarray}
% 
Essentially there are two ways to use this metric to reproduce
cosmological spacetimes: One is based on physical explosion, the other
on rapid variations in the ``effective speed of light''.

Following~\cite{Barcelo:2003wu, Fedichev:2003bv, Kagan:1996, Castin:1996, Kagan:1997tc} one can
take an homogeneous system $\rho(t), c_s(t)$ and a radial profile for
the velocity $\v=(\dot b/b) {\mathbf r}$, with $b$ a scale factor
depending only on $t$. Then, defining a new radial coordinate as
$r_b=r/b$ the metric can be expressed as
%
\begin{eqnarray}
\d s^2={\rho \over c_s}
\left[-c_s^2\,\d t^2+b^2(\d r_b^2+r_b^2\; \d\Omega_2^2)\right].
\label{fluid-geometry2}
\end{eqnarray}
%
Introducing a Hubble-like parameter,
%
\begin{equation}
H_b(t) =  {\dot b(t)\over b(t)}, 
\end{equation}
%
the equation of continuity can be written as 
%
\begin{equation}
\dot \rho + 3 H_b(t) \; \rho = 0; 
\qquad \Rightarrow \qquad 
\rho(t)= {\rho_0\over b^3(t)},
\label{eq:rho}
\end{equation}
%
with $\rho_0$ constant. Finally we arrive at the metric of a flat FLRW
geometry
%
\begin{eqnarray}
\d s^2=-T^2(t)\; \d t^2+a_s^2(t) \; (\d r_b^2+r_b^2\;\d\Omega_2^2),
\end{eqnarray}
%
with
%
\begin{eqnarray}
T(t)\equiv \sqrt{\rho \, c_s}; ~~~~~a_s(t) \equiv \sqrt{\rho \over c_s} \;b. 
\end{eqnarray}
%
The proper Friedmann time, $\tau$, is related to the laboratory time,
$t$, by 
\begin{equation}
\tau=\int T(t) \;\d t.
\end{equation}

The other avenue starts from a fluid at rest $v=0$ with respect to the 
laboratory at all times:
%
\begin{eqnarray}
\d s^2=-\rho \,c_s \; \d t^2+{\rho \over c_s} \; 
{\mathbf{dx}}^2.
\end{eqnarray}
%
Now it is not difficult to imagine situation in which $\rho$ remains
constant, in a sufficiently large region of space, while the speed of
sound decreases with time (this can be made in BECs for example
by changing with time the value of the scattering length
\cite{Barcelo:2003et,Barcelo:2003wu}). This again reproduces 
a expanding flat FLRW Universe.

Models considered to date focus on variants of the BEC-inspired
analogues:

\begin{itemize}

\item Fedichev and Fischer~\cite{Fedichev:2003bv,Fedichev:2003id} have
investigated WKB estimates of the cosmological particle production rate 
and $(1+1)$ dimensional cosmologies, both in expanding BECs.

\item Lidsey~\cite{Lidsey:2003ze}, and Fedichev and
Fischer~\cite{Fedichev:2003dj} have focussed on the behaviour of
cigar-like condensates in grossly asymmetric traps.

\item Barcelo \emph{et al.}~\cite{Barcelo:2003et,Barcelo:2003wu} have
focussed on the central region of the BEC and thereby tried (at least
locally) to mimic FLRW behaviour as closely as possible.

\item In  \cite{Fischer:2004bf} Fischer and Schutzhold  propose the use
of two-component BECs to simulate cosmic inflation.

\item Weinfurtner~\cite{Weinfurtner:2004in,Weinfurtner:2004mu} has
concentrated on the approximate simulation of de~Sitter spacetimes.

\end{itemize}

In all of these models the general expectations of the relativity
community have been borne out --- theory definitely predicts particle
production, and the very interesting question is the extent to which the
formal predictions are going to be modified when working with real
systems experimentally~\cite{Barcelo:2003wu}. We expect that these
analogue models provide us with new insights as to how their inherent
modified dispersion relations affect cosmological process such as the
generation of a primordial spectrum of perturbations (see for example
\cite{Brandenberger:2000as, Brandenberger:2000zm,  
 Brandenberger:2002sr, Brandenberger:2003vk,  Brandenberger:2002ty, Brandenberger:2000wr,
Brandenberger:2002hs,   Casadio:2002dj,  
Easther:2001fi,  Hassan:2002qk, Kempf:2001fa, Lemoine:2001ar, Lemoine:2002rf,  Martin:2003kp, Martin:2000bv,  Martin:2000xs, Martin:2002kt, 
Niemeyer:2002ze, Niemeyer:2001qe, 
Shankaranarayanan:2002ax, Paddy4} where analogue-like ideas are applied to
cosmological inflation).


An interesting side-effect of the original investigation, is that
birefringence can now be used to model ``variable speed of light'' [VSL]
geometries~\cite{Bassett:2000wj, Ellis:2003pw}. Since analogue models
quite often lead to two or more ``excitation cones'', rather than one,
it is quite easy to obtain a bimetric or multi-metric model. If one of
these metric is interpreted as the ``gravitational'' metric and the
other as the ``photon'' metric, then VSL cosmologies can be given a
mathematically well-defined and precise
meaning~\cite{Bassett:2000wj,Ellis:2003pw}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{--------------------Old version------------------------}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Analogue model techniques have also been applied to cosmology. In a
%cosmological framework the key items of interest are the
%Friedmann--Robertson--Walker [FRW] geometries, more properly called the
%Friedmann--\Lemaitre--Robertson--Walker [FLRW] geometries. The
%simulation of such geometries has been considered in various works such
%as~\cite{Barcelo:2003et, Barcelo:2003wu, Fedichev:2003dj,
%Fedichev:2003bv, Fedichev:2003id, Lidsey:2003ze, Calzetta:2002jz,
%Calzetta:2002zz, Weinfurtner:2004mu} with a specific view to enhancing
%our understanding of ``cosmological particle production'' driven by the
%expansion of the universe. Depending on whether we use physical time or
%conformal time the FLRW geometries can be cast into the form
%
%\begin{equation} 
%\d s^2 = - \d t^2 + a^2(t) \left\{ {\d r^2\over 1-kr^2}
%+ r^2[\d\theta^2+\sin^2\theta\;\d\phi^2] \right\} 
%\end{equation} 
% 
%or the equivalent
%
%\begin{equation} 
%\d s^2 = a(\eta)^2 \left\{ - \d \eta^2 + {\d r^2\over
%1-kr^2} + r^2[\d\theta^2+\sin^2\theta\;\d\phi^2] \right\} 
%\end{equation}
% 
%where the time parameterizations are related by
%
%\begin{equation} 
%\eta = \int {\d t\over a(t)}\; \qquad t =  \int a(\eta)
%\; \d\eta \end{equation} 
%
%It is a textbook result (see, for example,~\cite{Birrell}\,) that the
%expansion of the universe, that is changes in the scale factor $a$, will
%generically (for non-conformally coupled fields) result in the
%excitation and production of particles.
%
%Several ways of implementing \emph{approximations} to FRW universes in a
%analogue model context have been explored. Models considered to date
%focus on variants of the BEC-inspired analogues, subject either to
%physical explosion (\eg, a Bose-nova), or rapid variations in the
%``effective speed of light''.
%
%\begin{itemize}
%
%\item Calzetta and Hu~\cite{Calzetta:2002jz,Calzetta:2002zz} have
%specifically investigated the ``Bose--nova'' explosion scenario where
%the scattering length changes sign and destabilizes the condensate.
%(Equivalently the quartic self-interaction is forced to change sign by
%tuning the system through a Feschbach resonance).
%
%\item Lidsey~\cite{Lidsey:2003ze}, and Fedichev and
%Fischer~\cite{Fedichev:2003dj} have focussed on the behaviour of
%cigar-like condensates in grossly asymmetric traps.
%
%\item Barcelo \emph{et al}~\cite{Barcelo:2003et,Barcelo:2003wu} have
%focussed on the central region of the BEC and thereby tried (at least
%locally) to mimic FLRW behaviour as closely as possible.
%
%\item Fedichev and Fischer~\cite{Fedichev:2003bv,Fedichev:2003id} have
%also investigated $(1+1)$ dimensional cosmologies and WKB estimates of
%the particle production rate.
%
%\item Weinfurtner~\cite{Weinfurtner:2004in,Weinfurtner:2004mu} has
%concentrated on the approximate simulation of de~Sitter spacetimes.
%
%\end{itemize}
%
%In all of these models the general expectations of the relativity
%community have been borne out --- theory definitely predicts particle
%production, and the really interesting question it the extent to which
%we will ultimately be able to test these ideas
%experimentally~\cite{Barcelo:2003wu}.
%
%
%
%An interesting side-effect of the original investigation, is that
%birefringence can now be used to model ``variable speed of light'' [VSL]
%geometries~\cite{Bassett:2000wj,Ellis:2003pw}. Since analogue models
%quite often lead to two or more ``excitation cones'', rather than one,
%it is quite easy to obtain a bimetric or multi-metric model. If one of
%these metric is interpreted as the ``gravitational'' metric and the
%other as the ``photon'' metric, then VSL cosmologies can be given a
%mathematically well-defined and precise
%meaning~\cite{Bassett:2000wj,Ellis:2003pw}.

%---------------------------------------------- 
\subsection{Bose nov\ae: an example of the reverse flow of information?}
%----------------------------------------------

As we have seen in the previous sections, analogue models have in the
past been very useful in providing new, condensed matter physics
inspired, ideas about how to solve longstanding problems of
semiclassical gravity.  In closing this section it is interesting to
briefly discuss what perhaps represents, so far, the only tentative
attempt to use analogue models in the reverse direction; that is to
import well known concepts of semiclassical gravity into condensed
matter frameworks.

The phenomenon we are referring to is the so called ``Bose
nova''~\cite{Donley}. This is an experiment dealing with a gas of a few
million ${}^{85}$Rb atoms at a temperature of about 3 nK. The condensate
is rendered unstable by exploiting the possibility of tuning the
interaction (more precisely the scattering length) between the atoms via
a magnetic field. Reversing the sign of the interaction, making it
attractive, destabilizes the condensate. After a brief waiting time
(generally called $t_\mathrm{collapse}$), the condensate implodes and
loses a sizeable fraction of its atoms in the form of a ``nova burst''.
If left to evolve undisturbed, the number of atoms in the burst
stabilizes and a remnant condensate is left. However if the condensate
interaction is again made repulsive after some time $t_\mathrm{evolve}$,
before the condensate has sufficient time to stabilize, then the
formation of ``jets'' of atoms is observed, these jets being 
characterized by lower kinetic energy and a distinct shape with respect
to the burst emission.

Interestingly, an elegant explanation of such a phenomenology was
proposed in~\cite{Calzetta:2002jz, Calzetta:2002zz}, based on the well
known semiclassical gravity analysis of particle creation in an
expanding universe. In fact the dynamics of quantum excitations over the
collapsing BEC was shown to closely mimic that for quantum excitations
in a time-reversed (collapsing instead of expanding) scenario for
cosmological particle creation. This is not so surprising as the quantum
excitations above the BEC ground state feel a time-varying background
during the collapse, and as a consequence one then expects squeezing of
the vacuum state and mode mixing which are characteristic of quantum
field theory in variable external fields.

However the analogy is even deeper than this. In fact
in~\cite{Calzetta:2002jz, Calzetta:2002zz} a key role in explaining the
observed burst and jets is played by the cosmology borrowed concepts of
``frozen'' versus ``oscillating'' modes (although with a reverse
dynamics with respect to the standard [expanding] cosmological case). In
the case of Bose nov\ae{} the modes which are amplified are those for
which the physical frequency is smaller than the collapse rate, while
modes with higher frequencies remain basically unaffected and their
amplitudes obey a harmonic oscillator equation.  As the collapse rate
decreases, more and more modes stop growing and start oscillating, which
is equivalent to a creation of particles from the quantum vacuum. In the
case of a sudden stop of the collapse by a new reversal of the sign of
the interaction all of the previously growing modes are suddenly
converted into particles, explaining in this way the generation of jets
and their lower energy (they correspond to modes with lower frequencies
with respect to those generating the bursts).

Although this simple model cannot explain all the details of the Bose
nov\ae{} phenomenology, we think it is remarkable how far it can go in
explaining several observed features by exploiting the language and
techniques so familiar to quantum cosmology. In this sense the analysis
presented in~\cite{Calzetta:2002jz, Calzetta:2002zz} primarily shows a
possible new application of analogue models where they could be used to
lend ideas and techniques developed in the context of gravitational
physics to the explanation of condensed matter phenomena.




%---------------------------------------------- 
\subsection{Going further} 
%----------------------------------------------

For more details on the trans-Planckian problem some of the key papers
are the relatively early papers of Unruh~\cite{Unruh:1994zw,
Unruh:1995je}, and Jacobson~\cite{Jacobson:1991gr, Jacobson:1993hn}. 
For super-radiance and cosmological issues there seems to be
considerable ongoing interest, and one should carefully check {\sf
Spires} for the most recent articles.




%---------------------------------------------- 
% EOF
%----------------------------------------------

references.tex
%----------------------------------------------
% references.tex
%----------------------------------------------
% BOF
%----------------------------------------------




%----------------------------------------------
\section*{Acknowledgements}
%----------------------------------------------

The work of MV was supported by the Marsden fund administered by the
Royal Society of New Zealand. CB has been supported by a Marie Curie
European Reintegration Grant, and by the Education and Science Council of
Junta de Andaluc\'{\i}a (Spain). We also wish to thank Enrique Arilla
for figures 1 and 2 of the art work.

%----------------------------------------------
\section*{Comment on citations}
%----------------------------------------------

\bigskip

Note that this review article uses a slightly non-standard  bibliography
style file which includes back-references from the bibliography to the
section and subsection, or footnote, where the article was cited.


%----------------------------------------------
%----------------------------------------------
%----------------------------------------------


%----------------------------------------------
\nocite{*}
%----------------------------------------------



\bibliography{background,modern,recent,historic,books,unpublished,stefano-bib,carlos-bib,add-ons}

% background for general background articles
% modern for Unruh-inspired articles articles in period 1981-2002
% recent for  Unruh-inspired articles articles in 2003-2004
% historic for older articles, pre-Unruh phase
% books for ALL books of whatever vintage
% unpublished for older unpublished articles

% stefano-bib = stefano's bibliography
% carlos-bib  = carlos' bibliography


%----------------------------------------------
% Information for the Slac-Spires crew:
%----------------------------------------------
% These commented lines have been harvested from the bibtex files.
% All these articles have been cired in the text of the review.
% They were cited in alphabetic order based on first author.
% and no attempt has been made to order the list below...
%----------------------------------------------

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%-----------------------
% EOF
%-----------------------
simple-fluid.tex
 %%%%%%%%%%%%%%%%%%%%
 %%% simple-fluid.tex  
 %%%%%%%%%%%%%%%%%%%%
 %%% BOF
 %%%%%%%%%%%%%%%%%%%%
 
 
 
%--------------------------------------------------------------

\def\d{{\mathrm d}}
\def\R{{\rm I\!R}}
\def\v{{\mathbf v}}
\def\x{{\mathbf x}}
\def\n{{\mathbf n}}
\def\f{{\mathbf f}}
\def\F{{\mathbf F}}
\def\a{{\mathbf a}}
\def\A{{\mathbf A}}
\def\K{{\mathbf K}}
\def\V{{\mathbf V}}
\def\im{{\rm i}}
\def\det{{\mathrm{det}}}
\def\tr{{\mathrm{tr}}}
\def\bnabla{\mbox{\boldmath$\nabla$}}

%--------------------------------------------------

\def\ie{{\em i.e.\/}}%
\def\eg{{\em e.g.\/}}%
\def\etc{{\em etc.\/}}%
\def\etal{{\em et al.\/}}%

%--------------------------------------------------------------
 
 Acoustics in a moving fluid is the simplest and cleanest example of an
 analogue model~\cite{Unruh:1981cg, Visser:1993ub, Visser:1998qn,
 Visser:1997ux}. The basic physics is simple, the conceptual framework
 is simple, and specific computations are often simple (whenever, that
 is, they are not impossibly hard).\footnote{The need for a  certain
 degree of caution regarding the allegedly straightforward physics of
 simple fluids might be inferred from the fact that the Clay Mathematics
 Institute is currently offering a US\$1,000,000 Millennium Prize for
 significant progress on the question of existence and uniqueness of
 solutions to the  Navier--Stokes equation. See {\sf
 http://www.claymath.org/millennium/} for details.}
 
%--------------------------------------------
\subsection{Background}
%--------------------------------------------
 
 The basic physics is this: A moving fluid will drag sound waves along
 with it, and if the speed of the fluid ever becomes supersonic, then in
 the supersonic sound waves will never be able to fight their way back
 upstream~\cite{Unruh:1981cg,Visser:1993ub,Visser:1998qn,Visser:1997ux}.
  This implies the existence of a ``dumb hole'', a region from which
 sound can not escape.\footnote{In correct English, the word ``dumb"
 means ``mute'', as in ``unable to speak''. The word ``dumb'' does
 \emph{not} mean ``stupid'', though even many native English speakers
 get this wrong.} Of course this sounds very similar, at the level of a
 non-mathematical \emph{verbal} analogy, to the notion of a ``black
 hole'' in general relativity. The real question is whether this verbal
 analogy can be turned into a precise \emph{mathematical} and
 \emph{physical} statement --- it is only after we have a precise
 mathematical and physical connection between (in this example) the
 physics of acoustics in a fluid flow and at least some significant
 features  of general relativity that we can claim to have an ``analogue
 model of (some aspects of) gravity". We (and the community at large)
 often abuse language by referring to such a model as ``analogue
 gravity'' for short.

%-----------------------------------------------------------------------
\begin{figure}[htb]
\vbox{
\vskip 20 pt

\centerline{{\includegraphics[width=3in]{F-matt-moving-fluid}}}

\caption[Moving fluid]{ {\sl A moving fluid will drag sound pulses along
with it.  \smallskip \label{F:moving0} }}

}
\end{figure}
%-----------------------------------------------------------------------



Now the features of general relativity that one typically captures in an
``analogue model'' are the \emph{kinematic} features that have to do
with how fields (classical or quantum) are defined on curved spacetime,
and the \emph{sine qua non} of any analogue model is the existence of
some ``effective metric'' that captures the notion of the curved
spacetimes that arise in general relativity. (At the very least, one
might wish to capture the notion of the Minkowski geometry of special
relativity.) Indeed, the verbal description above (and its
generalizations in other physical frameworks) \emph{can} be converted
into a precise mathematical and physical statement, which ultimately is
the  reason that analogue models are of physical interest. The analogy
works at two levels; %

\begin{itemize}
\item Geometrical acoustics.
\item Physical acoustics.
\end{itemize}
%

The advantage of geometrical acoustics is that the derivation of the
precise mathematical form of the analogy is so simple as to be almost
trivial, and that the derivation is extremely general. The disadvantage
is that in the geometrical acoustics limit one can deduce only the
causal structure of the spacetime, and does not obtain a unique
effective metric . The advantage of physical acoustics is that while the
derivation of the analogy holds in a more restricted regime, the analogy
can do more for you in that it can now specify a specific effective
metric and  accommodate a wave equation for the sound waves.
 
 
 
 %-------------------------------------------------
 \subsection{Geometrical acoustics}
 %-------------------------------------------------
 
 At the level of geometrical acoustics we need only assume that:

%

\begin{itemize}%

\item The speed of sound $c$, relative to the fluid, is well
defined.%

\item The velocity of the fluid $\v$, relative to the
laboratory, is well defined.%

\end{itemize}%

%

Then, relative to the laboratory, the velocity of a sound ray
propagating, with respect to the fluid, along the direction
defined by the unit vector $\n$ is%
%
\begin{equation}%
\frac{\d\x}{\d t}=c\,\n+\v.%
\end{equation}%
%
This defines a sound cone in spacetime given by the condition
$\n^2=1$, i.e.,%
%
\begin{equation}%
- c^2 \; \d t^2 + \left(\d\x - \v \,\d t \right)^2 = 0\;.%
\end{equation}%
That is
\begin{equation}
- [c^2-v^2] \;\d t^2 - 2 \v \cdot \d\x \;\d t 
+\d\x\cdot\d\x=0
\end{equation}

%-----------------------------------------------------------------------
\begin{figure}[htb]
\vbox{
\vskip 20 pt

\centerline{{\includegraphics[width=5in]{F-soundcones2}}}

\caption[Sound cones]{ {\sl A moving fluid will tip the ``sound cones''
as it moves. Supersonic flow will tip the sound cones past the vertical.
 \smallskip\label{F:cones}}}

}
\end{figure}
%-----------------------------------------------------------------------






Solving this quadratic equation for $\d \x$ as a function of $\d t$
provides a double cone associate with each point in space and time. This
is associated with a conformal class of Lorentzian
metrics~\cite{Unruh:1981cg,Visser:1993ub,Visser:1998qn,Visser:1997ux,
Novello:2002qg}%

%
\begin{equation}%
{\mbox{\sl g}}  = \Omega^2\left[ \begin{array}{c|c}
-(c^2-v^2) & -{{\v}\,}^T\\
\hline -{\v} & {\bf I}
\end{array}\right]\;,%
\label{g-matrix}%
\end{equation}%
%
where $\Omega$ is an unspecified but non-vanishing function.

The virtues of the geometric approach are its extreme simplicity and the
fact that the basic structure is dimension-independent. Moreover this
logic rapidly (and relatively easily)  generalizes to more complicated
physical situations.\footnote{For instance, whenever one has a system of PDEs that can be written in first-order quasi-linear symmetric hyperbolic form, then it is an exact non-perturbative result that the matrix of coefficients for the first-derivative terms can be used to construct a conformal class of metrics that encodes the causal structure of the system of PDEs. For barotropic hydrodynamics this is briefly discussed in~\cite{Piotr}. This analysis is related to the behaviour of characteristics of the PDEs, and ultimately can be linked back to the Fresnel equation that appears in the eikonal limit.}

%-------------------------------------------------
\subsection{Physical acoustics}
%-------------------------------------------------

It is well known that for a static homogeneous inviscid fluid
the propagation of sound waves is governed by the simple wave
equation~\cite{Lamb,Landau-Lifshitz,Milne-Thomson,Skudrzyk}
%
\begin{equation}
\partial_t^2 \phi = c^2 \; \nabla^2 \phi.
\end{equation}
%
Generalizing this result to a fluid that is non-homogeneous, or to a
fluid that is in motion, possibly even in non-steady motion, is more
subtle than it at first would appear. To derive a wave equation in this
more general situation we shall start by adopting a few simplifying
assumptions to allow us to derive the following theorem.

%----------------------------
\paragraph{Theorem:}%\emph{
%----------------------------
If a fluid is barotropic and inviscid, and the flow is irrotational
(though possibly time dependent) then the equation of motion for the
velocity potential describing an acoustic disturbance is identical to
the d'Alembertian equation of motion for a minimally coupled massless
scalar field propagating in a $(3+1)$--dimensional Lorentzian
geometry
\begin{equation}
\Delta \phi \equiv 
{1\over\sqrt{-g}} 
\partial_\mu 
\left( \sqrt{-g} \; g^{\mu\nu} \; \partial_\nu \phi \right) = 0.
\end{equation}
%
Under these conditions, the propagation of sound is governed by an
{\em acoustic metric} --- $g_{\mu\nu}(t,\x)$. This acoustic
metric describes a $(3+1)$--dimensional Lorentzian (pseudo--Riemannian)
geometry. The metric depends algebraically on the density, velocity
of flow, and local speed of sound in the fluid. Specifically
%
\begin{equation}
g_{\mu\nu}(t,\x) 
\equiv {\rho\over c} 
\left[ \matrix{-(c^2-v^2)&\vdots&-{\v}^T\cr
               \cdots\cdots\cdots\cdots&\cdot&\cdots\cdots\cr
	       -{\v}&\vdots& \mathbf{I}\cr } \right].
\end{equation}
%
(Here $\mathbf{I}$ is the $3\times3$ identity matrix.) In general, when
the fluid is non-homogeneous and flowing, the {\em acoustic Riemann
tensor} associated with this Lorentzian metric will be nonzero.
\hfill{$\diamondsuit$}
% end theorem

%-----------------------
\paragraph{Comment:}
%-----------------------

It is quite remarkable that even though the underlying fluid dynamics is
Newtonian, nonrelativistic, and takes place in flat space plus time, the
fluctuations (sound waves) are governed by a curved $(3+1)$--dimensional
Lorentzian (pseudo-Riemannian) spacetime geometry. For practitioners of
general relativity this observation describes a very simple and concrete
physical model for certain classes of Lorentzian spacetimes, including
(as we shall later see) black holes. On the other hand, this discussion
is also potentially of interest to practitioners of continuum mechanics
and fluid dynamics in that it provides a simple concrete introduction to
Lorentzian differential geometric techniques.

%-----------------------
\paragraph{Proof:}
%-----------------------

The fundamental equations of fluid
dynamics~\cite{Lamb,Landau-Lifshitz,Milne-Thomson,Skudrzyk} are the
equation of continuity 
%
\begin{equation}
\partial_t \rho + \bnabla\cdot(\rho \; \v) = 0,
\label{E-continuity}
\end{equation}
%
and Euler's equation (equivalent to $\F= m\;\a$ 
applied to small lumps of fluid)
%
\begin{equation}
\rho {d \v\over dt} \equiv 
\rho \left[ \partial_t \v + (\v \cdot \bnabla) \v \right] =
\f.
\label{E-euler}
\end{equation}
%
Start the analysis by assuming the fluid to be inviscid (zero
viscosity), with the only forces present being those due to
pressure.\footnote{It is straightforward to add external forces, at least
conservative body forces such as Newtonian gravity.} Then for the force
density we have
%
\begin{equation}
\f = - \bnabla p.
\label{E-force}
\end{equation}
%
Via standard manipulations the Euler equation can be rewritten as
%
\begin{equation}
\partial_t {\v} = 
\v \times ( \bnabla \times \v)  - {1\over\rho} \;\bnabla p
- \bnabla\left( {{1\over2}} \; v^2\right).
\label{E-euler-2}
\end{equation}
%
Now take the flow to be {\em vorticity free}, that is, {\em locally
irrotational}. Introduce the
velocity potential $\phi$ such that $\v = -\bnabla \phi$, at
least locally. If one further takes
the fluid to be {\em barotropic} (this
means that $\rho$ is a function of $p$ only), it becomes possible
to define
%
\begin{equation}
h(p) = \int_0^p {dp'\over\rho(p')}; 
\qquad \hbox{so that} \qquad
\bnabla h = {1\over\rho} \; \bnabla p.
\end{equation}
%
Thus the specific enthalpy, $h(p)$, is a function of $p$ only.
Euler's equation now reduces to
%
\begin{equation}
-\partial_t \phi + h + {1\over2} (\bnabla\phi)^2  = 0.
\label{E-bernoulli}
\end{equation}
This is a version of Bernoulli's equation.

Now linearize these equations of motion around some assumed background
$(\rho_0,p_0,\phi_0)$.  Set 
\begin{eqnarray}
\rho &=& \rho_0 + \epsilon \rho_1 +O(\epsilon^2),
\\
p &=& p_0 + \epsilon p_1 +O(\epsilon^2),
\\
\phi &=&
\phi_0 + \epsilon \phi_1 + O(\epsilon^2).   
\end{eqnarray}
Sound is {\em defined} to
be these linearized fluctuations in the dynamical quantities.  Note that
this is the {\em standard definition} of [linear] sound and more
generally of acoustical disturbances. In principle, of course, a fluid
mechanic might really be interested in solving the complete equations of
motion for the fluid variables $(\rho, p, \phi)$. In practice, it is
both traditional and extremely useful to separate the exact motion,
described by the exact variables, $(\rho, p, \phi)$, into some average
bulk motion, $(\rho_0,p_0,\phi_0)$, plus low amplitude acoustic
disturbances, $(\epsilon \rho_1,\epsilon p_1,\epsilon \phi_1)$. See, for
example~\cite{Lamb,Landau-Lifshitz,Milne-Thomson,Skudrzyk}. 

Since this
is a subtle issue that we have seen cause considerable confusion in the
past, let us be even more explicit by asking the rhetorical question:
{\em ``How can we tell the difference between a wind gust and a sound
wave?''} The answer is that the difference is to some extent a matter of
convention --- sufficiently low-frequency long-wavelength disturbances
(wind gusts) are conventionally lumped in with the average bulk motion.
Higher-frequency, shorter-wavelength disturbances are conventionally
described as acoustic disturbances. If you wish to be hyper-technical,
we can introduce a high-pass filter function to define the bulk motion
by suitably averaging the exact fluid motion. There are no deep physical
principles at stake here --- merely an issue of convention. The place
where we are making a specific physical assumption that restricts the
validity of our analysis is in the requirement that the amplitude of the
high-frequency short-wavelength disturbances be small. This is the
assumption underlying the linearization programme, and this is why
sufficiently high-amplitude sound waves must be treated by direct
solution of the full equations of fluid dynamics.

Linearizing the continuity equation results in the pair of equations
%
\begin{eqnarray}
&&
\partial_t \rho_0 + 
\bnabla\cdot(\rho_0 \; \v_0) = 0,
\\
&&
\partial_t \rho_1 + 
\bnabla\cdot(\rho_1 \; \v_0 + \rho_0 \; \v_1) = 0.
\end{eqnarray}
%
Now, the barotropic condition implies
\begin{equation}
h(p) = 
h(p_0 + \epsilon p_1 +  O(\epsilon^2)) = 
h_0 + \epsilon \; {p_1\over\rho_0} + O(\epsilon^2).
\end{equation}
%
Use this result in linearizing the Euler equation. We obtain the pair
%
\begin{eqnarray}
&&-\partial_t \phi_0 + h_0 + {{1\over2}} (\bnabla\phi_0)^2  = 0.
\\
&&-\partial_t \phi_1 + {p_1\over\rho_0} - \v_0 \cdot \bnabla\phi_1 = 0.
\end{eqnarray}
%
This last equation may be rearranged to yield
%
\begin{equation}
p_1 =  \rho_0\; \left( \partial_t \phi_1 + \v_0 \cdot \bnabla\phi_1\right).
\label{E-linear-euler}
\end{equation}
%
Use the barotropic assumption to relate
%
\begin{equation}
\rho_1 = 
{\partial \rho\over\partial p} \; p_1 = 
{\partial \rho\over\partial p} \; \rho_0  \;
( \partial_t \phi_1 + \v_0 \cdot \bnabla\phi_1 ). 
\label{E-linear-barotropic}
\end{equation}
%
Now substitute this consequence of the linearized Euler equation into
the linearized equation of continuity. We finally obtain, up to an
overall sign, the wave equation:
\begin{eqnarray}
&-& \partial_t  
     \left( {\partial\rho\over\partial p} \; \rho_0 \; 
            ( \partial_t \phi_1 + \v_0 \cdot \bnabla\phi_1 ) 
     \right)
\nonumber\\
&+& \bnabla \cdot 
     \left( \rho_0 \; \bnabla\phi_1 
            - {\partial\rho\over\partial p} \; \rho_0 \; \v_0 \;
	      ( \partial_t \phi_1 + \v_0 \cdot \bnabla\phi_1 ) 	    
     \right)
=0.
\label{E-wave-physical}
\end{eqnarray}
%
This wave equation describes the propagation of the linearized
scalar potential $\phi_1$. Once $\phi_1$ is determined, equation
(\ref{E-linear-euler}) determines $p_1$, and equation
(\ref{E-linear-barotropic}) then determines $\rho_1$.  Thus this
wave equation  completely determines the propagation of acoustic
disturbances.  The background fields $p_0$, $\rho_0$ and $\v_0
= - \bnabla \phi_0$, which appear as time-dependent and
position-dependent coefficients in this wave equation, are
constrained to solve the equations of fluid motion for a
barotropic, inviscid, and irrotational flow.
Apart from these constraints, they are otherwise permitted to have
{\em arbitrary} temporal and spatial dependencies.

Now, written in this form, the physical import of this wave equation
is somewhat less than pellucid. To simplify things algebraically,
observe that the local speed of sound is defined by
%
\begin{equation}
c^{-2} \equiv {\partial\rho\over\partial p}. 
\end{equation}
%
Now construct the symmetric $4\times4$ matrix
%
\begin{equation}
f^{\mu\nu}(t,\x) \equiv 
{\rho_0\over c^2}
\left[ \matrix{-1&\vdots&-v_0^j\cr
               \cdots\cdots&\cdot&\cdots\cdots\cdots\cdots\cr
	       -v_0^i&\vdots&( c^2\;\delta^{ij} - v_0^i v_0^j )\cr } 
\right].
\label{E-explicit}	       
\end{equation}
%
(Greek indices run from $0$--$3$, while Roman indices run from
$1$--$3$.)  Then, introducing $(3+1)$--dimensional space-time
coordinates, which we write as $x^\mu \equiv (t; x^i)$ ,  the above wave equation
(\ref{E-wave-physical}) is easily rewritten as
%
\begin{equation}
\partial_\mu ( f^{\mu\nu} \; \partial_\nu \phi_1) = 0.
\end{equation}
%
This remarkably compact formulation is completely equivalent to
equation (\ref{E-wave-physical}) and is a much more promising
stepping-stone for further manipulations. The remaining steps are
a straightforward application of the techniques of curved space
$(3+1)$--dimensional Lorentzian geometry.

Now in any Lorentzian (that is, pseudo--Riemannian) manifold the curved
space scalar d'Alembertian is given in terms of the metric
$g_{\mu\nu}(t,\x)$ by (see, for
example,~\cite{Fock,Moller,Synge,MTW,Hawking-Ellis,Wald})
%
\begin{equation}
\Delta \phi \equiv 
{1\over\sqrt{-g}} 
\partial_\mu \left( \sqrt{-g} \; g^{\mu\nu} \; \partial_\nu \phi \right).
\end{equation}
%
The inverse metric, $g^{\mu\nu}(t,\x)$, is pointwise the matrix
inverse of $g_{\mu\nu}(t,\x)$, while $g \equiv \det(g_{\mu\nu})$.
Thus one can rewrite the physically derived wave equation
(\ref{E-wave-physical}) in terms of the d'Alembertian provided one
identifies
%
\begin{equation}
\sqrt{-g} \; g^{\mu\nu} = f^{\mu\nu}.
\end{equation}
%
This implies, on the one hand 
%
\begin{equation}
\det(f^{\mu\nu}) = (\sqrt{-g})^4 \; g^{-1} = g.
\end{equation} 
%
On the other hand, from the explicit expression (\ref{E-explicit}),
expanding the determinant in minors yields
%
\begin{equation}
\det(f^{\mu\nu}) 
= 
\left({\rho_0\over c^2}\right)^4 \cdot
\left[(-1) \cdot (c^2 - v_0^2) - (-v_0)^2\right] \cdot
\left[c^2\right] \cdot
 \left[c^2\right]
=
- {\rho_0^4\over c^2}.
\end{equation} 
%
Thus
\begin{equation}
g = - {\rho_0^4\over c^2}; \qquad \sqrt{-g} = {\rho_0^2\over c}.
\end{equation}
%
We can therefore pick off the coefficients of the \emph{inverse} acoustic
metric
%
\begin{equation}
g^{\mu\nu}(t,\x) \equiv 
{1\over \rho_0 c}
\left[ \matrix{-1&\vdots&-v_0^j\cr
               \cdots\cdots&\cdot&\cdots\cdots\cdots\cdots\cr
	       -v_0^i&\vdots&(c^2 \; \delta^{ij} - v_0^i \; v_0^j )\cr } 
\right].	       
\end{equation}
%
We could now determine the metric itself simply by inverting this
$4\times4$ matrix (and if the reader is not a general relativist, proceeding in this direct manner is definitely the preferred option). On the other hand, for general relativists it is even easier to recognize
that one has in front of one a specific example of the Arnowitt--Deser--Misner
split of a $(3+1)$--dimensional  Lorentzian spacetime metric into
space + time, more commonly used in discussing initial value data
in general relativity. (See, for
example, \cite{MTW} pp 505--508.) The acoustic metric is then read off by inspection
%
\begin{equation}
g_{\mu\nu} \equiv 
{\rho_0 \over  c}
\left[ \matrix{-(c^2-v_0^2)&\vdots&-v_0^j\cr
               \cdots\cdots\cdots\cdots&\cdot&\cdots\cdots\cr
	       -v_0^i&\vdots&\delta_{ij}\cr } 
\right].	       
\end{equation}
%
Equivalently, the acoustic interval can be expressed as
%
\begin{equation}
\d s^2 \equiv g_{\mu\nu} \; \d x^\mu \; \d x^\nu =
{\rho_0\over c} 
\left[
- c^2 \; \d t^2 + (\d x^i - v_0^i \; \d t) \; \delta_{ij} \; (\d x^j - v_0^j \; \d t )
\right].
\end{equation}
This completes the proof of the theorem. We have presented the theorem and proof, which closely follows the discussion in~\cite{Visser:1997ux}, in considerable detail because it is a standard template that can be readily generalised in many ways.  This discussion can then be used as a starting point to initiate the analysis of  numerous and diverse physical models.

%-------------------------------------------------
\subsection{General features of the acoustic metric}
%-------------------------------------------------
A few brief comments should be made before proceeding further:
\begin{itemize}
\item
Observe that the signature of this effective metric is indeed $(-,+,+,+)$, as
it should be to be regarded as Lorentzian.
\item
Observe that in physical acoustics it is the inverse metric density,
\begin{equation}
f^{\mu\nu} = \sqrt{-g} \; g^{\mu\nu}
\end{equation}
that is of more fundamental significance for deriving the wave equation
than is the metric $g_{\mu\nu}$ itself.  (This observation continues to hold in more general situations where it is often significantly easier to calculate the tensor density $f^{\mu\nu}$ than it is to calculate the effective metric $g_{\mu\nu}$.)
\item
It should be emphasized that there are two distinct metrics relevant
to the current discussion:
\begin{itemize}
\item
The {\em physical spacetime metric} is in this case just the usual flat metric
of Minkowski space:
\begin{equation}
\eta_{\mu\nu} \equiv 
({\rm diag}[-c_\mathrm{light}^2,1,1,1])_{\mu\nu}. 
\end{equation}
(Here $c_\mathrm{light} = \hbox{speed of light in vacuum}$.) The fluid particles couple
only to the physical metric $\eta_{\mu\nu}$. In fact the fluid
motion is completely non--relativistic, so that $||v_0||   \ll  c_\mathrm{light}$, and it quite sufficient to consider Galilean relativity for the underlying fluid mechanics.
\item
Sound waves on the other hand, do not ``see'' the physical metric
at all. Acoustic perturbations couple only to the effective {\em acoustic
metric} $g_{\mu\nu}$.
\end{itemize}

\item The geometry determined by the acoustic metric does however
inherit some key properties from the existence of the underlying flat
physical metric. For instance, the \emph{topology} of the manifold does
not depend on the particular metric considered.  The acoustic geometry
inherits the underlying topology of the physical metric --- ordinary
$\Re^4$ --- with possibly a few regions excised (due to whatever
hard-wall boundary conditions one might wish to impose on the fluid). In
systems constrained to have effectively less than 3 spacelike dimensions
one can reproduce more complicated topologies (consider for example an
effectively one-dimensional flow in a tubular ring). 

\item
Furthermore, the acoustic geometry  automatically inherits from the underlying Newtonian time parameter, the important 
property of ``stable causality''~\cite{Hawking-Ellis,Wald}. Note that
%
\begin{equation}
g^{\mu\nu} \, (\nabla_\mu t) \, (\nabla_\nu t) = -{1\over\rho_0 \; c} < 0.
\end{equation}
%
This precludes some of the more entertaining causality-related
pathologies that sometimes arise in general relativity. 
(For a general
discussion of causal pathologies in general relativity, see for example~\cite{Hawking-Ellis,Hawking:1991nk,Hawking:1991pk,Cassidy:1997kv,Hawking:2002yr,Visser:2002ua}).

\item
Other concepts that translate immediately are those of ``ergo-region'',
``trapped surface'', ``apparent horizon'', and ``event horizon''.
These notions will be developed more fully in the following subsection.
\item
The properly normalized four-velocity of the fluid is
%
\begin{equation}
V^\mu = {(1; \; v^i_0)\over\sqrt{\rho_0 \; c}},
\end{equation}
so that
\begin{equation}
g_{\mu\nu}\, V^\mu\, V^\nu = g(V,V) = -1.
\end{equation}
%
This four-velocity is related to the gradient of the natural time parameter by
%
\begin{equation}
\nabla_\mu t = (1,0,0,0); \qquad \qquad 
\nabla^\mu t =  -{(1; \; v^i_0)\over\rho_0 \; c} 
             = -{V^\mu\over\sqrt{\rho_0 \; c}}.
\end{equation}
%
Thus the integral curves of the fluid velocity field are orthogonal
(in the Lorentzian metric) to the constant time surfaces. The
acoustic proper time along the fluid flow lines (streamlines) is
%
\begin{equation}
\tau = \int \sqrt{\rho_0 \; c} \; dt,
\end{equation}
%
and the integral curves are geodesics of the acoustic metric if
and only if $\rho_0 \; c$ is position independent.
\item
Observe that in a completely general $(3+1)$--dimensional Lorentzian
geometry the metric has 6 degrees of freedom per point in spacetime.
($4\times4$ symmetric matrix $\Rightarrow$ $10$ independent
components; then subtract $4$ coordinate conditions). 

 In contrast,
the acoustic metric is more constrained. Being specified completely by
the three scalars $\phi_0(t, \x)$, $\rho_0(t, \x)$, and $c(t,
\x)$, the acoustic metric has at most $3$ degrees of freedom per
point in spacetime. The equation of continuity actually reduces this to
$2$ degrees of freedom, which can be taken to be $\phi_0(t, \x)$
and $c(t, \x)$. 

\emph{Thus the simple acoustic metric of this section can at best reproduce some subset of the generic metrics of interest in general relativity.}
\item
A point of notation: Where the general relativist uses the word
``stationary'' the fluid dynamicist uses the phrase ``steady flow''.
The general-relativistic word ``static'' translates to a rather messy
constraint on the fluid flow (to be discussed more fully below).
\item
Finally, we should emphasise that in Einstein gravity the spacetime metric is
related to the distribution of matter by the non-linear
Einstein--Hilbert differential equations. In contrast, in the present
context, the acoustic metric is related to the distribution of matter
in a simple algebraic fashion.
\end{itemize}


%-------------------------------------------------
\subsection{Dumb holes --- ergoregions, horizons, and surface gravity}
%-------------------------------------------------


Let us start with the notion of an ergo-region: Consider integral
curves of the vector 
%
\begin{equation}
K^\mu \equiv (\partial/\partial t)^\mu = (1,0,0,0)^\mu. 
\end{equation}
%
If the flow is steady then this is the time
translation Killing vector.  Even if the flow is not steady the
background Minkowski metric provides us with a natural definition of
``at rest''.  Then\footnote{Henceforth, in the interests of
notational simplicity, we shall drop the explicit subscript $0$ on
background field quantities unless there is specific risk of confusion.}
%
\begin{equation}
g_{\mu\nu} \; (\partial/\partial t)^\mu \;(\partial/\partial t)^\nu = 
g_{tt} = 
-[c^2 - v^2]. 
\end{equation}
%
This quantity changes sign when $ ||\v|| > c $.  Thus any region of
supersonic flow is an ergo-region.  (And the boundary of the
ergo-region may be deemed to be the ergo-surface.) The analogue of this
behaviour in general relativity is the ergosphere surrounding any
spinning black hole --- it is a region where space ``moves'' with
superluminal velocity relative to the fixed
stars~\cite{MTW,Hawking-Ellis,Wald}.





A trapped surface in acoustics is defined as follows: take any closed
two-surface. If the fluid velocity is everywhere inward-pointing and the
normal component of the fluid velocity is everywhere greater than the
local speed of sound, then no matter what direction a sound wave
propagates, it will be swept inward by the fluid flow and be trapped
inside the surface. The surface is then said to be outer-trapped. (For
comparison with the usual situation in general relativity see
\cite[pages 319-323]{Hawking-Ellis} or \cite[pages 310--311]{Wald}.)
Inner-trapped surfaces (anti-trapped surfaces) can be defined by
demanding that the fluid flow is everywhere outward-pointing with
supersonic normal component. It is only because of the fact that the
background Minkowski metric provides a natural definition of ``at rest''
that we can adopt such a simple and straightforward definition. In
ordinary general relativity we need to develop considerable additional
technical machinery, such as the notion of the ``expansion'' of bundles
of ingoing and outgoing null geodesics, before defining trapped
surfaces. That the above definition for acoustic geometries is a
specialization of the usual one can be seen from the discussion on pages
262--263 of Hawking and Ellis \cite{Hawking-Ellis}.  The acoustic
trapped region is now defined as the region containing outer trapped
surfaces, and the acoustic (future) apparent horizon as the boundary of
the trapped region. (We can also define anti-trapped regions and past
apparent horizons but these notions are of limited utility in general
relativity.)\footnote{This discussion naturally leads us to what is
perhaps the central question of analogue models --- just how much of the
standard ``laws of black hole mechanics''~\cite{Bardeen:1973gs,
Wald:1999vt} carry over into these analogue models? Quite a lot but not
everything --- that's our main topic for the rest of the review.}

%-----------------------------------------------------------------------
\begin{figure}[htb]
\vbox{
\vskip 20 pt

\centerline{{\includegraphics[width=3in]{F-trappedsurf2}}}

\caption[Trapped surfaces]{ {\sl A moving fluid can form ``trapped
surfaces'' when supersonic  flow tips the sound cones past the vertical.
\smallskip\label{F:trapped}}}

}
\end{figure}
%-----------------------------------------------------------------------




The event horizon (absolute horizon) is defined, as in general
relativity, by demanding that it be the boundary of the region from
which null geodesics (phonons) cannot escape.  This is actually
the future event horizon. A past event horizon can be defined in
terms of the boundary of the region that cannot be reached by
incoming phonons --- strictly speaking this requires us to define
notions of past and future null infinities, but we will simply take
all relevant incantations as understood.  In particular the event
horizon is a null surface, the generators of which are null geodesics.

In all stationary geometries the apparent and event horizons coincide,
and the distinction is immaterial. In time-dependent geometries the
distinction is often important.  When computing the surface gravity we
shall restrict attention to stationary geometries (steady flow). In
fluid flows of high symmetry, (spherical symmetry, plane symmetry) the
ergosphere may coincide with the acoustic apparent horizon, or even the
acoustic event horizon. This is the analogue of the result in general
relativity that for static (as opposed to stationary) black holes the
ergosphere and event horizon coincide. For many more details, including
appropriate null coordinates and Carter--Penrose diagrams, both in
stationary and time-dependent situations, see~\cite{Barcelo:2004wz}.


Because of the definition of event horizon in terms of phonons (null
geodesics) that cannot escape the acoustic black hole, the event horizon
is automatically a null surface, and the generators of the event horizon
are automatically null geodesics. In the case of acoustics there is one
particular parameterization of these null geodesics that  is ``most
natural'', which is the parameterization in terms of the Newtonian time
coordinate of the underlying physical metric.  This allows us to
unambiguously define a ``surface gravity'' even for non-stationary
(time-dependent) acoustic event horizons, by calculating the extent to
which this natural time parameter fails to be an affine parameter for
the null generators of the horizon. (This part of the construction fails
in general relativity where there is no universal natural
time-coordinate unless there is a timelike Killing vector --- this is
why extending the notion of surface gravity to non-stationary geometries
in general relativity is so difficult.)

When it comes to explicitly calculating the surface gravity in terms of
suitable gradients of the fluid flow, it is nevertheless very useful to
limit attention to situations of steady flow (so that the acoustic
metric is stationary). This has the added bonus that for stationary
geometries the notion of ``acoustic surface gravity'' in acoustics is
unambiguously equivalent to the general relativity definition. It is
also useful to take cognizance of the fact that the situation simplifies
considerably for static (as opposed to merely stationary) acoustic
metrics.

To set up the appropriate framework, write the general stationary
acoustic metric in the form
%
\begin{equation}
\d s^2 = {\rho\over c} 
\left[ - c^2 \; \d t^2 + (\d\x - \v \; \d t)^2 \right].
\end{equation}
%
The time translation Killing vector is simply 
$ K^\mu = (1;\vec 0\,)$, with 
%
\begin{equation}
K^2 \equiv g_{\mu\nu} K^\mu K^\nu \equiv - ||\K||^2 = -
{\rho\over c}[ c^2 - v^2]. 
\end{equation}
%
The metric can also be written as
%
\begin{equation}
\d s^2 = {\rho\over c} 
\left[ 
-(c^2-v^2) \; \d t^2  - 2 \v\cdot \d\x \; \d t + (\d\x\,)^2 
\right].
\end{equation}
%
Now suppose that the vector $\v/(c^2 - v^2)$ is integrable,
then we can define a new time coordinate by
%
\begin{equation}
\d\tau = \d t + {\v \cdot \d\x \over c^2 - v^2 }.
\end{equation}
%
Substituting this back into the acoustic line element gives
%\footnote{The corresponding formula in~\cite{Unruh94} is missing
%a factor of $c$ and a bracket.}
%
\begin{equation}
\d s^2 = {\rho\over c} 
\left[ 
-(c^2-v^2) \; \d\tau^2  + 
\left\{ \delta_{ij} + {v^i v^j \over c^2 - v^2} \right \}
\d x^i \; \d x^j
\right].
\end{equation}
%
In this coordinate system the absence of the time-space cross-terms
makes manifest that the acoustic geometry is in fact static (the
Killing vector is hypersurface orthogonal). The condition that an
acoustic geometry be static, rather than merely stationary, is thus
seen to be
%
\begin{equation}
\bnabla \times \left\{ {\v\over (c^2 - v^2)} \right\} = 0,
\end{equation}
%
that is, (since in deriving the existence of the effective 
metric we have already assumed the fluid to be irrotational),
%
\begin{equation}
\v \times \bnabla (c^2 - v^2) = 0.
\end{equation}
%
This requires the fluid flow to be parallel to another vector that is
not quite the acceleration but is closely related to it. (Note that,
because of the vorticity free assumption, ${1\over2} \bnabla v^2$ is
just the three-acceleration of the fluid, it is the occurrence of a
possibly position dependent speed of sound that complicates the
above.)

Once we have a static geometry, we can of course directly apply
all of the standard tricks~\cite{Membrane} for calculating the
surface gravity developed in general relativity. We set up a system
of fiducial observers (FIDOS) by properly normalizing the
time-translation Killing vector
%
\begin{equation}
\V_\mathrm{FIDO} \equiv 
{\K\over ||\K|| } = 
{\K\over\sqrt{(\rho/c)\;[c^2-v^2]}}.
\end{equation}
%
The four-acceleration of the FIDOS is defined as 
%
\begin{equation}
\A_\mathrm{FIDO} \equiv
(\V_\mathrm{FIDO} \cdot \bnabla) \V_\mathrm{FIDO}, 
\end{equation}
%
and using the fact that $\K$ is
a Killing vector, it may be computed in the standard manner
%
\begin{equation}
\A_\mathrm{FIDO} = +{1\over2} {\bnabla ||\K||^2 \over ||\K||^2 }.
\end{equation}
%
That is 
%
\begin{equation}
\A_\mathrm{FIDO} = {1\over2} 
\left[ 
{\bnabla (c^2-v^2) \over (c^2-v^2)}
+ {\bnabla (\rho/c) \over (\rho/c)}
\right].
\end{equation}
%
The surface gravity is now defined by taking the norm $||\A_\mathrm{FIDO}||$,
multiplying by the lapse function, $ ||\K|| = \sqrt{(\rho/c)\;[c^2-v^2]}$,
and taking the limit as one approaches the horizon: $|v|\to c$ (remember 
that we are currently dealing with the static case). The net result is
%
\begin{equation}
||\A_\mathrm{FIDO}|| \; ||\K|| = 
{1\over2} \v \cdot \bnabla (c^2-v^2) + O(c^2-v^2),
\end{equation}
%
so that the surface gravity is given in terms of a normal derivative
by\footnote{Because of the background Minkowski metric there can be no
possible confusion as to the definition of this normal derivative.}
%
\begin{equation}
g_{H} = {1\over2} {\partial (c^2-v^2)\over\partial n} = 
 c \; {\partial (c-v) \over \partial n}.
\end{equation}
%
This is not quite Unruh's result~\cite{Unruh:1981cg,Unruh:1994zw,Unruh:1995je}
since he implicitly took the speed of sound to be a position-independent
constant. The fact that prefactor $\rho/c$ drops out of the final result
for the surface gravity can be justified by appeal to the known
conformal invariance of the surface gravity~\cite{Jacobson:1993pf}.
Though derived in a totally different manner, this result is also
compatible with the expression for ``surface-gravity'' obtained in the
solid-state black holes of Reznik~\cite{Reznik:1997ag}, wherein a
position dependent (and singular) refractive index plays a role
analogous to the acoustic metric. As a further consistency check, one
can go to the spherically symmetric case and check that this reproduces
the results for ``dirty black holes'' enunciated
in~\cite{Visser:1992qh}.

Since this is a static geometry, the relationship between the Hawking
temperature and surface gravity may be verified in the usual fast-track
manner --- using the Wick rotation trick to analytically continue to
Euclidean space~\cite{Gibbons:1976ue}. If you don't like Euclidean
signature techniques (which are in any case only applicable to
equilibrium situations) you should go back to the original Hawking
derivations~\cite{Hawking:1974rv, Hawking:1974sw}.\footnote{There are a
few potential subtleties in the derivation of the existence Hawking
radiation which we are for the time being glossing over, see Section
\ref{S-lessons} for details.}

One final comment to wrap up this section: the coordinate transform we
used to put the acoustic metric into the explicitly static form is
perfectly good mathematics, and from the general relativity point of
view is even a simplification. However, from the point of view of the
underlying Newtonian physics of the fluid, this is a rather bizarre
way of deliberately de-synchronizing your clocks to take a perfectly
reasonable region --- the boundary of the region of supersonic flow
--- and push it out to ``time'' plus infinity. From the fluid dynamics
point of view this coordinate transformation is correct but perverse,
and it is easier to keep a good grasp on the physics by staying with
the original Newtonian time coordinate.


If the fluid flow does not satisfy the integrability condition which
allows us to introduce an explicitly static coordinate system, then
defining the surface gravity is a little trickier. 

Recall that by construction the acoustic apparent horizon is in general
defined to be a two-surface for which the normal component of the fluid
velocity is everywhere equal to the local speed of sound, whereas the
acoustic event horizon (absolute horizon) is characterized by the
boundary of those null geodesics (phonons) that do  not escape to
infinity. In the stationary case these notions coincide, and it is still
true that the horizon is a null surface, and that the horizon can be
ruled by an appropriate set of null curves. Suppose we have somehow
isolated the location of the acoustic horizon, then in the vicinity of
the horizon we can split up the fluid flow into normal and tangential
components

%
\begin{equation}
\v = \v_\perp+ \v_\parallel; 
\qquad \hbox{where} \qquad
\v_\perp = v_\perp  \; \hat \n .
\end{equation}
%
Here (and for the rest of this particular section) it is essential
that we use the natural Newtonian time coordinate inherited from
the background Newtonian physics of the fluid. In addition $\hat
\n$ is a unit vector field that at the horizon is perpendicular to
it, and away from the horizon is some suitable smooth extension.
(For example, take the geodesic distance to the horizon and consider
its gradient.) We only need this decomposition to hold in some open
set encompassing the horizon and do not need to have a global
decomposition of this type available. Furthermore, by definition
we know that $v_\perp = c$ at the horizon. Now consider the vector
field
%
\begin{equation}
L^\mu = (\,1; \; v_\parallel^i \,).
\end{equation}
%
Since the spatial components of this vector field are by definition
tangent to the horizon, the
integral curves of this vector field will be generators for the
horizon. Furthermore the norm of this vector (in the acoustic
metric) is
%
\begin{equation}
|| {\mathbf L} ||^2 = 
-{\rho\over c} 
\left[ 
\vphantom{\Big|}
-(c^2 - v^2)  
- 2\v_\parallel \cdot \v 
+ \v_\parallel \cdot \v_\parallel 
\right]
=
{\rho\over c} \;  (c^2 - v_\perp^2).
\end{equation}
%
In particular, on the acoustic horizon $L^\mu $ defines a null
vector field, the integral curves of which are generators for the
acoustic horizon. We shall now verify that these generators are
geodesics, though the vector field ${\mathbf L}$ is not normalized with an
affine parameter, and in this way shall calculate the surface
gravity. 


Consider the quantity $({\mathbf L}\cdot\bnabla){\mathbf L}$ and calculate
%
\begin{equation}
L^\alpha \nabla_\alpha L^\mu 
= 
L^\alpha (\nabla_\alpha L_\beta - \nabla_\beta L_\alpha) g^{\beta \mu}
+ {1\over2} \nabla_\beta (L^2) g^{\beta \mu}.
\end{equation}
%
To calculate the first term note that
%
\begin{equation}
L_\mu = {\rho\over c} \; (-[c^2-v_\perp^2]; \v_\perp).
\end{equation}
%
Thus 
%
\begin{equation}
L_{[\alpha,\beta]} =
-\left[ 
\matrix{
0&\vdots&-\nabla_i\left[{\rho\over c}(c^2-v_\perp^2)\right]\cr
\cdots\cdots\cdots\cdots&\cdot&\cdots\cdots\cr
+\nabla_j\left[{\rho\over c}(c^2-v_\perp^2)\right]&\vdots&
\left({\rho\over c} \; v^\perp\right){}_{[i,j]}\cr 
} 
\right].	       
\end{equation}
%
And so:
%
\begin{equation}
L^\alpha L_{[\beta,\alpha]} = 
\left(
\v_\parallel \cdot \bnabla\left[{\rho\over c}(c^2-v_\perp^2)\right]; 
\nabla_j\left[{\rho\over c}(c^2-v_\perp^2)\right] 
+ v_\parallel^i \left({\rho\over c} \; v^\perp\right)_{[j,i]} 
\right).
\end{equation}
%
On the horizon, where $c=v_\perp$, this simplifies tremendously
%
\begin{equation}
(L^\alpha L_{[\beta,\alpha]})|_\mathrm{horizon} = 
-{\rho\over c} \; \left(0; \nabla_j(c^2-v_\perp^2)\right) = 
-{\rho\over c} \; {\partial(c^2-v_\perp^2)\over\partial n} \; \left(0;\hat n_j\right).
\end{equation}
%
Similarly, for the second term we have
%
\begin{equation}
\nabla_\beta (L^2) = 
\left(0; \nabla_j\left[{\rho\over c}(c^2-v_\perp^2)\right] \right)
\end{equation}
%
On the horizon this again simplifies
%
\begin{equation}
\nabla_\beta (L^2)|_\mathrm{horizon} = 
+{\rho\over c} \; \left(0; \nabla_j(c^2-v_\perp^2) \right) 
= +
{\rho\over c} \; {\partial(c^2-v_\perp^2)\over\partial n} \; \left(0;\hat n_j\right).
\end{equation}
%
There is partial cancellation between the two terms, and so
%
\begin{equation}
(L^\alpha \nabla_\alpha L_\mu)_\mathrm{horizon}
=  
+{1\over2} \; {\rho\over c} \; {\partial(c^2-v_\perp^2)\over\partial n} \; \left(0;\hat n_j\right).
\end{equation}
%
while
\begin{equation}
(L_\mu)_\mathrm{horizon} = {\rho\over c} \; \left(0; c\;\hat n_j\right).
\end{equation}
%
%
%%
%\begin{equation}
%L^\alpha \nabla_\alpha L^\mu 
%=  
%+{1\over2 c^2} 
%\left(v^j \nabla_j[(c^2-v_\perp^2)] ; 
%   (c^2 \delta^{ij} - v^i v^j) \nabla_j[(c^2-v_\perp^2)] \right).
%\end{equation}
%%
%But, as we have already seen, at the horizon the gradient term is
%purely normal. Thus
%%
%\begin{equation}
%L^\alpha \nabla_\alpha L^\mu 
%=  +{1\over2 c} {\partial(c^2-v_\perp^2)\over \partial n} \;
%    (1; v_\parallel^i). 
%\end{equation}
Comparing this with the standard definition of surface
gravity~\cite{Wald}~\footnote{There is an issue of normalization
here. On the one hand we want to be as close as possible to general
relativistic conventions. On the other hand, we would like the
surface gravity to really have the dimensions of an acceleration. The
convention adopted here, with one explicit factor of $c$, is the best compromise we have come up with. (Note that in an acoustic setting, where the speed of sound is not necessarily a constant, we cannot simply set $c\to1$ by a choice of units.)}
%
\begin{equation}
(L^\alpha \nabla_\alpha L_\mu)_\mathrm{horizon} 
=  + {g_H\over c}  \; (L_\mu)_\mathrm{horizon}. 
\end{equation}
%
we finally have
%
\begin{equation}
g_H = 
{1\over2} {\partial(c^2-v_\perp^2)\over \partial n} =
 c \; {\partial(c-v_\perp)\over\partial n}.
\end{equation}
%

This is in agreement with the previous calculation for static acoustic
black holes, and insofar as there is overlap, is also consistent with
results of Unruh~\cite{Unruh:1981cg, Unruh:1994zw, Unruh:1995je},
Reznik~\cite{Reznik:1997ag}, and  the results for ``dirty black
holes''~\cite{Visser:1992qh}. From the construction it is clear that the
surface gravity is a measure of the extent to which the Newtonian time
parameter inherited from the underlying fluid dynamics fails to be an
affine parameter for the null geodesics on the horizon.\footnote{There
are situations in which this surface gravity is a lot larger than one
might naively expect~\cite{Liberati:2000pt}.}

Again, the justification for going into so much detail on this specific
model is that this style of argument can be viewed as a template --- it
will (with suitable modifications) easily generalize to more complicated
analogue models.



%-------------------------------------------------
\subsubsection{Example: Vortex geometry}
%-------------------------------------------------

As an example of a fluid flow where the distinction between ergosphere
and acoustic event horizon is critical consider the ``draining bathtub''
fluid flow. We shall model a draining bathtub by a $(3+1)$ dimensional
flow with a linear sink along the $z$ axis. Let us start with the
simplifying assumption that  the background density $\rho$ is a
position-independent constant throughout the flow (which automatically
implies that the background pressure $p$ and speed of sound $c$ are also
constant throughout the fluid flow). The equation of continuity then
implies that for the radial component of the fluid velocity we must have

%
\begin{equation}
v^{\hat r} \propto {1\over r}.
\end{equation}
%
In the tangential direction, the requirement that the flow be
vorticity free (apart from a possible delta-function contribution at
the vortex core) implies, via Stokes' theorem, that 
%
\begin{equation}
v^{\hat t} \propto  {1\over r}.
\end{equation}
%

(If these flow velocities are nonzero, then following the discussion of
~\cite{Visser:2004zs} there must be some external force present to set
up and maintain the background flow. Fortunately it is easy to see that
this external force affects only the background flow and does not
influence the linearized fluctuations we are interested in.) %

For the background velocity potential we must then have
%
\begin{equation}
\phi(r,\theta) = -A\; \ln(r/a) - B \;\theta.
\end{equation}
%
Note that, as we have previously hinted, the velocity potential is not a
true function (because it has a discontinuity on going through $2\pi$
radians). The velocity potential must be interpreted as being defined
patch-wise on overlapping regions surrounding the vortex core at
$r=0$. The velocity of the fluid flow is
%
\begin{equation}
\v = - \bnabla \phi = {(A \; \hat r + B \; \hat\theta)\over r}.
\end{equation}
%

%-----------------------------------------------------------------------
\begin{figure}[htb]
\vbox{
\vskip 20 pt

\centerline{{\includegraphics[width=3in]{F-matt-drainhole}}}

\caption[Vortex geometry]{ {\sl A collapsing vortex geometry (draining
bathtub): The green spirals denote streamlines of the fluid flow. The
outer circle represents the ergosurface while the inner circle
represents the [outer] event horizon.  \smallskip\label{F:vortex}}}
    
}
\end{figure}
%-----------------------------------------------------------------------




Dropping a position-independent prefactor, the acoustic metric for a draining
bathtub is explicitly given by
%
\begin{equation}
\d s^2 = 
- c^2 \; \d t^2 
+ \left(\d r - {A\over r} \d t\right)^2 
+ \left(r \, \d\theta - {B\over r} \d t\right)^2 + \d z^2.
\end{equation}
%
Equivalently
%
\begin{equation}
\d s^2 = 
- \left(c^2 -{A^2+B^2\over r^2}\right) \d t^2   
- 2{A\over r} \, \d r \, \d t  - 2 B \, \d\theta \, \d t + \d r^2 + r^2 \d\theta^2 + \d z^2.
\end{equation}
%
A similar metric, restricted to $A=0$ (no radial flow), and
generalized to an anisotropic speed of sound, has been exhibited by
Volovik~\cite{Volovik:1996qw}, that metric being a model for the acoustic
geometry surrounding physical vortices in superfluid ${}^3$He. (For a
survey of the many analogies and similarities between the physics of
superfluid ${}^3$He and the Standard Electroweak Model
see~\cite{Volovik:1995yn}, this reference is also useful as
background to understanding the Lorentzian geometric aspects of
${}^3$He fluid flow.) Note that the metric given above is {\em not}
identical to the metric of a spinning cosmic string, which would
instead take the form~\cite{Visser:1995cc}
%
\begin{equation}
\d s^2 = 
- c^2 (\d t - {\tilde A} \, \d\theta)^2  + \d r^2 + (1-{\tilde B}) r^2 \d\theta^2 +\d z^2.
\end{equation}
%
In conformity with previous comments, the vortex fluid flow is seen to
possess an acoustic metric that is stably causal and which does not
involve closed timelike curves. (At large distances it is possible to
{\em approximate} the vortex geometry by a spinning cosmic
string~\cite{Volovik:1996qw}, but this approximation becomes progressively
worse as the core is approached.)

The ergosphere forms at
%
\begin{equation}
r_\mathrm{ergosphere} = {\sqrt{A^2 + B^2}\over c}.
\end{equation}
%
Note that the sign of $A$ is irrelevant in defining the ergosphere and
ergo-region: it does not matter if the vortex core is a source or a
sink.

The acoustic event horizon forms once the radial component of the
fluid velocity exceeds the speed of sound, that is at
%
\begin{equation}
r_\mathrm{horizon} = {|A|\over c}.
\end{equation}
%
The sign of $A$ now makes a difference. For $A<0$ we are dealing with
a future acoustic horizon (acoustic black hole), while for $A>0$ we
are dealing with a past event horizon (acoustic white hole).



%-------------------------------------------------
\subsubsection{Example: Slab geometry}
%-------------------------------------------------



A popular model for the investigation of event horizons in the
acoustic analogy is the one-dimensional slab geometry where the
velocity is always along the $z$ direction and the velocity profile
depends only on $z$. The continuity equation then implies that
$\rho(z) \; v(z)$ is a constant, and the acoustic metric becomes
%
\begin{equation}
\d s^2  \propto  {1\over v(z) \; c(z) }
\left[ 
- c(z)^2 \; \d t^2 + \left\{ \d z - v(z) \; \d t \right\}^2 + \d x^2 + \d y^2 
\right].
\end{equation}
%
That is
%
\begin{equation}
\d s^2  \propto  {1\over v(z) \; c(z) }
\left[ 
- \left\{c(z)^2-v(z)^2\right\}  \d t^2 - 2 v(z) \; \d z\;  \d t+ \d x^2 + \d y^2 + \d z^2
\right].
\end{equation}
%

If we set $c=1$ and ignore the conformal factor we have the toy model
acoustic geometry discussed by Unruh~\cite[page 2828, equation
(8)]{Unruh:1995je} Jacobson~\cite[page 7085, equation
(4)]{Jacobson:1996zs}, Corley and Jacobson~\cite{Corley:1996ar}, and
Corley~\cite{Corley:1997nw}. (In this situation one must again invoke an
external force to set up and maintain the fluid flow. Since the
conformal factor is regular at the event horizon, we know that the
surface gravity and Hawking temperature are independent of this
conformal factor~\cite{Jacobson:1993pf}.) In the general case it is
important to realise that the flow can go supersonic for either of two
reasons: the fluid could speed up, or the speed of sound could decrease.
 When it comes to calculating the ``surface gravity'' both of these
effects will have to be taken into account.


%-------------------------------------------------
\subsubsection{Example: \Painleve--Gullstrand geometry}
%-------------------------------------------------


To see how close the acoustic metric can get to reproducing the
Schwarzschild geometry it is first useful to introduce one of the
more exotic representations of the Schwarzschild geometry: the
\Painleve--Gullstrand line element, which is simply an unusual
choice of coordinates on the Schwarzschild spacetime.\footnote{The
\Painleve--Gullstrand line element is sometimes called the \Lemaitre\ 
line element.} In modern notation the Schwarzschild geometry in
ingoing ($+$) and outgoing ($-$) \Painleve--Gullstrand coordinates
may be written as:  % {\bf *** check signs ***}
%
\begin{equation}
\d s^2 = 
- \d t^2 +
\left( \d r \pm \sqrt{2GM\over r} \d t \right)^2 
+ r^2\left( \d\theta^2 + \sin^2\theta \; \d\phi^2 \right).
\end{equation}
%
Equivalently
%
\begin{equation}
\d s^2 = 
- \left(1-{2GM\over r}\right) \d t^2 
\pm \sqrt{2GM\over r}\, \d r \, \d t 
+ \d r^2 + r^2\left( \d\theta^2 + \sin^2\theta \; \d\phi^2 \right).
\end{equation}
%

This representation of the Schwarzschild geometry was not (until the
advent of the analogue models) particularly well-known, and it has been
independently rediscovered several times during the 20th century. See
for instance \Painleve~\cite{Painleve}, Gullstrand~\cite{Gullstrand},
\Lemaitre~\cite{Lemaitre}, the related discussion by
Israel~\cite{Hawking:300}, and more recently, the paper by Kraus and
Wilczek~\cite{Kraus:1994fh}.  The \Painleve--Gullstrand coordinates are
related to the more usual Schwarzschild coordinates by %

\begin{equation}
t_{PG} = t_{S} \pm 
\left[ 
4 M \; \hbox{\rm arctanh}\left(\sqrt{2GM\over r}\right) - 2 \; \sqrt{2GMr} 
\right].
\end{equation}
%
Or equivalently
%
\begin{equation}
\d t_{PG} = \d t_{S} \pm {\sqrt{2GM/r} \over 1-2GM/r}\; \d r.
\end{equation}
%
With these explicit forms in hand, it becomes an easy exercise to
check the equivalence between the \Painleve--Gullstrand line element
and the more usual Schwarzschild form of the line element. It should
be noted that the $+$ sign corresponds to a coordinate patch that
covers the usual asymptotic region plus the region containing the
future singularity of the maximally extended Schwarzschild spacetime.
It thus covers the future horizon and the black hole singularity.  On
the other hand the $-$ sign corresponds to a coordinate patch that
covers the usual asymptotic region plus the region containing the past
singularity.  It thus covers the past horizon and the white hole
singularity.

As emphasized by Kraus and Wilczek, the \Painleve--Gullstrand line
element exhibits a number of features of pedagogical interest. In
particular the constant time spatial slices are completely flat ---
the curvature of space is zero, and all the spacetime curvature of the
Schwarzschild geometry has been pushed into the time--time and
time--space components of the metric.

Given the \Painleve--Gullstrand line element, it might seem trivial to
force the acoustic metric into this form: simply take $\rho$ and $c$
to be constants, and set $v=\sqrt{2GM/r}$? While this certainly forces
the acoustic metric into the \Painleve--Gullstrand form the problem
with this is that this assignment is incompatible with the continuity
equation $\bnabla\cdot(\rho\v)\neq 0$ that was used in deriving
the acoustic equations.

The best we can actually do is this: Pick the speed of sound $c$
to be a position independent constant, which we normalize to unity
($c=1$). Now set $v=\sqrt{2GM/r}$, and use the continuity equation
$\bnabla\cdot(\rho\v)= 0$ to deduce $\rho|\v| \propto
1/r^2$ so that $\rho \propto r^{-3/2}$. Since the speed of sound
is taken to be constant we can integrate the relation $c^2 =
dp/d\rho$ to deduce the equation of state must be $p = p_\infty +
c^2 \rho$ and that the background pressure satisfies $p-p_\infty
\propto c^2 r^{-3/2}$. Overall the acoustic metric is now
%
\begin{equation}
ds^2  \propto  r^{-3/2}
\left[ - dt^2 +
\left( dr \pm \sqrt{2GM \over r} dt \right)^2 
+ r^2\left( d\theta^2 + \sin^2\theta \; d\phi^2 \right) 
\right].
\end{equation}
%

So we see that the net result is conformal to the \Painleve--Gullstrand
form of the Schwarzschild geometry but not identical to it. For many
purposes this is quite good enough: we have an event horizon, we can
define surface gravity, we can analyze Hawking
radiation.\footnote{Similar constructions work for the
Reissner--Nordstrom geometry~\cite{Liberati:2000pt}, as long as one does
not get too close to the singularity. Likewise certain aspects of the
Kerr geometry can be emulated in this way~\cite{Visser:2004zs}.} Since
surface gravity and Hawking temperature are conformal
invariants~\cite{Jacobson:1993pf} this is sufficient for analyzing basic
features of the Hawking radiation process. The only way in which the
conformal factor can influence the Hawking radiation is through
backscattering off the acoustic metric.  (The phonons are minimally
coupled scalars, not conformally coupled scalars so there will in
general be effects on the frequency-dependent greybody factors.)

If we focus attention on the region near the event horizon, the
conformal factor can simply be taken to be a constant, and we can
ignore all these complications.




%----------------------------------------------------------------
\subsection{Regaining geometric acoustics}
%----------------------------------------------------------------

Up to now, we have been developing general machinery to force acoustics
into Lorentzian form. This can be justified either with a view to using
fluid mechanics to teach us more about general relativity, or to using
the techniques of Lorentzian geometry to teach us more about fluid
mechanics.

For example, given the machinery developed so far, taking the short
wavelength/high frequency limit to obtain geometrical acoustics is now
easy. Sound rays (phonons) follow the {\em null geodesics} of the
acoustic metric. Compare this to general relativity where in the
geometrical optics approximation light rays (photons) follow {\em null
geodesics} of the physical spacetime metric. Since null geodesics are
insensitive to any overall conformal factor in the
metric~\cite{MTW,Hawking-Ellis,Wald} one might as well simplify life by
considering a modified conformally related metric
%
\begin{equation}
h_{\mu\nu} \equiv 
\left[ \matrix{-(c^2-v_0^2)&\vdots&-v_0^j\cr
               \cdots\cdots\cdots\cdots&\cdot&\cdots\cdots\cr
	       -v_0^i&\vdots&\delta^{ij}\cr } 
\right]_.	       
\end{equation}
%
This immediately implies that, in the geometric acoustics limit,
sound propagation is insensitive to the density of the fluid. In
this limit, acoustic propagation depends only on the local speed
of sound and the velocity of the fluid. It is only for specifically
wave related properties that the density of the medium becomes
important.

We can rephrase this in a language more familiar to the acoustics
community by invoking the Eikonal approximation. Express the
linearized velocity potential, $\phi_1$, in terms of an amplitude,
$a$, and phase, $\varphi$, by $\phi_1 \sim a e^{i\varphi}$. Then,
neglecting variations in the amplitude $a$, the wave equation
reduces to the {\em Eikonal equation}
%
\begin{equation}
h^{\mu\nu} \; \partial_\mu \varphi \; \partial_\nu \varphi = 0.
\end{equation}
%
This Eikonal equation is blatantly insensitive to any overall
multiplicative prefactor (conformal factor).

As a sanity check on the formalism, it is useful to re-derive some
standard results.  For example, let the null geodesic be parameterized
by $x^\mu(t) \equiv (t; \x(t))$. Then the null condition implies
%
\begin{eqnarray}
&& h_{\mu\nu} {dx^\mu \over dt} {dx^\nu \over dt} = 0
\nonumber\\
&&\iff
-(c^2 - v_0^2) - 2 v_0^i {dx^i \over dt} 
+ {dx^i \over dt} {dx^i \over dt} = 0
\nonumber\\
&&\iff
\left\Vert {d{\x} \over dt} - \v_0 \right\Vert = c.
\end{eqnarray}
%
Here the norm is taken in the flat physical metric. This has the
obvious interpretation that the ray travels at the speed of sound,
$c$, relative to the moving medium.

Furthermore, if the geometry is stationary one can do slightly
better. Let $x^\mu(s) \equiv (t(s); \x(s))$ be some null path
from $\x_1$ to $\x_2$, parameterized in terms of physical
arc length ({\em i.e.} $|| d{\x}/ds || \equiv 1$). Then the
tangent vector to the path is
%
\begin{equation}
{dx^\mu\over ds} = \left( {dt\over ds}; {dx^i\over ds} \right).
\end{equation}
%
The condition for the path to be null (though not yet necessarily
a null geodesic) is
%
\begin{equation}
g_{\mu\nu} {dx^\mu\over ds} {dx^\nu\over ds} = 0.
\end{equation}
%
Using the explicit algebraic form for the metric, this can be
expanded to show
%
\begin{equation}
-(c^2 - v_0^2) \left({dt\over ds}\right)^2 
- 2 v_0^i \left({dx^i\over ds}\right) \left({dt\over ds}\right)
+1 = 0.
\end{equation}
%
Solving this quadratic 
%
\begin{equation}
\left({dt\over ds}\right) 
= { - v_0^i \left({dx^i\over ds}\right)
    + \sqrt{ c^2 - v_0^2 + \left(v_0^i \; {dx^i\over ds}\right)^2 }
   \over
   c^2 - v_0^2}.
\end{equation}
%
Therefore, the total time taken to traverse the path is  
%
\begin{eqnarray}
T[\gamma] 
&=& \int_{\x_1}^{\x_2} (dt/ds) \; ds 
\nonumber\\
&=& \int_\gamma {1\over c^2-v_0^2} 
                 \Big\{  
                 \sqrt{ (c^2 - v_0^2)ds^2 + (v_0^i \; dx^i)^2 } 
		 - v_0^i \; dx^i 
                 \Big\}.
\end{eqnarray}
%
If we now recall that extremizing the total time taken is Fermat's
principle for sound rays, we see that we have checked the formalism
for stationary geometries (steady flow) by reproducing the discussion
on page 262 of Landau and Lifshitz~\cite{Landau-Lifshitz}.



%-------------------------------------------------
\subsection{Generalizing the physical model}

%-------------------------------------------------

There are a large number of ways in which the present particularly
simple analogue model can be generalized. Obvious issues within the
current physical framework are:

\begin{itemize}
\item Adding external forces.
\item Working in (1+1) or (2+1) dimensions.
\item Adding vorticity, to go beyond the irrotational constraint.
%\item Working with multiple fields, beyond simple pressure fluctuations.
\end{itemize}
Beyond these immediate questions, we could also seek similar effects in other 
physical or mathematical frameworks.  

%-------------------------------------------------
\subsubsection{External forces}

%-------------------------------------------------

Adding external forces is easy, an early discussion can be found
in~\cite{Visser:1997ux} and more details are available
in~\cite{Visser:2004zs}. The key point is that with an external force
one can to some extent shape  the background flow (see for example the
discussion on \cite{Giovanazzi:2004pm}). Upon linearization, the
fluctuations are however insensitive to any external force.



%-------------------------------------------------
\subsubsection{The role of dimension}

%-------------------------------------------------


The role of spacetime dimension in these acoustic geometries is
sometimes a bit surprising and potentially confusing. This is important
because there is a real physical distinction, for instance, between
truly (2+1)-dimensional systems and effectively (2+1)-dimensional
systems in the form of (3+1)-dimensional systems with cylindrical
symmetry.  Similarly there is a real physical distinction between a
truly (1+1)-dimensional system and a (3+1)-dimensional system with
transverse symmetry. We emphasise that in cartesian coordinates the wave
equation

\begin{equation} 
\label{wavef_eq}
\frac{\partial}{\partial {x^{\mu}}} 
\left( f^{\mu \nu} \frac{\partial}{\partial {x^{\nu}}}\; \phi \right)=0,
\end{equation}
where
\begin{equation}
f^{\mu \nu}=\left[
\begin{array}{ccc}
-{\rho}/{c^2}            &|& -{\rho}\;{v^j} /{c^2}\\
%&|& \\
\hline
%&|& \\
-{\rho}\; {v^i}/{c^2}  &|& 
\rho \;\{\delta^{ij} - v^i v^j /{c^2}\}
\end{array}
\right],
\end{equation}
holds \emph{independent} of the dimensionality of spacetime. It
depends only on the Euler equation, the continuity equation, a
barotropic equation of state, and the assumption of irrotational
flow~\cite{Unruh:1981cg,Visser:1993ub,Visser:1998qn,Visser:1997ux}.

Introducing the inverse acoustic metric $g^{\mu \nu}$, defined by
\begin{equation}
f^{\mu \nu}=\sqrt{-g} \, g^{\mu \nu}; 
\qquad
g=\frac{1}{\det(g^{\mu \nu})}
\end{equation} 
the wave equation (\ref{wavef_eq}) corresponds to the d'Alembertian wave equation in a curved
space-time with contravariant metric tensor:
\begin{equation}
g^{\mu \nu}=\left({\rho\over c}\right)^{-2/(d-1)}
\left[
\begin{array}{c|c}
-1/c^2            & -\v^{\,T}/c^2 \\
\hline
%--- & & ---------- \\
\vphantom{\Big|}
- \v/c^2  & 
\mathbf{I}_{d\times d} - \v\otimes \v^{\,T}/c^2
\end{array}
\right],
\end{equation}
where $d$ is the dimension of \emph{space} (not spacetime).
The covariant acoustic metric is then
\begin{equation}
g_{\mu \nu}=\left( \frac{\rho}{c} \right)^{2/(d-1)}
\left[
\begin{array}{ccc}
-\left( c^2-v^2 \right)       &|& -\v^{\,T} \\
\hline
%--- & & ---------- \\
-\v  &|& \mathbf{I}_{d\times d}
\end{array}
\right]. 
\end{equation}
%Note that in all these cases the (fundamental) dimensionality of
%spacetime is put in by hand --- in the present formalism there is no
%way to determine the fundamental dimensionality dynamically. (Of
%course in a Kaluza--Klein framework the {\emph{effective}}
%dimensionality can change if some dimensions become small for
%dynamical reasons.) 


%--------------------------------------------------------------
\paragraph{$d=3$:}
%--------------------------------------------------------------

The acoustic line element for three space and one time dimension reads
\begin{equation} 
g_{\mu \nu}=\left( \frac{\rho}{c} \right)
\left[
\begin{array}{ccc}
-\left( c^2-v^2 \right)       &|& -\v^{\,T} \\
\hline
%--- & & ---------- \\
-\v  &|& \mathbf{I}_{3\times 3}
\end{array}
\right]. 
\end{equation}
%This is the primary case of interest in this article.


%--------------------------------------------------------------
\paragraph{$d=2$:}
%--------------------------------------------------------------

The acoustic line element for two space and one time dimension reads
\begin{equation} 
g_{\mu \nu}=\left( \frac{\rho}{c} \right)^2
\left[
\begin{array}{ccc}
-\left( c^2-v^2 \right)       &|&- \v^{\,T} \\
\hline
%--- & & ---------- \\
-\v  &|& \mathbf{I}_{2 \times 2}
\end{array}
\right]. 
\end{equation}
This situation would be appropriate, for instance, when dealing with
surface waves or excitations confined to a particular substrate.

%--------------------------------------------------------------
\paragraph{$d=1$:}
%--------------------------------------------------------------

The naive form of the acoustic metric in (1+1) dimensions is
ill-defined, because the conformal factor is raised to a formally
infinite power. This is a side effect of the well-known conformal
invariance of the Laplacian in 2 dimensions. The wave equation in terms
of the densitized inverse metric $f^{\mu\nu}$ continues to make good
sense; it is only the step from $f^{\mu\nu}$ to the effective metric
that breaks down.

Note that this issue only presents a difficulty for physical systems
that are \emph{intrinsically} one-dimensional. A three-dimensional system with
plane symmetry, or a two-dimensional system with line symmetry, provides
a perfectly well behaved model for (1+1) dimensions, as in the cases
$d=3$ and $d=2$ above.





%-------------------------------------------------
\subsubsection{Adding vorticity}
%-------------------------------------------------

%Note that the irrotational condition is automatically satisfied for the
%superfluid component of physical superfluids. This point has been
%emphasised by Comer~\cite{Comer}, who has also pointed out that in
%superfluids there will be multiple acoustic metrics (and multiple
%acoustic horizons) corresponding to first and second sound. Even for
%normal fluids, vorticity free flows are common, especially in situations
%of high symmetry.
%
%Furthermore, for the preceding analysis to hold it is sufficient that
%the flow \emph{locally} be vorticity free, $\bnabla\times\v=0$,
%so that velocity potentials exist on an atlas of open patches --- this
%enables us to handle vortex filaments, where the vorticity is
%concentrated into a thin vortex core, provided we do not attempt to
%probe the vortex core itself. It is not necessary for the velocity
%potential $\phi$ to be globally defined. %


For the preceding analysis to hold it is necessary and sufficient that
the flow locally be vorticity free, $\bnabla \times \v = 0$, so
that velocity potentials exist on an atlas of open patches. Note that
the irrotational condition is automatically satisfied for the
super-fluid component of physical superfluids. (This point has been
emphasised by Comer~\cite{Comer}, who has also pointed out that in
superfluids there will be multiple acoustic metrics ---and multiple
acoustic horizons--- corresponding to first and second sound). Even for
normal fluids, vorticity free flows are common, especially in situations
of high symmetry. Furthermore, the previous condition enables us to
handle vortex filaments, where the vorticity is concentrated into a thin
vortex core, provided we do not attempt to probe the vortex core itself.
It is not necessary for the velocity potential $\phi$ to be globally
defined.


Though physically important, dealing with situations of distributed vorticity is much more difficult,
and the relevant wave equation is more complicated in that the velocity
scalar is now insufficient to completely characterize the fluid flow.~\footnote{Vorticity is automatically generated, for instance, whenever the background fluid is non-barotropic, and in particular when $\nabla \rho \times \nabla p \neq 0$. Furthermore, it has been argued in~\cite{Schutzhold:2005ex} that quantum back-reaction can also act as a source for vorticity.}
An
approach similar to the spirit of the present discussion, but in terms
of Clebsch potentials, can be found in~\cite{PerezBergliaffa:2001nd}.
The eikonal approximation (geometrical acoustics) leads to the same
conformal class of metrics previously discussed, but in the realm of
physical acoustics the wave equation is considerably more complicated
than a simple d'Alembertian.  (Roughly speaking, the vorticity becomes a
source for the d'Alembertian, while the vorticity evolves in response to
gradients in a generalized scalar potential. This seems to  take us
outside the realm of models of direct interest to the general relativity
community.)~\footnote{In references \cite{GarciadeAndrade:2004au,
GarciadeAndrade:2005ap, GarciadeAndrade:2004fe, GarciadeAndrade:2004fs,
GarciadeAndrade:2004kv, GarciadeAndrade:2005tx, GarciadeAndrade:2004zm,
GarciadeAndrade:2004zn} the author has attempted to argue that vorticity
can be related to the concept of torsion in a general affine connexion.
We disagree. Although deriving a wave equation in the presence of
vorticity very definitely moves one beyond the realm of a simple
Riemannian spacetime, adding torsion to the connexion is not sufficient
to capture the relevant physics.}


%-------------------------------------------------
\subsection{Simple Lagrangian meta-model}
%-------------------------------------------------

\def\L{{\mathcal{L}}}

As a first (and rather broad) example of the very abstract ways in which
the notion of an acoustic metric can be generalized, we start from the
simple observation that irrotational barotropic fluid mechanics can be
described by a Lagrangian, and ask if we can extend the notion of an
acoustic metric to all (or at least some wide class of) Lagrangian
systems?

Indeed, suppose we have a single scalar field $\phi$ whose dynamics is
governed by some generic Lagrangian $\L(\partial_\mu\phi, \phi)$,
which is some arbitrary function of the field and its first
derivatives (here we will follow the notation and ideas of
\cite{Barcelo:2001ah}). In the general analysis that follows the
previous irrotational and inviscid fluid system is included as a
particular case; the dynamics of the scalar field $\phi$ is now much
more general. We want to consider linearized fluctuations around some
background solution $\phi_0(t,\x)$ of the equations of motion, and to
this end we write

%
\begin{equation}
\phi(t,\x) = \phi_0(t,\x) + \epsilon  \phi_1(t,\x) + 
 {\epsilon^2\over2}  \phi_2(t,\x) + O(\epsilon^3).
\end{equation}
%
Now use this to expand the Lagrangian around the classical solution
$\phi_0(t,\x)$:
%
\begin{eqnarray}
{\L}(\partial_\mu \phi,\phi) 
&=& 
{\L}(\partial_\mu \phi_0,\phi_0)
%\nonumber
%\\
%&+& 
+
\epsilon \left[ 
{\partial \L\over\partial(\partial_\mu \phi)} \; \partial_\mu \phi_1
+
{\partial \L\over\partial\phi} \; \phi_1
\right]
\nonumber
\\
&+&
%+
{\epsilon^2\over2} \left[ 
{\partial \L\over\partial(\partial_\mu \phi)} \; \partial_\mu \phi_2
+
{\partial \L\over\partial\phi} \;  \phi_2
\right]
\nonumber
\\
&+&
{\epsilon^2\over2} \Bigg[ 
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\; \partial_\mu \phi_1 \; \partial_\nu \phi_1
%\nonumber
%\\
%&&\qquad 
+
2 {\partial^2 \L\over \partial(\partial_\mu \phi)\; \partial \phi} 
\;\partial_\mu \phi_1 \; \phi_1
\nonumber
\\
&&\qquad 
+
{\partial^2 \L\over \partial\phi\; \partial\phi} 
\; \phi_1 \; \phi_1
\Bigg]
%\nonumber
%\\
%&+&
+
O(\epsilon^3).
\end{eqnarray}
%
It is particularly useful to consider the action
%
\begin{equation}
\label{E:bare-action}
S[\phi] = \int \d^{d+1} x \; \L(\partial_\mu\phi,\phi),
\end{equation}
%
since doing so allows us to integrate by parts. (Note that the
Lagrangian $\L$ is taken to be a scalar density, not a true scalar.)  We
can now use the Euler--Lagrange equations for the background field
%
\begin{equation}
\partial_\mu \left( {\partial \L \over \partial(\partial_\mu \phi)} \right) 
- {\partial \L\over\partial \phi} = 0,
\end{equation}
%
to discard the linear terms (remember we are linearizing around a
solution of the equations of motion) and so we get
%
\begin{eqnarray}
S[\phi] &=& S[\phi_0] 
%\nonumber
%\\
%&+&
+
{\epsilon^2\over2}
\int \d^{d+1} x \Bigg[
\left\{
  {\partial^2 \L\over
   \partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
   \right\} 
\; \partial_\mu \phi_1 
\; \partial_\nu \phi_1 
\nonumber
\\
&&
\qquad
+
\left(
{\partial^2 \L\over \partial\phi\; \partial \phi} - 
\partial_\mu \left\{{\partial^2 \L\over \partial(\partial_\mu \phi)\; 
\partial \phi} \right\}
\right) 
\;
\phi_1 \; \phi_1
\Bigg]
%\nonumber
%\\
%&+&
+
O(\epsilon^3).
\end{eqnarray}
%
Having set things up this way, the equation of motion for the
linearized fluctuation is now easily read off as
%
\begin{eqnarray}
&&  \partial_\mu \left(\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
   \right\}
 \partial_\nu \phi_1 \right)
%\nonumber
%\\
%&&
%\qquad\qquad
- \left(
{\partial^2 \L\over \partial\phi\; \partial \phi} - 
\partial_\mu \left\{{\partial^2 \L\over \partial(\partial_\mu \phi)\; 
\partial \phi} \right\}
\right)
\phi_1
= 0. 
\nonumber
\\
&&
\end{eqnarray}
%
This is a second-order differential equation with position-dependent
coefficients (these coefficients all being implicit functions of the
background field $\phi_0$). 

This can
be given a nice clean geometrical interpretation in terms of a
d'Alembertian wave equation --- provided we \emph{define} the
effective spacetime metric by
%
\begin{equation}
\sqrt{-g} \; g^{\mu\nu} \equiv  f^{\mu\nu} \equiv 
\left.\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}\right|_{\phi_0}.
\end{equation}
%
Note that this is another example of a situation in which calculating 
the inverse metric density is easier than calculating the metric itself.

Suppressing the $\phi_0$ except when necessary for clarity, this
implies [in (d+1) dimensions, $d$ space dimensions plus $1$ time
dimension]
%
\begin{equation}
(-g)^{(d-1)/2} = -\det   
\left\{ 
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}.
\end{equation}
%
Therefore
%
\begin{equation}
 g^{\mu\nu}(\phi_0) =  
\left.
\left(
- \det   
\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}
\right)^{-1/(d-1)}
\right|_{\phi_0}
\;
\left.
\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}
\right|_{\phi_0}.
\end{equation}
%
And, taking the inverse
%
\begin{equation}
g_{\mu\nu}(\phi_0) =    
\left.
\left( - 
\det\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}
\right)^{1/(d-1)}
\right|_{\phi_0}
 \; \; 
\left.
\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}^{-1}
\right|_{\phi_0}.
\end{equation}
%
We can now write the equation of motion for the linearized
fluctuations in the geometrical form
%
\begin{equation}
\label{E:geometrical}
\left[\Delta(g(\phi_0)) - V(\phi_0)\right] \phi_1 = 0,
\label{laeone}
\end{equation}
%
where $\Delta$ is the d'Alembertian operator associated with the
effective metric $g(\phi_0)$, and $V(\phi_0)$ is the
background-field-dependent  (and so in general position-dependent) ``mass term'':
%
\begin{eqnarray}
V (\phi_0)&=&
{1\over\sqrt{-g}} \;
\left(
{\partial^2 \L\over \partial\phi\; \partial \phi} - 
\partial_\mu 
\left\{
{\partial^2 \L\over \partial(\partial_\mu \phi)\; \partial \phi} 
\right\}
\right)
\\
&=&
 \left(
- \det   
\left\{
{\partial^2 \L\over\partial(\partial_\mu \phi) \; \partial(\partial_\nu \phi)} 
\right\}
\right)^{-1/(d-1)}
\nonumber\\
&& \qquad \times
\left(
{\partial^2 \L\over \partial\phi\; \partial \phi} - 
\partial_\mu 
\left\{
{\partial^2 \L \over \partial(\partial_\mu \phi)\; \partial \phi} 
\right\}
\right). 
\end{eqnarray}
%
Thus $V(\phi_0)$ is a true scalar (not a density).  Note that the
differential equation (\ref{E:geometrical}) is automatically formally
self-adjoint (with respect to the measure $\sqrt{-g} \; \d^{d+1}x$).



It is important to realise just how general the result is (and where the
limitations are): it works for {\em any} Lagrangian depending only on a
single scalar field and its first derivatives. The linearized PDE will
be {\emph{hyperbolic}} (and so the linearized equations will have
wave-like solutions) if and only if the effective metric $g_{\mu\nu}$
has Lorentzian signature $\pm[-,+^d]$. Observe that if the Lagrangian
contains nontrivial second derivatives you should not be too surprised
to see terms beyond the d'Alembertian showing up in the linearized
equations of motion.


As a specific example of the appearance of effective metrics due to
Lagrangian dynamics we reiterate the fact that inviscid irrotational
barotropic hydrodynamics naturally falls into this scheme (which is why,
with hindsight, the derivation of the acoustic metric presented earlier
in this review was so relatively straightforward). In inviscid
irrotational barotropic hydrodynamics the lack of viscosity
(dissipation) guarantees the existence of a Lagrangian; which a priori
could depend on several fields. Since the flow is irrotational $\v =
 -\bnabla \phi$ is a function only of the velocity
potential, and the Lagrangian is a function only of this potential and
the density. Finally the equation of state can be used to eliminate the
density leading to a Lagrangian that is a function only of the single
field $\phi$ and its derivatives.~\cite{Barcelo:2001ah}


%----------------------------------------------
\subsection{Going further}
%----------------------------------------------
\label{SS:going-further-2}
%----------------------------------------------


The class of analogue models based on fluid mechanics is now quite large
and the literature is extensive. Most of the relevant discussion will be
deferred until subsequent sections, so for the time being we shall just
mention reasonably immediate generalizations such as:


\begin{itemize}

\item Working with specific fluids.

\begin{itemize}

\item Superfluids.
%~\cite{Volovik:1998pf,Volovik:1998pf,Fischer:2000hg,
%Volovik:2000ua,Carter:2001tg}.   

%{\bf *** Many more needed here ***}


\item Bose--Einstein condensates.
%\cite{Garay:1999sk,Garay:2000jj,Barcelo:2000tg,Visser:2001jd,
%Barcelo:2001ca,Fedichev:2003bv,Fedichev:2003id,Lidsey:2003ze,
%Barcelo:2003wu,Fischer:2004hr,Fischer:2004bf}.

%{\bf *** Many more needed here ***} 
\end{itemize}
 
\item Abstract generalizations.

\begin{itemize}

\item Normal modes in generic systems.
%~\cite{Barcelo:2001ah,Barcelo:2001cp,Barcelo:2001tb} .

%{\bf *** Many more needed here ***}

\item Multiple signal speeds.
%~\cite{Barcelo:2001cp,Visser:2002sf,Bassett:2000wj}.

%{\bf *** Many more needed here ***}
\end{itemize}
\end{itemize}
We next turn to a brief historical discussion, seeking to place the 
work of the last decade into its proper historical perspective.


%---------------
% EOF
%---------------

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