#  K-essence Emergent Spacetime as Generalized Vaidya Geometry

Goutam Manna<EMAIL_ADDRESS>Department of Physics, Prabhat Kumar
College, Contai, Purba Medinipur-721404, India  Parthasarathi Majumdar
<EMAIL_ADDRESS>School of Physical Sciences, Indian Association for the
Cultivation of Science, Kolkata-700032, India  Bivash Majumder
<EMAIL_ADDRESS>Department of Mathematics, Prabhat Kumar College,
Contai, Purba Medinipur-721404, India

###### Abstract

We establish a formal connection between the K-essence emergent gravity
scenario and generalizations of Vaidya spacetime. Choosing the K-essence
action to be of the Dirac-Born-Infeld variety, the physical spacetime to be a
general static spherically symmetric black hole and restricting the K-essence
scalar field to be a function solely of the advanced or the retarded time, we
show that the emergent gravity metric resembles closely the generalized Vaidya
metrics for null fluid collapse proposed by Husain. Imposing null energy
conditions on the emergent energy-momentum tensor derived from the emergent
Einstein equation, restrictions are obtained on the functions characterizing
the emergent metric for consistent identification with generalized Vaidya
spacetimes. We discuss the possibility of dynamical horizons in the K-essence
emergent Vaidya spacetime. Admissible explicit black hole background metrics
are discussed as examples.

Emergent gravity, k-essence, Vaidya spacetimes

###### pacs:

04.20.-q; 04.20.Jb; 04.70.-s

##  I Introduction

The identification of specific K-essence emergent gravity models with known
spacetimes has been one approach babi1 -babi5  to better understand the
physical origin of Dark Energy, if the phenomenon of Late Time Acceleration of
the Universe is assumed to truly exist, on the basis of recent cosmological
observations. An investigation undertaken to identify emergent K-essence
spacetimes with known gravitational configurations, of course, goes beyond the
original intent of unravelling Dark Energy, since new motivations from analog
gravity make such an investigation a useful enterprise. Based on a specific
Dirac-Born-Infeld born1 -born3  model for the K-essence scalar field and
specific standard physical black hole spacetimes, one of us (GM) has co-
authored papers gm1 ; gm2 , establishing conformal invariance of the emergent
spacetime with Barriola-Vilenkin and Robinson-Trautman spacetimes. To explore
general properties of such a ‘map’ between emergent spacetimes corresponding
to physical black hole geometries, and unexpected curved geometries apparently
unrelated to those black holes, we focus in this paper on Dirac-Born-Infeld
type K-essence scalar field models, in the background of general static
spherically symmetric black hole spacetimes. Following the standard
construction of the emergent composite metric as a function of the physical
spacetime metric and the K-essence scalar, we obtain a class of emergent
metrics which, under certain restrictions on the scalar field, resemble null
fluid collapse models as generalizations husain , wang  of the Vaidya
spacetime vai1 -jg . We do not address the question as to whether such a
relationship between seemingly unconnected spacetime geometries has any deeper
significance in either clarifying any aspect of Dark Energy, or indeed
providing any insight into gravitational collapse. As alluded to in the
abstract, at this point the mapping between the disparate spacetimes is quite
formal. However, detailed computations for the explicit examples of black hole
spacetimes may provide some extra insight for the discerning reader.

The paper is organized as follows : In section 2, we follow ref.s babi1 -babi5
to briefly review the construction of the composite emergent metric for a very
general K-essence scalar field sector in an arbitrary physical spacetime
background, not necessarily stationary. Towards the end of this section, we
specialize to the precise K-essence scalar field action gm1 ; gm2  which we
actually use in the present work. In the next section, we construct the
emergent spacetime metric for a general static spherically symmetric black
hole background, with a scalar field restricted to be an arbitrary function of
the advanced or retarded Eddington-Finkelstein time, and to be independent of
the other variables of the four dimensional spacetime. Such a choice implies
that the composite emergent metric will violate Lorentz invariance. So, at
this point it is not clear if the construction will at all lead to anything
useful. To investigate this question of utility, we construct the Einstein
tensor corresponding to our emergent metric, and compute the components of the
emergent energy-momentum tensor by direct substitution into an emergent
Einstein equation. This emergent tensor must obey energy conditions if the
emergent geometry is to have any interpretation as a curved spacetime. This
requirement is shown to lead to certain restrictions on the functions
characterizing the composite metric, i.e., on the function characterizing the
background spacetime, as also on the function of the advanced/retarded time
characterising the K-essence sector. The possibility that the emergent
spacetime due to the K-essence scalar field can admit dynamical horizons (DH)
is discussed, following earlier work on the Vaidya spacetime admitting a
dynamical horizon. This is followed in section 4 by results of computations
involving several explicit black hole metrics corresponding to the spacetime
background. The last section is discussion of our work.

##  II Review of K-essence and Emergent Gravity

In this section, we present a short review of the construction of the
effective metric for the emergent spacetime corresponding to a general
background geometry and a very general K-essence scalar field sector. The
K-essence scalar field $\phi$ minimally coupled to the background spacetime
metric $g_{\mu\nu}$ has action babi1 -babi5

$\displaystyle S_{k}[\phi,g_{\mu\nu}]=\int d^{4}x{\sqrt{-}g}L(X,\phi)$ (1)

where $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$. The energy-
momentum tensor is

$\displaystyle T_{\mu\nu}\equiv{2\over\sqrt{-g}}{\delta S_{k}\over\delta
g^{\mu\nu}}=L_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}L$ (2)

$L_{\mathrm{X}}={dL\over dX},~{}~{}L_{\mathrm{X}X}={d^{2}L\over
dX^{2}},~{}~{}L_{\mathrm{\phi}}={dL\over d\phi}$ and $\nabla_{\mu}$ is the
covariant derivative defined with respect to the gravitational metric
$g_{\mu\nu}$. The scalar field equation of motion is

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\tilde{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+2XL_{X\phi}-L_{\phi}=0$
(3)

where

$\displaystyle\tilde{G}^{\mu\nu}\equiv
L_{X}g^{\mu\nu}+L_{XX}\nabla^{\mu}\phi\nabla^{\nu}\phi$ (4)

and $1+{2XL_{XX}\over L_{X}}>0$.

We make the conformal transformation $G^{\mu\nu}\equiv{c_{s}\over
L_{x}^{2}}\tilde{G}^{\mu\nu}$, with $c_{s}^{2}(X,\phi)\equiv{(1+2X{L_{XX}\over
L_{X}})^{-1}}$. Then the inverse metric of $G^{\mu\nu}$ is

$\displaystyle G_{\mu\nu}={L_{X}\over
c_{s}}[g_{\mu\nu}-{c_{s}^{2}}{L_{XX}\over
L_{X}}\nabla_{\mu}\phi\nabla_{\nu}\phi].$ (5)

A further conformal transformation gm1 ; gm2
$\bar{G}_{\mu\nu}\equiv{c_{s}\over L_{X}}G_{\mu\nu}$ gives

$\displaystyle\bar{G}_{\mu\nu}={g_{\mu\nu}-{{L_{XX}}\over{L_{X}+2XL_{XX}}}\nabla_{\mu}\phi\nabla_{\nu}\phi}$
(6)

Here one must always have $L_{X}\neq 0$ for $c_{s}^{2}$ to be positive
definite and only then equations $(1)-(4)$ will be physically meaningful.

It is clear that, for non-trivial spacetime configurations of $\phi$, the
emergent metric $G_{\mu\nu}$ is, in general, not conformally equivalent to
$g_{\mu\nu}$. So $\phi$ has properties different from canonical scalar fields,
with the local causal structure also different from those defined with
$g_{\mu\nu}$. Further, if $L$ is not an explicit function of $\phi$ then the
equation of motion $(3)$ reduces to;

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\bar{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ (7)

We shall take the Lagrangian as $L=L(X)=1-V\sqrt{1-2X}$ with $V$ is a
constant. This is a particular case of the DBI lagrangian gm1 ; gm2 , born1
-born3

$\displaystyle L(X,\phi)=1-V(\phi)\sqrt{1-2X}$ (8)

for $V(\phi)=V=constant$ and $kinetic~{}energy~{}of~{}\phi>>V$
i.e.$(\dot{\phi})^{2}>>V$. This is typical for the K-essence fields where the
kinetic energy dominates over the potential energy. Then
$c_{s}^{2}(X,\phi)=1-2X$. For scalar fields
$\nabla_{\mu}\phi=\partial_{\mu}\phi$. Thus (6) becomes

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\partial_{\mu}\phi\partial_{\nu}\phi$
(9)

Note the rationale of using two conformal transformations: the first is used
to identify the inverse metric $G_{\mu\nu}$, while the second realises the
mapping onto the metric given in $(9)$ for the lagrangian
$L(X)=1-V\sqrt{1-2X}$.

##  III Emergent Spacetime for general spherically symmetric black holes

The line element corresponding to a general spherically symmetric static
(black hole) spacetime is

$\displaystyle ds^{2}=f(r)dt^{2}-f^{-1}(r)dr^{2}-r^{2}d\Omega^{2}$ (10)

where $d\Omega^{2}=d\theta^{2}+d\Phi^{2}$.

We define the tortoise coordinate by the relation $dr^{*}=f^{-1}(r)dr$ and
$v=t+\epsilon r^{*}$. With these definitions, the line element (10) reduces to
the Eddington-Finkelstein line element

$\displaystyle ds^{2}=f(r)dv^{2}-2\epsilon dvdr-r^{2}d\Omega^{2}.$ (11)

When $\epsilon=+1$, the null coordinate $v$ represents the Eddington advanced
time (outgoing), while when $\epsilon=-1$, it represents the Eddington
retarded time (incoming).

Now, from (9) the emergent spacetime is described by the line element

$\displaystyle dS^{2}=ds^{2}-\partial_{\mu}\phi\partial_{\nu}\phi
dx^{\mu}dx^{\nu}.$ (12)

We make the assumptions that the scalar field $\phi(x)=\phi(v)$, so that the
emergent spacetime line element is

$\displaystyle dS^{2}=[f(r)-\phi_{v}^{2}]dv^{2}-2\epsilon
dvdr-r^{2}d\Omega^{2}$ (13)

where $\phi_{v}=\frac{\partial\phi}{\partial v}$. Notice that this assumption
on $\phi$ actually violates local Lorentz invariance, since in general,
spherical symmetry would only require that $\phi(x)=\phi(v,r)$. The additional
assumption that $\phi(v,r)=\phi(v)$ i.e., it is independent of $r$ implies
that outside of this particular choice of frame, a spherically symmetric
$\phi$ is actually a function of both $v,r$.

We now compare the emergent spacetime (13) with the metric husain ; wang  of
the generalized Vaidya spacetimes corresponding to gravitational collapse of a
null fluid (take $\epsilon=+1$)

$\displaystyle
dS_{V}^{2}=\left(1-\frac{2m(v,r)}{r}\right)dv^{2}-2dvdr-r^{2}d\Omega^{2}$ (14)

yielding the mass function

$\displaystyle m(v.r)=\frac{1}{2}r\left[1+\phi_{v}^{2}-f(r)\right].$ (15)

One can now compute the emergent Einstein tensor following husain , with the
notation that subscripts on $m$ and $f$ denote derivatives with respect to
those subscripts.

$\displaystyle{\cal G}^{0}_{0}={\cal G}^{1}_{1}$ $\displaystyle=$
$\displaystyle-\frac{2m_{r}}{r^{2}}=\frac{1}{r^{2}}\left[rf_{r}+f-1-\phi_{v}^{2}\right],~{}$
(16) $\displaystyle{\cal G}^{1}_{0}$ $\displaystyle=$ $\displaystyle
2\frac{m_{v}}{r^{2}}=\frac{2\phi_{v}\phi_{vv}}{r},$ (17) $\displaystyle{\cal
G}^{2}_{2}={\cal G}^{3}_{3}$ $\displaystyle=$
$\displaystyle-\frac{m_{rr}}{r}=\frac{1}{2r}\left[2f_{r}+rf_{rr}\right].$ (18)

Recourse to the ‘emergent’ Einstein equation

$\displaystyle{\cal G}^{\mu}_{\nu}=\kappa{\cal T}^{\mu}_{\nu}$ (19)

where, $\kappa\equiv 8\pi G$ leads to the components ${\cal T}^{\mu}_{\nu}$,
which can be parametrized exactly as in ref. husain ; wang  in terms of the
components $\gamma,\rho$ and $P$ given by

$\displaystyle{\cal T}_{\mu\nu}={\cal T}_{\mu\nu}^{(n)}+{\cal
T}_{\mu\nu}^{(m)}=\left[\begin{array}[]{cccc}(\gamma/2+\rho)&\gamma/2&0&0\\\
\gamma/2&(\gamma/2-\rho)&0&0\\\ 0&0&P&0\\\ 0&0&0&P\end{array}\right]$ (24)
(25)

where ${\cal T}_{\mu\nu}^{(n)}=\gamma l_{\mu}l_{\nu}$; ${\cal
T}_{\mu\nu}^{(m)}=(\rho+P)(l_{\mu}n_{\nu}+l_{\nu}n_{\mu})+P\bar{G}_{\mu\nu}$
with $l_{\mu}$ and $n_{\mu}$ are two null vectors. Contractions of all indices
are performed through the emergent metric $\bar{G}_{\mu\nu}$. The expressions
for the three independent components are given by,

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{2\phi_{v}\phi_{vv}}{\kappa r}$ (26) $\displaystyle\rho$
$\displaystyle=$ $\displaystyle\frac{1}{\kappa
r^{2}}\left[1+\phi_{v}^{2}-f-rf_{r}\right]$ (27) $\displaystyle P$
$\displaystyle=$ $\displaystyle\frac{1}{2\kappa
r}\left[2f_{r}+rf_{rr}\right]..$ (28)

It is obvious that energy conditions imposed on ${\cal T}^{\mu\nu}$ will in
turn constrain $f(r)$ and $\phi(v)$ and their derivatives. Thus,

$\displaystyle\gamma>0$ $\displaystyle\Rightarrow$
$\displaystyle\phi_{v}\phi_{vv}>0,$ (29) $\displaystyle\rho>0$
$\displaystyle\Rightarrow$ $\displaystyle 1+\phi_{v}^{2}>f+rf_{r},$ (30)
$\displaystyle P>0$ $\displaystyle\Rightarrow$ $\displaystyle
2f_{r}+rf_{rr}>0.$ (31)

###  III.1 Possible Dynamical Horizon in the Emergent Spacetime

Having demonstrated that the emergent spacetime is a generalized Vaidya
spacetime, obeying the Einstein equation with an emergent energy-momentum
tensor ${\cal T}^{\mu\nu}$ obeying the energy conditions (29)-(31), it stands
to reason to enquire, if, just like null hypersurfaces in the ingoing
Schwarzschild-Vaidya spacetime discussed in ref. ash2003  as an example of
dynamical horizons, our emergent spacetime also admits dynamical horizons. We
rewrite eqn (13), given in terms of ingoing Eddington-Finkelstein coordinates
$v,r,\theta,\phi$, as

$\displaystyle dS^{2}=F(v,r)dv^{2}-2dvdr-r^{2}d\Omega^{2}$ (32)

where, $F(v,r)\equiv f(r)-\phi^{2}_{v}(v)$. We follow ref. ash2002 ; ash2003 ;
ash2004 ; hayward ; blau ; poisson  in the discussion of this subsection.

The spherical symmetry of the emergent spacetime (32) implies that this
spacetime can be decomposed locally into the Cartesian product of a two
dimensional Lorentzian spacetime spanned by outgoing and ingoing null vectors
$l^{\mu}$ and $n^{\mu}$ respectively, and a spacelike 2-sphere. With our
choice of coordinates above, at any fixed point of the Lorentzian spacetime,
i.e., on the 2-spheres $v=const.~{},~{}r=const$, the null vectors are chosen
as

$\displaystyle l$ $\displaystyle=$
$\displaystyle\partial_{v}+\frac{1}{2}F(v,r)\partial_{r}$ $\displaystyle n$
$\displaystyle=$ $\displaystyle-\partial_{r}$ (33)

so that, $l^{2}=0=n^{2}~{},~{}l^{\mu}n_{\mu}=1$. One can compute the expansion
of these vectors on the chosen 2-spheres, to obtain,

$\displaystyle\Theta_{l}=\frac{F(v,r)}{r}~{},~{}\Theta_{n}=-\frac{2}{r}$ (34)

It is clear from eqn (34) that the hypersurface $F(v,r)=0$ foliated by the
chosen 2-spheres, is a candidate dynamical horizon, since the 2-spheres
correspond to marginal outer trapped surfaces. Note that $F(v,r)=0\Rightarrow
r=r(v)$ implies that the background metric function $f(r(v))=\phi^{2}_{v}(v)$,
the square of the $K$-essence scalar field derivative with respect to $v$.
Now, the energy condition eqn (29) implies that $\phi_{v}^{2}$ must be a non-
decreasing function of $v$. Further, it is not difficult to show that ${\cal
L}_{n}\Theta_{l}<0$ on the chosen 2-spheres. Spherical symmetry in this case
implies hypersurface orthogonality (and hence zero rotation) and also
vanishing shear. Thus, the $F(v,r)=0$ hypersurface does indeed correspond to a
dynamical horizon, as per the definition given in ash2003 . The derivatives of
$F(v,r)$ are $F^{\prime}=\frac{\partial F}{\partial r}=f^{\prime}(r)$ and
$\dot{F}=\frac{\partial F}{\partial v}=-2\phi_{v}\phi_{vv}<0$ with the
condition (24). There is also the possibility of reaching an equilibrium
isolated horizon, but that would correspond to the $K$-essence scalar field
turning completely non-dynamical.

What is especially interesting in this case, in contrast to the usual Vaidya
spacetime, is that the $K$-essence scalar naturally plays the role of the null
dust in the generalized Vaidya spacetime. On the dynamical horizon, for every
background spacetime characterized by $f(r(v))$, the presence of the
$K$-essence scalar field predicts a dynamical horizon with the scalar field
$\phi(v)$ strongly constrained by $\phi^{2}_{v}=f(r(v))$. So, if the
background is a vacuum or an electrovac black hole spacetime, only for very
restricted scalar field solutions can the emergent spacetime admit a dynamical
horizon. Thus, since for spherically symmetric black hole backgrounds,
$f(r(v))>0$ for the exterior black hole spacetime, there is no restriction as
such on $\phi_{v}^{2}$ from the above equality between $\phi_{v}^{2}$ and
$f(r(v))$. However, as we see below. for specific choices of $f(r(v))$, more
specific restrictions on $\phi_{v}^{2}$ appear. These restrictions, while
consistent with energy conditions, are specific to dynamical horizons.

##  IV Explicit Examples of Background Spacetime

###  IV.1 Schwarzschild Black Hole as background

Now, we may choose $f(r)=(1-2M/r)$, i.e., the physical spacetime is an
exterior Schwarzschild spacetime. In this case, the emergent spacetime has the
line element

$\displaystyle
dS^{2}=[1-\frac{2M}{r}-\phi_{v}^{2}]dv^{2}-2dvdr-r^{2}d\Omega^{2}.$ (35)

From (15), the mass function is

$\displaystyle m(v,r)=M+\frac{r}{2}\phi_{v}^{2}.$ (36)

Therefore, the non-vanishing components of the Einstein tensors are

$\displaystyle\mathcal{G}^{0}_{0}=\mathcal{G}_{1}^{1}=-\frac{2m_{r}}{r^{2}}=-\frac{\phi_{v}^{2}}{r^{2}};~{}\mathcal{G}_{0}^{1}=\frac{2m_{v}}{r^{2}}=\frac{2\phi_{v}\phi_{vv}}{r};$
$\displaystyle\mathcal{G}_{2}^{2}=\mathcal{G}_{3}^{3}=-\frac{m_{rr}}{r}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(37)

where all subscripts designate derivaties as in the last section. For the case
of Schwarzschild background (35) the values of $\gamma$, $\rho$ and $P$ are

$\displaystyle\gamma=\frac{2\phi_{v}\phi_{vv}}{\kappa
r};~{}\rho=\frac{\phi_{v}^{2}}{\kappa r^{2}}~{}and~{}P=0$ (38)

which satisfies the weak and strong energy conditions haw-ellis ; wang

$\displaystyle\gamma\geq 0,~{}\rho\geq 0,~{}P\geq 0~{}(\gamma\neq 0)$ (39)

and the dominant energy condition

$\displaystyle\gamma\geq 0,~{}\rho\geq~{}P\geq 0~{}(\gamma\neq 0)$ (40)

provided $\phi_{v}\phi_{vv}>0$. Therefore, energy-momentum tensor becomes

$\displaystyle{\cal T}_{\mu\nu}=\left[\begin{array}[]{cccc}\frac{1}{\kappa
r}(\phi_{v}\phi_{vv}+\frac{\phi_{v}^{2}}{r})&\frac{\phi_{v}\phi_{vv}}{\kappa
r}&0&0\\\ \frac{\phi_{v}\phi_{vv}}{\kappa r}&\frac{1}{\kappa
r}({\phi_{v}\phi_{vv}}-\frac{\phi_{v}^{2}}{r})&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right]$ (45) (46)

This type of energy-momentum tensor belongs to type-II class haw-ellis  which
have a double null vector.

The dynamical horizon character for this case with the definition (28) is
clear: we get the expansion of the outgoing and ingoing null normals are

$\displaystyle\Theta_{l}=\frac{1}{r}(1-\frac{2M}{r}-\phi_{v}^{2});~{}\Theta_{n}=-\frac{2}{r}$
(47)

Thus, for the dynamical horizon to exist, it is clear that $\phi_{v}^{2}<1$ in
addition to being a non-decreasing function of $v$ as per the energy
conditions. This is a very stringent restriction on what K-essence scalar
field configurations are permissible for the generalized Vaidya solution
discerned by us, to have a dynamical horizon. Also, at the marginally trapped
surfaces,

$\displaystyle\mathcal{L}_{n}\Theta_{l}\mid_{F=0}=-\frac{1}{r^{2}}(1-\phi_{v}^{2})<0~{},~{}\phi_{v}^{2}<1$
(48)

In this case, $f(r)=1-\frac{r_{S}}{r}$, where $r_{S}(=2M)$ is the
Schwarzschild radius. For the DH we have

$\displaystyle f(r(v),v)\equiv 1-\frac{r_{S}}{r(v)}=\phi_{v}^{2}$
$\displaystyle\Rightarrow r(v)=\frac{r_{S}}{1-\phi_{v}^{2}}\simeq
r_{S}(1+\phi_{v}^{2}+...)$ (49)

From (39), it is clear that the dynamical horizon corresponding to the
emergent metric lies outside the event horizon of the background black hole.
The dynamical horizon location $r(v)$ [FIG.1] is plotted as a function of the
K-essence scalar field kinetic energy $\phi_{v}^{2}$ for the admissible range
$[0,1]$ and is given in FIG.1.

Figure 1: (Color online): Plot of the DH location ($r(v)$) (red line) in units
of $r_{S}$ vs. $\phi_{v}^{2}$.

FIG.1 clearly shows the stretching out of the dynamical horizon of the
emergent spacetime beyond the event horizon of the background black hole. This
stretching increases with the magnitude of the kinetic energy $\phi_{v}^{2}$
of the K-essence scalar field, reaching out to asymptopia as the kinetic
energy approaches its limiting value of unity. This delineates the impact of
the scalar field on the background spacetime. It also suggests that
cosmological effects might begin to arise as this approach to asymptopia of
the dynamical horizon occurs. A detailed discussion of these interesting
implications is, unfortunately, beyond the scope of the present investigation,
where the similarity of the emergent spacetime to a generalized Vaidya
spacetime is all we have focused on.

Next, we consider $\phi_{v}^{2}$ as an explict function of $v$ as

$\displaystyle\phi_{v}^{2}=e^{-v/v_{0}}\theta(v)$ (50)

where $v_{0}$ is a positive constant. This choice of the K-essence scalar
field kinetic energy appears to be a reasonable one subject to the restriction
on $\phi_{v}^{2}$. The idea is to examine the effect of small departures from
the background spacetime on the generalized Vaidya spacetime, due to the
scalar field interactions. At the dynamical horizon, the mass function (31),
in units of the Schwarzschild mass $M$ becomes

$\displaystyle\frac{m(r(v),v)}{M}=(1-e^{-v/v_{0}})^{-1}.$ (51)

A plot the mass function in units of the Schwarzschild black hole mass $M$, as
an explicit function of $v$ (with $v_{0}=2$), is given in FIG. 2.

Figure 2: (Color online): The mass function (blue line), at the horizon, in
unit of $M$ vs $v$ for $v_{0}=2$.

The behaviour of the mass function of the generalized Vaidya spacetime is seen
to be significantly impacted in the region of large kinetic energy of the
scalar field, thus tallying with the behaviour of the location of the
dynamical horizon with this kinetic energy. This happens at vanishingly small
values of $v$, for the explicit choice depicted in eqn (50). Other reasonable
choices for the explicit functional form of the scalar field ought to lead to
the same conclusions.

###  IV.2 Reissner-Nordstrom Black Hole as background

Again, we choose $f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$ i.e., the physical
spacetime is Reissner-Nordstrom (RN) where $Q$ is the charge of the RN black
hole. In this case the line element of the emergent spacetime is

$\displaystyle
dS^{2}=[1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\phi_{v}^{2}]dv^{2}-2dvdr-r^{2}d\Omega^{2}$
(52)

and the mass function is

$\displaystyle m(v,r)=M-\frac{Q^{2}}{2r}+\frac{r}{2}\phi_{v}^{2}$ (53)

For this case the non-vanishing components of Einstein tensors are

$\displaystyle\mathcal{G}^{0}_{0}=\mathcal{G}_{1}^{1}=-\frac{Q^{2}}{r^{4}}-\frac{\phi_{v}^{2}}{r^{2}};~{}\mathcal{G}_{0}^{1}=\frac{2\phi_{v}\phi_{vv}}{r};~{}\mathcal{G}_{2}^{2}=\mathcal{G}_{3}^{3}=\frac{Q^{2}}{r^{4}}.$
(54)

Using the relation (53) we get the values of $\gamma$, $\rho$ and $P$ are

$\displaystyle\gamma=\frac{2\phi_{v}\phi_{vv}}{\kappa
r};~{}\rho=\frac{1}{\kappa
r^{2}}[\frac{Q^{2}}{r^{2}}+\phi_{v}^{2}]~{}and~{}P=\frac{Q^{2}}{\kappa r^{4}}$
(55)

which have also satisfied the weak and strong energy condition (39) and
dominant energy condition (35) as $\gamma\geq 0,~{}Q^{2}+r^{2}\phi_{v}^{2}\geq
Q^{2}\geq 0$.

In this case, from (28), the expansions are

$\displaystyle\Theta_{l}=\frac{1}{r}(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\phi_{v}^{2});~{}\Theta_{n}=-\frac{2}{r}$
(56)

and

$\displaystyle\mathcal{L}_{n}\Theta_{l}\mid_{F=0}=-\frac{1}{r^{2}}(1-\frac{Q^{2}}{r^{2}}-\phi_{v}^{2})<0$
(57)

At the horizon, $F(v,r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\phi_{v}^{2}=0$, we
get, exactly as in the Schwarzschild example, the restriction $\phi_{v}^{2}<1$
for an Reissner-Nordstrom electrovac black hole background. As before, this is
the stringent requirement on the K-essence scalar field configuration for the
emergent RN-Vaidya spacetime to admit a dynamical horizon. In this case,
$f(r)=\frac{(r-r_{+})(r-r_{-})}{r^{2}}$, where $r_{\pm}$ are respectively the
outer (event) and inner (Cauchy) horizon locations. For the dynamical horizon
we have

$\displaystyle\frac{(r(v)-r_{+})(r(v)-r_{-})}{r^{2}(v)}=\phi_{v}^{2}$
$\displaystyle\Rightarrow$ $\displaystyle
r_{\pm}(v)=\frac{1}{2}[(r_{+}+r_{-})\pm\sqrt{(r_{+}-r_{-})^{2}+4r_{+}r_{-}\phi_{v}^{2}}](1+\phi_{v}^{2}+...)$

Assuming that the black hole is far from extremality, $r_{+}>>r_{-}$ and also
that $\phi_{v}^{2}<1$, we have, the locations of the dynamical horizons as

$\displaystyle
r_{\pm}(v)=r_{\pm}(1+\phi_{v}^{2})\pm\frac{r_{+}r_{-}}{r_{+}-r_{-}}\phi_{v}^{2}\pm...$
(59)

Once again, we notice that the presence of the K-essence scalar field pushes
the dynamical horizon of the emergent metric to beyond the event horizon of
the background RN black hole. The dynamical horizon location ($r_{+}(v)$) in
units of radius of the event horizon of the background black hole $r_{+}$ can
be written as

$\displaystyle\frac{r_{+}(v)}{r_{+}}$ $\displaystyle=$
$\displaystyle\frac{1+(r_{-}/r_{+})}{2(1-\phi_{v}^{2})}$ (60) $\displaystyle+$
$\displaystyle\frac{1}{2(1-\phi_{v}^{2})}\sqrt{(1-\frac{r_{-}}{r_{+}})^{2}+4\frac{r_{-}}{r_{+}}\phi_{v}^{2}};~{}$
$\displaystyle\frac{m(r_{+}(v),v)}{r_{+}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\frac{r_{+}(v)}{r_{+}})(1+\phi_{v}^{2})$
$\displaystyle-$
$\displaystyle\frac{1}{2(r_{+}(v)/r_{+})}(\frac{r_{+}(v)}{r_{+}}-1)(\frac{r_{+}(v)}{r_{+}}-\frac{r_{-}}{r_{+}})$

The graphs of the dynamical horizon ($r_{+}(v)$) in units of $r_{+}$ vs
$\phi_{v}^{2}$ are given in FIG3, for the non-extremal Reissner-Nordstrom
background with two well-separated values of $r_{-}/r_{.}$.

Figure 3: (Color online): Plot of the DH location $r_{+}(v)$ in units of
$r_{+}$ vs $\phi_{v}^{2}$, with $\frac{r_{-}}{r_{+}}=0.1$ (red solid line) and
$\frac{r_{-}}{r_{+}}=0.001$ (blue dashed line).

The graphs show a remarkable similarity for both non-extremal ratios, showing
that the inner horizon has little influence on the location of the dynamical
horizon, stretched beyond the event horizon of the background black hole
spacetime.

Now, putting the expression of $\phi_{v}^{2}$ (40) in the equation (51),
yields the plot of the corresponding mass function at the horizon
$m(r_{+}(v),v)$ in unit of $r_{+}$, as an explicit function of $v$, this is
given in FIG4. Once again, the location of the inner horizon of background
appears to have little impact on the mass function.

Figure 4: (Color online): The mass function, at the DH, vs $v$ graph in unit
of $r_{+}$, considering $\frac{r_{-}}{r_{+}}=0.01$ (red solid line) and
$\frac{r_{-}}{r_{+}}=0.001$ (blue dashed line) and $v_{0}=2$.

Extremal Reissner-Nordstrom black hole background: In this case,
$r_{+}=r_{-}\equiv r_{0}$, so that $f(r)=(r-r_{0})^{2}/r^{2}$. It is easy to
see that, even though the background metric does not have a bifurcate horizon,
the DH of the corresponding emergent spacetime appears to have a bifurcate
structure, given by the locations

$\displaystyle\frac{(r(v)-r_{0})^{2}}{r^{2}(v)}=\phi_{v}^{2}$
$\displaystyle\Rightarrow r_{\pm}(v)=r_{0}(1\mp\phi_{v})^{-1}$ (62)

The outer horizon, once again has been pushed out compared to the event
horizon of the background black hole. The plot of $r_{+}(v)$ in units of
$r_{0}$ vs $\phi_{v}$ is given in FIG5.

Figure 5: (Color online): The extremal DH location ($r_{+}(v)$) in units of
$r_{0}$ vs. $\phi_{v}$ graph.

The mass function in this extremal case can be expressed in terms of the
explicit form of $\phi_{v}^{2}$, given by eqn (50)

$\displaystyle\frac{m(r_{+}(v),v)}{r_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{2(1-e^{-v/2v_{0}})}$ (63)

The plot of the mass function (53) in unit of $r_{0}$, at the extremal outer
DH, against $v$ is given in FIG. 6.

Figure 6: (Color online): The mass function at the outer DH location
($r_{+}(v)$) in unit of $r_{0}$ vs. $v$ graph considering $v_{0}=2$.

###  IV.3 de-Sitter Reissner-Nordstrom black hole as background

If we consider
$f(r)=(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2})$ where
$\Lambda$ is the cosmological constant i.e., the physical spacetime is
Reissner-Nordstrom-de Sitter (RNdS) ($\Lambda>0$) then the emergent spacetime
is

$\displaystyle
dS^{2}=[1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}-\phi_{v}^{2}]dv^{2}-2dvdr-r^{2}d\Omega^{2}$
(64)

and the mass function becomes

$\displaystyle
m(v,r)=M-\frac{Q^{2}}{2r}+\frac{\Lambda}{6}r^{3}+\frac{r}{2}\phi_{v}^{2}$ (65)

and non-zero components of Einstein tensors are

$\displaystyle\mathcal{G}^{0}_{0}=\mathcal{G}_{1}^{1}=-[\frac{Q^{2}}{r^{4}}+\Lambda+\frac{\phi_{v}^{2}}{r^{2}}];~{}\mathcal{G}_{0}^{1}=\frac{2\phi_{v}\phi_{vv}}{r};$
$\displaystyle\mathcal{G}_{2}^{2}=\mathcal{G}_{3}^{3}=\frac{Q^{2}}{r^{4}}-\Lambda.$
(66)

In this case the values of $\gamma$, $\rho$ and $P$ are

$\displaystyle\gamma=\frac{2\phi_{v}\phi_{vv}}{\kappa
r};~{}\rho=\frac{1}{\kappa r^{2}}[\frac{Q^{2}}{r^{2}}+\Lambda
r^{2}+\phi_{v}^{2}]~{}and~{}$ $\displaystyle
P=\frac{1}{\kappa}[\frac{Q^{2}}{r^{4}}-\Lambda]$ (67)

which have to satisfy the energy conditions (39) provided
$\frac{Q^{2}}{r^{4}}>\Lambda$ and dominant energy condition (35) as
$\gamma\geq 0,~{}Q^{2}+\Lambda r^{4}+r^{2}\phi_{v}^{2}\geq Q^{2}-\Lambda
r^{4}\geq 0$.

In this case, from (28), the expansions are

$\displaystyle\Theta_{l}=\frac{1}{r}(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}-\phi_{v}^{2});~{}\Theta_{n}=-\frac{2}{r}$
(68)

and

$\displaystyle\mathcal{L}_{n}\Theta_{l}\mid_{F=0}=-\frac{1}{r^{2}}(1-\frac{Q^{2}}{r^{2}}-\Lambda
r-\phi_{v}^{2})<0$ (69)

We have not given any plots in this case, because it is difficult to extract
the location of the dynamical horizon from the quartic equation which comes
out of the computations.

##  V Discussion

In the cosmologial context of dark energy, if the kinetic energy of the
K-essence scalar field $\phi_{v}^{2}$ is a real and positive constant $K$,
then the following scenarios arise: (a) for a Schwarzschild background, the
emergent spacetime analogoues to the Barriola-Vilenkin type spacetime gm1  for
a particular solution of K-essence scalar field. In this case, the global
monopole charge is replaced by the constant kinetic energy $(K)$ of the scalar
field with $K\in\\{0,1\\}$. (b) For Reissner-Nordstrom and de-Sitter Reissner-
Nordstrom backgrounds, the emergent spacetime is similar to the Robinson-
Trautman (RT) type spacetime gm2 ; gm3 , if $K$ is constrained to be unity for
a particular solution of K-essence scalar field. The event horizons are
changed in the above three cases in the pres- ence of constant kinetic energy
of the K-essence scalar field.

We hasten to add, however, that the link discerned by us between a specific
K-essence emergent gravity model and generalized Vaidya models characterizing
gravitational collapse of null fluids with a large class of mass functions is
interesting from a purely gravitational theory standpoint, rather than in the
somewhat mystifying cosmological context of dark energy whose very existence
may now be in some doubt subir , on the basis of the latest analysis of data
from the Planck consortium planck . However, the connection found in this
paper may have uses in another context, namely the context of analogue gravity
models where gravitational phenomena difficult to observe in real spacetime
may still be scrutinized in terrestrial laboratories.

The most striking of analogue models is the class of acoustic black hole
analogues visser  where inviscid, barotropic fluids with specific flow
parameters, perturbed by linear acoustic perturbations, have been shown to
reproduce experimentally accessible laboratory analogues of Hawking radiation
unruh , superradiance bm2003  and inertial frame dragging with Lense-Thirring
precession cgm2017 . Some of these analogue phenomena have been claimed to
have been observed stein , wein . However, the context of such observation is
not always within the theoretical premise of vanishing viscosity set down in
the incipient works. In fact, nonzero, albeit small viscosity has been shown
visser  to lead to violation of Lorentz invariance, and hence the
impossibility of extracting a cogent acoustic metric in these more realistic
situations. Even so, it has been shown that superradiance may still be viable
if the inclusion of effects due to it can be treated perturbatively in the
viscosity coefficient oindrila . In this case, it is most meaningful to
investigate whether an approximate analogue acoustic metric can indeed be
constructed. This is perhaps an arena where the formal connection found in the
text above can be of substantial utility, since the configurations of
perturbations in such fluids have similarities to the scalar fields in the
K-essence scenarios discussed above. We hope to report on this interesting
relation in the near future.

Acknowledgement: The authors would like to thank the referees for illuminating
suggestions to improve the manuscript.

## References

  * (1) M.Visser,C.Barcelo and S.Liberati, Gen.Rel.Grav. 34 1719 (2002).
  * (2) R.J. Scherrer, Phys.Rev.Lett.93 011301 (2004).
  * (3) L.P.Chimento, Phys.Rev.D69 123517 (2004).
  * (4) E.Babichev, V.Mukhanov and A.Vikman, JHEP 09, 061 (2006).
  * (5) E.Babichev,M.Mukhanov and A.Vikman, JHEP 0802 101 (2008).
  * (6) E.Babichev,M.Mukhanov and A.Vikman, WSPC-Proceedings, February 1, 2008.
  * (7) Alexander Vikman, K-essence: Cosmology, causality and Emergent Geometry, Dissertation an der Fakultat fur Physik,Arnold Sommerfeld Center for Theoretical Physics, der Ludwig-Maximilians-Universitat Munchen, Munchen, den 29.08.2007.
  * (8) M.Born and L.Infeld,Proc.Roy.Soc.Lond A144(1934) 425
  * (9) W.Heisenberg, Zeitschrift fur Physik A Hadrons and Nuclei 113 no.1-2 (1939)
  * (10) P.A.M.Dirac, Royal Society of London Proceedings Series A 268 (1962) 57.
  * (11) D.Gangopadhyay and Goutam Manna, Euro.Phys.Lett. 100 49001 (2012).
  * (12) Goutam Manna and Debashis Gangopadhyay, Eur. Phys. J. C 74 2811 (2014).
  * (13) Goutam Manna, Bivash Majumder and Ashoke Das, Eur. Phys. J. Plus 135, 107, (2020)
  * (14) Viqar Husain, Phys. Rev. D 53 4, (1996)
  * (15) Anzhong Wang and Yumei Wu, General Relativity and Gravitation, 31 1, (1999).
  * (16) P.C.Vaidya, Physical Review, 83 1, (1951)
  * (17) P.C.Vaidya and I.M.Pandya, Progress of Theoretical Physics, 35 1, (1966)
  * (18) P.C.Vaidya, Physical Review, 174 5, (1968)
  * (19) P.C.Vaidya, APJ, 144 3, (1966)
  * (20) P.C.Vaidya, General Relativity and Gravitation, 31 1, (1999).
  * (21) Jerry B. Griffiths and Jiri Podolsky, Exact Space-Times in Einstein’s General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (2009).
  * (22) Hawking, S. W ., and Ellis, G. F. R. (1973) The Large Scale Structure of Spacetime, Cambridge University Press, Cambridge
  * (23) Abhay Ashtekar and Badri Krishnan, Dynamical Horizons: Energy, Angular Momentum, Fluxes, and Balance Laws, Phys. Rev. Lett., 89, 26, (2002).
  * (24) A. Ashtekar and B. Krishnan, Dynamical horizons and their properties, Phys. Rev. D 68, 104030 (2003).
  * (25) Abhay Ashtekar and Badri Krishnan, Isolated and Dynamical Horizons and Their Applications, Living Reviews in Relativity, 2004
  * (26) Sean A. Hayward, General laws of black-hole dynamics, Phys. Rev. D 49, 6467, (1994)
  * (27) Matthias Blau, “Lecture Notes on General Relativity”, [http://www.blau.itp.unibe.ch/GRLecturenotes.html], (2020)
  * (28) Eric Poisson, A Relativist’s Toolkit-The Mathematics of Black-Hole Mechanics, Cambridge University press, (2004)
  * (29) J T Nielsen, A Guffanti and S Sarkar, Marginal evidence for cosmic acceleration from Type Ia supernovae, Sci.Rep. 6 (2016) 3559.
  * (30) Planck Collaboration : P R Ade et. al., Planck 2015 results. XX. Constraints on inflation , Astron. Astrophys.( Inflation-driver : September 2017).
  * (31) M. Visser. Acoustic black holes: Horizons, ergospheres, and Hawking radiation in: Class.Quant.Grav. 15, pp. 1767–1791 (1998). Preprint arXiv: 9712010 [gr-qc]
  * (32) W. M. Unruh, Experimental black hole evaporation, Phys. Rev. Lett. 46, pp. 1351–1353 (1981).
  * (33) S. Basak and P. Majumdar, ‘Superresonance’ from a rotating acoustic black hole, Class. Quant. Grav. 20, 3907 (2003). e-print-arXiv: gr-qc/0203059 [gr-qc] .
  * (34) C. Chakraborty, O. Ganguly, and P. Majumdar, Intertial Frame Dragging in an Acoustic Analogue spacetime, Ann. Phys. (Berlin) 529, 1700231 (2017), e-Print: arXiv:1510.01436 [gr-qc] .
  * (35) J. R. Munõz de Nova, K. Golubkov, V. I. Kolobov, J. Steinhauer, Observation of thermal Hawking radiation and its temperature in an analogue black hole, Nature 569, 688-691 (2019). e-Print: arXiv:1809.00913 [gr-qc].
  * (36) V. Cardoso, A. Coutant, M. Richartz, and S. Weinfurtner, Detecting Rotational Superradiance in Fluid Laboratories, Phys. Rev. Lett., 117(27), 271101 (2016).
  * (37) O. Ganguly, Acoustic superradiance in a slightly viscous fluid, e-print arXiv: 1705.04395[gr-qc].