# Conformal Dilaton Gravity and Warped Spacetimes in 5D

Reinoud Jan Slagter  Asfyon, Astronomisch Fysisch Onderzoek Nederland
and former
University of Amsterdam, The Netherland<EMAIL_ADDRESS>
###### Abstract

An exact time-dependent solution of a black hole is found in conformally
invariant gravity on a warped Randall-Sundrum spacetime, by writing the metric
$g_{\mu\nu}=\omega^{\frac{4}{n-2}}\tilde{g}_{\mu\nu}$. Here
$\tilde{g}_{\mu\nu}$ represents the ”un-physical” spacetime and $\omega$ the
dilaton field, which will be treated on equal footing as any renormalizable
scalar field. In the case of a five-dimensional warped spacetime, we write
${}^{(4)}{\tilde{g}_{\mu\nu}}=\bar{\omega}^{2}{{}^{(4)}\bar{g}_{\mu\nu}}$ Both
$\omega$ and $\bar{\omega}$ can be used to describe the different notion the
in-going and outside observers have of the Hawking radiation by using
different conformal gauge freedom. The disagreement about the interior of the
black hole is explained by the antipodal map of points on the horizon. The
free parameters of the solution can be chosen in such a way that
$\bar{g}_{\mu\nu}$ is singular-free and topologically regular, even for
$\omega\rightarrow 0$. It is remarkable that the 5D and 4D effective field
equations for the metric components and dilaton fields can be written in
general dimension $n=4,5$. It is conjectured that, in context of quantization
procedures in the vicinity of the horizon, unitarity problems only occur in
the bulk at large extra-dimension scale. The subtraction point in an effective
theory will be in the UV only in the bulk, because the use of a large extra
dimension results in a fundamental Planck scale comparable with the
electroweak scale.

  * Dec 2020

Keywords: conformal invariance, dilaton field, black hole complementarity,
brane world models

##  1 Introduction

Our universe could be a 4-surface (”brane”) embedded in a 5-dimensional
spacetime, where the extra dimension represents the ”bulk”. We will focus on
the ”warped variant of these so-called brane-worlds gravity models, originally
introduced by Randall and Sundrum (RS)[1, 2] and extended by Shiromizu, et
al.[3]. See also the overview work of Maartens[4]. At the same time, Arkani-
Hamed, et al. (ADD)[5, 6] considered also higher dimensional models which put
forward the idea that a large volume for the compact extra dimensions will
lower the Planck scale. Their models are not restricted to 5 dimensions, as is
the case in the RS model with one ”preferred” extra dimension. The advantage
of these models is that they could solve the hierarchy problem, i.e., why
there is such a huge gap between the electroweak and Planck scale. It has also
promising results for achieving late-time acceleration of the universe without
dark energy or matter (DGP models)[7].However, one should be careful in
changing general relativity theory (GRT). If one adapt that Nature will choose
the most simple Einstein-Hilbert (EH) action, then, for example, the
cosmological constant should not be included in the EH action. One avoids in
this way the huge discrepancy between the contribution from zero-point
fluctuations in quantum field theory and the predicted value in GRT (some 120
orders of magnitude).

Another unsolved issue, is the complementarity near a black hole. In the
vicinity of the horizon, curvature will be huge, so quantum effects will
become important. Up till now, no convincing theory of quantum gravity is
available. Many attempts were made in order to make a renormalizable and
unitary quantum gravity model. I refer to the standard works on these
subjects, for example Parker, et al[8] and the fundamental work of ’t Hooft,
et al.[9].

A well know method in constructing a renormalizable model[10], is to add
fourth order derivative terms in the curvature tensor (Euler-term). However,
one looses unitarity. Also the ”old” effective field theory (EFT) has its
problems. One ignores what is going on at high energy. In order to solve the
anomalies one encounters in calculating the effective action, one can apply
the so-called conformal dilaton gravity (CDG) model[11, 12, 13, 14, 16, 18,
17]. CDG is a promising route to tackle the problems arising in quantum
gravity model, such as the loss of unitarity close to the horizon. One assumes
local conformal symmetry, which is spontaneously broken (for example by a
quartic self-coupling of the Higgs field). Changing the symmetry of the action
was also successful in the past, i.e., in the SM of particle physics. A
numerical investigation of the coupling was recently done[19]. The key feature
in CDG, is the splitting of the metric tensor
$g_{\mu\nu}=\omega^{\frac{4}{n-2}}(x)\tilde{g}_{\mu\nu}$, with $\omega$ the
dilaton field. Applying perturbation techniques (and
renormalization/dimensional regularization), in order to find the effective
action and its divergencies, one first integrate over $\omega$ (shifted to the
complex contour), considered as a conventional renormalizable scalar field and
afterwards over $\tilde{g}_{\mu\nu}$ and matter fields. The dilaton field is
locally unobservable. It is fixed when we choose the global spacetime and
coordinate system. If one applies this principle to a black hole spacetime,
then the energy-momentum tensor of $\omega$ determines the Hawking radiation.
We shall see that it can be calculated in our model. When $\tilde{g}_{\mu\nu}$
is flat, then the handling of the anomalies simplifies considerably[33]. When
$\tilde{g}_{\mu\nu}$ is non-flat, the problems are more deep-seated.

In this context, the modification of GRT by an additional spacetime dimension
could be an alternative compromise, because Einstein gravity on the brane will
be modified by the very embedding itself and opens up a possible new way to
address the dark energy problem[21, 22]. These models can be applied to the
standard Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime and the
modification on the Friedmann equations can be investigated[25]. Recently,
Maldacena, et al.[26], applies the RS model to two black hole spacetimes and
could construct a traversable macroscopic wormhole solution by adding only a
5D U(1) gauge field (see also Maldacena[27]). However, an empty bulk would be
preferable. In stead, one can investigate the contribution of the projected 5D
Weyl tensor on the 4D brane. It carries information of the gravitational field
outside the brane. If one writes the 5D Einstein equation in CDG setting, it
could be possible that an effective theory can be constructed without an UV
cutoff, because the fundamental scale $M_{5}$ can be much less than the
effective scale $M_{Pl}$ due to the warp factor. The physical scale is
therefore not determined by $M_{Pl}$

In section 2 we review the 5D RS model. In section 3 we outline the principles
of CDG. In section 4 we apply the model to a black hole spacetime. In section
5 we present a physical interpretation. All the calculations were done by
computer-aided software.111the interested reader can obtain from the author
the Maple file with all the calculations used in this manuscript

##  2 Brane-world models

###  2.1 The Randall-Sundrum model

In the brane-world scenario under consideration (the RS1 model), the
4-dimensional world is described by a domain wall $({{}^{(4)}{\cal
M}},{{}^{(4)}g_{\mu\nu}})$ in a 5-dimensional spacetime $({{}^{(5)}{\cal
M}},{{}^{(5)}g_{\mu\nu}})$. All the standard model fields are confined to the
brane. On the other hand, gravity can propagate in the full spacetime. The
Planck scale is now an effective coupling constant, describing gravity on
scales much larger than the extra dimension ($d$) and related to the
fundamental scale via the volume of the extra dimension $M_{pl}^{2}\sim
M_{4+d}^{2+d}l^{d}$, with L the the length scale of the extra dimension. If
$L\sim\frac{1}{M_{pl}}$, then $M_{4+d}\sim M_{pl}$. If the extra dimensional
volume is far above the Planck scale, then the true fundamental scale
$M_{4+d}$ can be much less than the effective scale $\sim 10^{19}GeV$. From a
different point of view, one could say that the dynamics of the higher
dimensional gravitational fields influence the brane dynamics. We used the
notations of Shiromizu, et al.[3] and Maartens[4]. The 5-dimensional Einstein
equations are

${{}^{(5)}}{G_{\mu\nu}}=-\Lambda_{5}{{}^{(5)}g_{\mu\nu}}+\kappa_{5}^{2}{{}^{(5)}T_{\mu\nu}},$
(1)

with $\kappa_{5}^{2}=8\pi G_{5}=\frac{8\pi}{M_{5}^{3}}$. The negative bulk
tension $\Lambda_{5}$ is offset by the positive brane tension $\lambda$
(vacuum energy). One writes the 5D metric

${{}^{(5)}g_{\mu\nu}}={{}^{(4)}g_{\mu\nu}}+n_{\mu}n_{\nu}\qquad{{}^{(5)}ds^{2}}={{}^{(4)}g_{\mu\nu}}(x^{\alpha},y)dx^{\mu}dx^{\nu}+\Gamma(y)dy^{2},$
(2)

with $n^{\mu}$ the unit normal to the brane and $y$ the extra dimension. The
extrinsic curvature of $y=const$ surfaces is given by

$K_{\mu\nu}={{}^{(4)}g_{\mu}^{\alpha}}{{}^{(5)}}\nabla_{\alpha}n_{\nu}.$ (3)

The Gauss-Codazzi equation determines the change of $K_{\mu\nu}$ along
$y=const.$

${{}^{(4)}\nabla}_{\mu}K^{\mu}_{\nu}-{{}^{(4)}\nabla}_{\nu}K={{}^{(5)}R_{\mu\sigma}}{{}^{(4)}{g_{\nu}^{\mu}}}n^{\sigma}$
(4)

The relation between the 4D and 5D Ricci tensors is given by the Gauss
equation

${{}^{(4)}R_{\mu\nu}}={{}^{(5)}{R_{\rho\sigma}}}{{}^{(4)}}g_{\mu}^{\rho}{{}^{(4)}}g_{\nu}^{\sigma}-{{}^{(5)}{R^{\alpha}_{\beta\gamma\delta}}}n_{\alpha}{{}^{(4)}}g_{\mu}^{\beta}n^{\gamma}{{}^{(4)}}g_{\nu}^{\delta}+KK_{\mu\nu}-K_{\mu}^{\alpha}K_{\nu\alpha}.$
(5)

The second term on the right-hand side is Eq.(1) is zero, if there are no
matter fields in the bulk $({{}^{(5)}T_{\mu\nu}}=0$). Using the Israel’s
junction conditions and by imposing $Z_{2}$-symmetry, one obtains the
effective induced Einstein equations on the brane[3, 4]

${{}^{(4)}G_{\mu\nu}}=-\Lambda_{eff}{{}^{(4)}g_{\mu\nu}}+\kappa_{4}^{2}{{}^{(4)}T_{\mu\nu}}+\kappa_{5}^{4}{\cal
S}_{\mu\nu}+\frac{2}{3}\kappa_{5}^{2}{\cal F}_{\mu\nu}-{\cal E}_{\mu\nu},$ (6)

with ${{}^{(4)}T_{\mu\nu}}$ the energy-momentum tensor on the brane, ${\cal
S}_{\mu\nu}$ the quadratic contribution of the energy-momentum tensor
${{}^{(4)}T_{\mu\nu}}$ arising from the extrinsic curvature terms in the
projected Einstein tensor,

${\cal
S}_{\mu\nu}=\frac{1}{12}{{}^{(4)}}T{{}^{(4)}}T_{\mu\nu}-\frac{1}{4}{{}^{(4)}}T_{\mu\alpha}{{}^{(4)}}T{{}^{\alpha}_{\nu}}+\frac{1}{24}{{}^{(4)}g_{\mu\nu}}\Bigl{(}3{{}^{(4)}}T_{\alpha\beta}{{}^{(4)}}T^{\alpha\beta}-{{}^{(4)}}T^{2}\Bigr{)}$
(7)

and ${\cal F}_{\mu\nu}$ is the correction term from the bulk energy-momentum
tensor,

${\cal
F}_{\mu\nu}={{}^{(5)}T_{\alpha\beta}}{{}^{(4)}g_{\mu}^{\alpha}}{{}^{(4)}g_{\nu}^{\beta}}+{{}^{(4)}g_{\mu\nu}}\Bigl{(}{{}^{(5)}T_{\alpha\beta}}n^{\alpha}n^{\beta}-\frac{1}{4}{{}^{(5)}T}\Bigr{)}.$
(8)

In the RS model, the effective cosmological constant is
$\Lambda_{eff}=\frac{1}{2}(\Lambda_{5}+\frac{\kappa_{5}^{4}}{6}\lambda^{2})$,
with $\lambda$ the brane tension, $\sim\frac{M_{pl}^{2}}{l^{2}}$ and $l$ the
effective size of the extra dimension. Because there is no deviation of
Newton’s law down to ${\cal O}(10^{-1})$, we have
$\lambda>(1TeV)^{4},M_{5}>10^{5}TeV$. The 4 dimensional cosmological constant
is zero, when the RS balance between $\Lambda_{5}$ and $\lambda$ is broken.
The fundamental gravity scale is lowered, possibly even down to the
electroweak scale $\sim TeV$. For an empty bulk, ${\cal F}_{\mu\nu}=0$. The
last term in Eq.(6) represents the projected Weyl term and is a part of the 5D
Weyl tensor that carries information of the gravitational field outside the
brane.

${\cal
E}_{\mu\nu}={{}^{(5)}}C^{\alpha}_{\beta\rho\sigma}n_{\alpha}n^{\rho}{{}^{(4)}}g_{\mu}^{\beta}{{}^{(4)}}g_{\nu}^{\sigma}.$
(9)

Note that ${{}^{(4)}T_{\mu\nu}}$ is the energy-momentum tensor of the
particles and fields confined to the brane (plus the term
$\lambda{{}^{(4)}}g_{\mu\nu}$). The extrinsic curvature of the brane can then
be expressed as

$K_{\mu\nu}=-\frac{1}{2}\kappa_{5}^{2}\Bigl{(}{{}^{(4)}T_{\mu\nu}}-\frac{1}{3}{{}^{(4)}}T\Bigr{)}.$
(10)

The 4D contracted Bianchi identity becomes (for empty bulk)

${{}^{(4)}}\nabla^{\mu}{\cal
E}_{\mu\nu}=\kappa_{4}^{2}{{}^{(4)}\nabla^{\mu}T_{\mu\nu}}+\kappa_{5}^{4}\nabla^{\mu}{\cal
S}_{\mu\nu}.$ (11)

It relates ${{}^{(4)}}\nabla^{\mu}{\cal E}_{\mu\nu}$ to the several ”matter
terms” on the right-hand side of Eq.(11) and expresses the fact that 4D
spacetime variation in the matter-radiation can source KK-modes. We shall see
in an application in section 4, that this identity will lead to a third-order
partial differential equation, which forms with the effective Einstein
equations, a closed system of pde’s. Further, Eq.(4) becomes

${{}^{(4)}\nabla}_{\mu}K^{\mu}_{\nu}-{{}^{(4)}\nabla}_{\nu}K=\kappa_{5}^{2}{{}^{(4)}T_{\rho\sigma}}n^{\sigma}{{}^{(4)}}g_{\nu}^{\rho}.$
(12)

One must realize that in the conformal dilaton application,
${{}^{(4)}T_{\mu\nu}}$ contains also a contribution from the dilaton field
$\omega$, which is a ”scale”-term by the splitting
$g_{\mu\nu}=\omega^{2}\tilde{g}_{\mu\nu}$ (section 3). The dilaton will play
the role of a ”scalar field”. In general, the covariant splitting which is
applied in the warped spacetime model, has the advantage that it shows a clear
insight into the interplay between the 4D and 5D gravitational fields.
However, one needs to boundary conditions on the brane for a consistent
dynamical evolution.

###  2.2 Application to the FLRW spacetime

In a former study we investigated the brane-world model of Randall and Sundrum
on the warped 5 dimensional spacetime[23]

$ds^{2}=W(t,r,y)^{2}\Bigl{[}e^{2\gamma(t,r)-2\psi(t,r)}(-dt^{2}+dr^{2})+e^{2\psi(t,r)}dz^{2}+\frac{r^{2}}{e^{2\psi(t,r)}}d\varphi^{2}\Bigr{]}+\Gamma(y)dy^{2}.$
(13)

In first instance, we will take $\Gamma(y)=1$. From the 5D Einstein equations

${}^{(5)}{G_{\mu\nu}}+\Lambda_{5}^{(5)}{g_{\mu\nu}}=0,$ (14)

one obtains for the warpfactor

$W(t,r,y)=W_{1}(t,r)W_{2}(y),$ (15)

with

$W_{2}(y)=e^{\sqrt{-\frac{1}{6}\Lambda_{5}}(y-y_{0})}$ (16)

and

$W_{1}(t,r)^{2}=\pm\frac{1}{\tau
r}\Bigl{(}d_{1}e^{\sqrt{2\tau}t}-d_{2}e^{-\sqrt{2\tau}t}\Bigr{)}\Bigl{(}d_{3}e^{\sqrt{2\tau}r}-d_{4}e^{-\sqrt{2\tau}r}\Bigr{)},$
(17)

where $\tau$ and $d_{i}$ are constants. One recognizes the well-known RS
warpfactor $W_{2}(y)$. It turns out that $W_{1}$ cannot be isolated from the
effective 4D brane equations.

###  2.3 Application to a black hole spacetime

Let us now apply the equation on the spacetime (a 4D BTZ model[29, 30, 31, 24,
25])

$ds^{2}=W(t,r,y)^{2}\Bigl{[}-N(t,r)^{2}dt^{2}+\frac{1}{N(t,r)^{2}}dr^{2}+dz^{2}+r^{2}(d\varphi+N_{\varphi}(t,r)dt)^{2}\Bigr{]}+dy^{2}.$
(18)

Again, from the 5D Einstein equations, one obtains for
$W(t,r,y)=W_{1}(t,r)W_{2}(y)$, with $W_{2}(y)$ the same expression, i. e.,
Eq.(16). However, $W_{1}(t,r)$ cannot be isolated. One needs the 4D effective
Einstein equations. It will become clear, that $W_{1}$ is related to the scale
of spacetime of the geometry near horizon region. In order to make the
connection with the CDG model, one would like to write the metric as

$ds^{2}=W(t,r,y)^{2}\Bigl{[}-N(t,r)^{2}dt^{2}+\frac{1}{N(t,r)^{2}}dr^{2}+dz^{2}+r^{2}(d\varphi+N_{\varphi}(t,r)dt)^{2}+dy^{2}\Bigr{]}.$
(19)

From the 5D Einstein equations, however, we then obtain that $W_{2}(y)=const.$

##  3 Conformal dilaton gravity

###  3.1 The basics

Where Einstein’s theory of general relativity treats the large-scale behavior
of gravity, the Standard Model (SM) of particle physics is the theoretical
framework where, on small-scale, the other three forces of nature are
explained. Combining these two theories leads to fundamental difficulties
regarding the nature of spacetime. It is one of the greatest challenges of
theoretical physics. It is believed that the unification has taken place close
to the Planck scale, $1.6.10^{-35}m$, so far out of reach of experimental
verification. It is conjectured, however, that close to the horizon of a black
hole, quantum gravity effects will be manifest. So observations on black holes
could be a way to collect quantum-gravitational information. It is well known
that Einstein gravity, when quantized, is unitary, but nonrenormalizable[9].
In general, there is no obvious way to attain renormalizability without
violating unitarity. Higher order derivative theories turn out to be
renormalizable at the one-loop quantum level, but lacks unitarity of the
S-matrix. In the low energy domain ( well below the UV cutoff), one can
construct an effective field theory that can be quantized and is
renormalizable. However, just as in Standard Model (SM), one needs gauge-
fixing terms as well as Faddeev-Popov ghost fields. However, this route runs
into problems when one tries to incorporate dark mass, dark energy and
singularities. CDG could offer an alternative. Conformal gravity is part of
the covariant approach to quantum gravity. There are several methods to attack
the consequences of conformal symmetyry at the quantum level. We follow the
method initiated by ’t Hooft[11, 12, 13]. See also Alvarez, et al.[17] and
Codello, et al.[18] The fundamental calculation of the divergences of the
action involving the gravitational field, where already done long time ago[9,
10]. See also the overview work of Parker[8]. Let us summarize the CDG
idea[11, 12]. One can split the spacetime (in the 5D case)

${}^{(5)}{g_{\mu\nu}}=\omega^{4/3}{{}^{(5)}{\tilde{g}_{\mu\nu}}},$ (20)

where $\omega$ is a dilaton metascalar, encoding all the scale dependencies.
One can call the ”un-physical” $\tilde{g}_{\mu\nu}$ a metatensor, i.e., it
transforms as a tensor, but contains powers of the Jocobian of the coordinate
transformation. One can easily prove that the action (in n dimensions and
without matter terms)

$\displaystyle S=\int
d^{n}x\sqrt{-\tilde{g}}\Bigl{[}\frac{1}{2}\xi\omega^{2}\tilde{R}+\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}\omega\partial_{\nu}\omega+\Lambda\kappa^{\frac{4}{n-2}}\xi^{\frac{n}{n-2}}\omega^{\frac{2n}{n-2}}\Bigr{]}$
(21)

is conformal invariant under

$g_{\mu\nu}\rightarrow\Omega^{\frac{4}{n-2}}g_{\mu\nu},\quad\omega\rightarrow\Omega^{-\frac{n-2}{2}}\omega.$
(22)

The coupling constant $\Lambda$ is dimensionless in any dimension. It fulfills
also the off-shell Ward identity. The constant $\xi=\frac{n-2}{4(n-1)}$. One
re-scaled $\omega\rightarrow\kappa\sqrt{\xi}\omega$, with
$\frac{1}{\kappa^{2}}=M_{pl}^{n-2}=\frac{1}{16\pi G_{N}}$. So Newton’s
constant has disappeared completely. Variation of the action leads to the
field equations

$\displaystyle\xi\omega\tilde{R}-\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\omega-\frac{2n}{n-2}\Lambda\kappa^{\frac{4}{n-2}}\xi^{\frac{n}{n-2}}\omega^{\frac{n+2}{n-2}}=0$
(23)

and

$\displaystyle\omega^{2}\tilde{G}_{\mu\nu}=T_{\mu\nu}^{\omega}-\Lambda\tilde{g}_{\mu\nu}\kappa^{\frac{4}{n-2}}\xi^{\frac{2}{n-2}}\omega^{\frac{2n}{n-2}},$
(24)

with

$\displaystyle
T_{\mu\nu}^{\omega}=\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\omega^{2}-\tilde{g}_{\mu\nu}\tilde{\nabla}^{2}\omega^{2}+\frac{1}{\xi}\Bigl{(}\frac{1}{2}\tilde{g}_{\alpha\beta}\tilde{g}_{\mu\nu}-\tilde{g}_{\mu\alpha}\tilde{g}_{\nu\beta}\Bigr{)}\partial^{\alpha}\omega\partial^{\beta}\omega.$
(25)

The covariant derivatives are taken with respect to $\tilde{g}_{\mu\nu}$.

###  3.2 Notes on anomalies

In general, one conjectures that conformal invariance is an exact symmetry,
comparable with the Higgs mechanism. The effective action will stay locally
conformal invariant after the $\omega$ integration. The integration over
$\tilde{g}_{\mu\nu}$ (and matter terms) will not be performed before imposing
conformal constraints222We have still the freedom of $\Omega$ (Eq.(22)). So
different observers experience different $\bar{\omega}$ and also different
matter distribution (note that the Ricci tensor is not invariant under
Eq.(22):
$R\rightarrow\frac{1}{\Omega^{2}}R-\frac{6}{\Omega^{3}}\nabla^{\mu}\partial\Omega$).
. The problem is how to handle the functional integration over
$\tilde{g}_{\mu\nu}$ (the ”curved background anomaly”). On a flat background,
the conformal anomalies can be handled properly[13]. The choice of $\omega$ is
then unique. $\omega$ is a renormalizable field which could create the
spacetime twofold: an infalling and outside observer use different ways to fix
the conformal gauge in order to overcome unitarity problems. For non-flat
$\tilde{g}_{\mu\nu}$, standard QM does not apply, because there is no notion
of energy in a conformal invariant theory. Further, if one adds matter sources
to the Lagrangian, for example a scalar gauge field $(\Phi,A_{\mu})$, it will
break the conformal invariance, i.e., it spoils the tracelessness of the
energy momentum tensor. However, if one shifts $\omega$ to the complex contour
($\omega^{2}\rightarrow-\frac{\omega^{2}}{\kappa^{2}}$), then the scalar field
and dilaton field can be handled on equal footing (in order to obtain the same
unitarity properties). Using dimensional regularization and renormalization,
it was found that the scalar field Lagrangian

${\cal
L}=\sqrt{-g}\Bigl{(}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{12}R\phi^{2}\Bigr{)},$
(26)

generates an effective action, whose divergent part is

$\Gamma^{div}\sim\frac{\sqrt{-g}}{4-n}(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^{2}).$
(27)

One can apply this result directly to the Lagrangian Eq.(21), where $R$ and
$g$ are replaced by $\tilde{R}$ and $\tilde{g}$. So the effective action
becomes

$S^{eff}=C\int
d^{n}x\frac{\sqrt{-\tilde{g}}}{8\pi^{2}(4-n)}\Bigl{(}\tilde{R}^{\mu\nu}\tilde{R}_{\mu\nu}-\frac{1}{3}\tilde{R}^{2}\Bigl{)}.$
(28)

We will treat now $\tilde{g}_{\mu\nu}$ classical, so we will have no unitarity
problem. Further, we have still the infinite coefficient in front of the
effective action. It is conjectured[11], that no counter term is needed in
order to make the integral finite. The divergent coefficient in front of
$S^{eff}$ will not be a problem when $\tilde{g}_{\mu\nu}$ is treated
classical: $N$ and $N^{\varphi}$ can take large values and in the limit that
$\frac{C}{4-n}\rightarrow\infty$, quantum fluctuations would vanish and only
the classical parts would remain. Only $\omega$ (and matter fields) are
described by renormalizable Lagrangians. This means that $\tilde{g}_{\mu\nu}$
is flat beyond the Planck scale, but gets renormalized by $\omega$ and matter
fields at much lower scales.

Quantum amplitudes are obtained by integrating the exponent of the entire
action over all the components of $g_{\mu\nu}$, now written as
$\omega^{2}\tilde{g}_{\mu\nu}$ (or, in the 5D case,
${}^{(5)}{g_{\mu\nu}}=\omega^{4/3}{{}^{(5)}{\tilde{g}_{\mu\nu}}}$). The
integration over $\omega$ must be along a complex contour[11]. Further, we
consider the brane spacetime
${}^{(5)}{\tilde{g}_{\mu\nu}}=^{(4)}{\tilde{g}_{\mu\nu}}+n_{\mu}n_{\nu}$ with
now
${}^{(4)}{\tilde{g}_{\mu\nu}}=\bar{\omega}^{2}{{}^{(4)}{\bar{g}_{\mu\nu}}}$.
Our $\bar{g}_{\mu\nu}$ is given by the up-lifted BTZ black hole spacetime. The
evolution equation for $N$ will contain no intrinsic mass or length scale.
(for $\Lambda=0$) and reflects the fact that we are dealing here with
gravitational waves only. The scale is determined by $\omega$ and
$\bar{\omega}$333We shall see that in the vacuum case, we are dealing with two
different scales, $\omega$ and $\bar{\omega}$. Note, however, that
$\omega^{4/3}=\bar{\omega}^{2}$ as expected. One says that $\omega$ is part of
the gravitational sector. It fixes the scale at which quantum gravity effects
becomes important.

It is obvious, that close to the horizon and ergo-sphere of the spacetime
Eq.(26), the curvature is high and quantum effects will become important. We
already mentioned that the Lagrangian Eq.(21) with $\Lambda=0$ contains no
intrinsic mass or length scale. This is comparable with the classical
Lagrangian of the SM, if one omits the Higgs mass term. So it is challenging
to apply a spontaneous symmetry breaking mechanism of conformal invariance by
adding a mass term
$-\frac{1}{2}\sqrt{-\bar{g}}m^{2}\bar{\omega}^{2}\bar{\phi}^{2}$ (through the
quartic self-coupling of the Higgs field) and other contributions[13], just as
the Higgs mechanism in SM. Newton’s constant then reappears by the re-scaling
$\omega\rightarrow\kappa\sqrt{\xi}\omega$. 444For the renormalization problems
caused by the mass term, we refer to the literature. It is conjectured that it
will not spoil conformal invariance of the effective action. The application
to our model is currently under investigation. One conjectures that the
effective action (with matter fields and on the black hole spacetime)) will be
locally spontaneously broken. [33]

Now the RS idea can be applied: the renormalized counter terms in $S^{eff}$
will cause unitarity problems only in the bulk at the large extra-dimension-
scale ($l$), but are invisible on the brane. This means that the substraction
point will be in the UV region as seen by an observer in the bulk. Note that
this is possible, because we have in the induced 4D Einstein equations the KK-
contribution from the bulk, i.e., the electric part of the Weyl tensor ${\cal
E}_{\mu\nu}$. From the brane observer, these KK corrections are non-local,
since they incorporate 5D gravity wave modes. Note that if the extra
dimensional volume is significantly above the Planck scale, than the true
fundamental scale $M_{4+d}$ can be much less than the effective scale
$M_{pl}\sim 10^{19}GeV$. In the RS-1 model, $M_{5}^{3}=\frac{M_{pl}}{l}$, with
$l$ the effective length scale scale of the infinite large extra dimension
$y$. The warp factor (Eq.(16)) then causes a finite contribution to the 5D
volume

$\int d^{5}x\sqrt{-^{(5)}g}\sim\int d^{4}x\int dyW_{2}\sim l\int d^{4}x.$ (29)

So the effective size of the extra dimension probed by the 5D graviton, is
$l$. Note that the negative bulk tension is taken
$\Lambda_{5}\sim-\frac{1}{l^{2}}\sim-\mu^{2}$ in the RS-1 model, with $\mu$
the energy scale. So for low energy scale it prevents gravity leaking into the
extra dimension. As by-product, the model solves the hierarchy problem.
In order to solve the conformal anomaly, caused by the interaction of the
matter fields and gravity, one calculates the beta-functions (note that the
trace is proportional to the beta functions), which has to be zero (these
finite quantum theories are scale invariant) For a renormalizable theory,
there must be as many beta functions as adjustable parameters, thus allowing
to completely determine the theory. Another problem is the dynamics of
$\bar{g}_{\mu\nu}$. If the behavior is classical, then there is an interaction
with the quantum fields, i.e., the dilaton field. On a black hole spacetime,
one has to deal with Hawking radiation, even in the absent of matter terms, by
outgoing gravitational waves. This is represented by the dynamical equation
for the dilaton field, which decouples from $\bar{g}_{\mu\nu}$. It is clear
that $\bar{\omega}$ will not decouple when a scalar field is incorporated[19].

We shall see in section 4 that $\bar{\omega}$ determines the evolution of
$\bar{g}_{\mu\nu}$ (i.e., $N$ and $N^{\varphi}$).

##  4 The field equations of the warped CDG model

###  4.1 The 5D equations

Let us apply the equations of section 3 to the 5D spacetime
(${}^{(5)}{g_{\mu\nu}}=\omega^{4/3}{{}^{(5)}{\tilde{g}_{\mu\nu}}}$)

$ds^{2}=\omega(t,r,y)^{4/3}\Bigl{[}-N(t,r)^{2}dt^{2}+\frac{1}{N(t,r)^{2}}dr^{2}+dz^{2}+r^{2}(d\varphi+N_{\varphi}(t,r)dt)^{2}+dy^{2}\Bigr{]}.$
(30)

This metric is used for the 3D BTZ black hole solution (where the $dz^{2}$ is
removed) and can be up-lifted to 4D[25, 19]. From the 5D Einstein equations
Eq.(24), one obtains of course again
$\omega(t,r,y)=\omega_{2}(y)\omega_{1}(t,r)$, with $\omega_{2}(y)=l$ =
constant = length scale of the extra dimension.

The set of PDE’s for $\omega$ and $N$ now becomes (we omit the subscript 1 in
$\omega$)

$\ddot{\omega}=-N^{4}{\omega}^{\prime\prime}+\frac{5}{3\omega}(N^{4}{\omega^{\prime}}^{2}+\dot{\omega}^{2})$
(31)
$\ddot{N}=\frac{3\dot{N}^{2}}{N}-N^{4}\Bigl{(}N^{\prime\prime}+\frac{3N^{\prime}}{r}+\frac{N^{\prime
2}}{N}\Bigr{)}-\frac{2}{\omega}\Bigl{[}N^{5}\Bigl{(}\omega^{\prime\prime}-\frac{5\omega^{\prime
2}}{3\omega}+\frac{\omega^{\prime}}{r}\Bigr{)}+N^{4}\omega^{\prime}N^{\prime}+\dot{\omega}\dot{N}\Bigr{]}$
(32)

The equation for $N$ can also be presented without $\omega^{\prime\prime}$ on
the right hand side. Then there is an extra constraint equations in
$\omega^{\prime\prime}$:

$\omega^{\prime\prime}=-\frac{\Lambda\omega^{7/3}\kappa^{4/3}\sqrt[3]{18}l}{16N^{2}}-\frac{\omega^{\prime}N^{\prime}}{N}-\frac{\omega^{\prime}}{2r}+\frac{4\dot{\omega}^{2}}{3\omega
N^{4}}-\frac{\dot{\omega}\dot{N}}{N^{5}}.$ (33)

A non-trivial solution for $\omega$ becomes (compare with the warped spacetime
solution of Eq.(17)

$\omega=\Bigl{[}\frac{c_{1}}{(r+c_{2})t+c_{3}r+c_{4}}\Bigr{]}^{3/2}$ (34)

(or $r$ and $t$ interchanged). For $c_{2}c_{3}=c_{4}$, we find the exact
solution for $N$

$N^{2}=\frac{1}{5r^{2}}\frac{10c_{2}^{3}r^{2}+20c_{2}^{2}r^{3}+15c_{2}r^{4}+4r^{5}+D_{1}}{D_{2}(c_{3}+t)^{4}+D_{3}},$
(35)

with $c_{i},D_{i}$ constants. Note that $N$ can be written as
$N=N_{1}(r)N_{2}(t)$. The dilaton equation Eq.(23) is superfluous. One easily
checks that the trace of Eq.(24) is zero and that
$\tilde{\nabla}^{\mu}(T_{\mu\nu}^{\omega}-\Lambda\tilde{g}_{\mu\nu}\kappa^{\frac{4}{n-2}}\xi^{\frac{2}{n-2}}\omega^{\frac{2n}{n-2}})=0$.
Also, the Ward identity is fulfilled. The equations for $N^{\varphi}$ are

${\dot{N^{\varphi}}}^{\prime}=-\frac{2\dot{\omega}{N^{\varphi}}^{\prime}}{\omega}$
(36)
${{N^{\varphi}}^{\prime}}^{2}=-\frac{4N}{r^{3}}\Bigl{(}N^{\prime}+\frac{N\omega^{\prime}}{\omega}\Bigr{)}$
(37)
${N^{\varphi}}^{\prime\prime}=-{N^{\varphi}}^{\prime}\Bigl{(}\frac{2\omega^{\prime}}{\omega}+\frac{3}{r}\Bigr{)}$
(38)

Figure 1: Example of an exact solution of $\tilde{g}_{tt}$ (left) and
$\omega^{4/3}\tilde{g}_{tt}$

with solution

$N^{\varphi}=F_{1}(t)+\int\frac{1}{r^{3}\omega^{2}}dr$ (39)

with $F_{1}(t)$ an arbitrary function in t. A special solution of
$N^{\varphi}$ is

$N^{\varphi}=\frac{(t+c_{3})^{3}}{2r^{2}}\Bigl{(}6c_{2}r^{2}ln(r)-c_{2}^{3}-6c_{2}^{2}r+2r^{3}\Bigr{)}$
(40)

Further, we have the constraint

$N^{\prime}=-\frac{N\omega^{\prime}}{\omega}+\frac{e_{1}}{N\omega^{4}r^{3}}$
(41)

with $e_{1}$ a constant. This constraint can be used as boundary condition in
the numerical solution.

In figure 1 we plotted a typical exact solutions of $\omega^{4/3}g_{tt}$ and
$g_{tt}$. The Killing horizon is where the ergo region begins. Note however,
that the general exact solution for $\omega$ has one more free parameter
$c_{4}$.

###  4.2 The effective 4D Einstein equations on the brane

We now write

${}^{(5)}{\tilde{g}_{\mu\nu}}={{}^{(4)}}{\tilde{g}_{\mu\nu}}+n_{\mu}n_{\nu}$
(42)

with $n_{\mu}=[0,0,0,0,\sqrt{l}]$ the unit normal to the brane. Ones again, we
define

${{}^{(4)}}{\tilde{g}_{\mu\nu}}=\bar{\omega}^{2}\bar{g}_{\mu\nu}$ (43)

The effective Einstein equations on the brane are

${{}^{(4)}{\bar{G}_{\mu\nu}}}=\frac{1}{\bar{\omega}^{2}}\Bigl{(}-\Lambda_{eff}{{}^{(4)}{\bar{g}_{\mu\nu}}}-\Lambda\kappa^{2}{{}^{(4)}{\bar{g}_{\mu\nu}}}\bar{\omega}^{4}+{{}^{(4)}}{T_{\mu\nu}}^{(\bar{\omega})}\Bigr{)}-{\cal
E}_{\mu\nu},$ (44)

with ${\cal E}_{\mu\nu}$ the electric part of the Weyl tensor, given by

${\cal
E}_{\mu\nu}={{}^{(5)}{C_{\alpha\beta\gamma\delta}}}n^{\delta}{{}^{(4)}{\bar{g}_{\mu}^{\alpha}}}{{}^{(4)}{\bar{g}_{\nu}^{\beta}}}$
(45)

Note that ${\cal E}_{\mu\nu}$ is trace free. For the ”un-physical” metric
${{}^{(4)}}{\bar{g}_{\mu\nu}}$ we have now

$ds^{2}=-N(t,r)^{2}dt^{2}+\frac{1}{N(t,r)^{2}}dr^{2}+dz^{2}+r^{2}(d\varphi+N_{\varphi}(t,r)dt)^{2}\Bigr{]}.$
(46)

Again, an equation for $\bar{\omega}$ can be isolated (we take from now on
$\Lambda_{eff}=0$), i.e.,

$\ddot{\bar{\omega}}=-N^{4}{\bar{\omega}}^{\prime\prime}+\frac{2}{\bar{\omega}}(N^{4}{\bar{\omega}}^{{}^{\prime}2}+\dot{\bar{\omega}}^{2})$
(47)
$\displaystyle\ddot{N}=\frac{3\dot{N}^{2}}{N}-N^{4}\Bigl{(}N^{\prime\prime}+\frac{3N^{\prime}}{r}+\frac{N^{\prime
2}}{N}\Bigr{)}$ (48)
$\displaystyle-\frac{3}{\bar{\omega}}\Bigl{[}N^{5}\Bigl{(}\bar{\omega}^{\prime\prime}-\frac{2\bar{\omega}^{\prime
2}}{\bar{\omega}}+\frac{\bar{\omega}^{\prime}}{r}\Bigr{)}+N^{4}\bar{\omega}^{\prime}N^{\prime}+\dot{\bar{\omega}}\dot{N}\Bigr{]}$
(49)

Now the non-trivial solution of $\bar{\omega}$ becomes (compare with Eq.(34) )

$\bar{\omega}=\Bigl{[}\frac{b_{1}}{(r+b_{2})t+b_{3}r+b_{4}}\Bigr{]},$ (50)

with $b_{i}$ constants. The solution for N is the same as Eq.(35) (with
$c_{i}$ replaced by $b_{i}$ and $D_{i}$ by $C_{i}$)

$N^{2}=\frac{1}{5r^{2}}\frac{10b_{2}^{3}r^{2}+20b_{2}^{2}r^{3}+15b_{2}r^{4}+4r^{5}+C_{1}}{C_{2}(b_{3}+t)^{4}+C_{3}},$
(51)

The constraint for $\bar{\omega}^{\prime\prime}$ becomes now

$\bar{\omega}^{\prime\prime}=-\frac{\Lambda\bar{\omega}^{3}\kappa^{2}l}{9N^{2}}-\frac{\omega^{\prime}N^{\prime}}{N}-\frac{\omega^{\prime}}{2r}+\frac{2\dot{\bar{\omega}}^{2}}{\bar{\omega}N^{4}}-\frac{\dot{\bar{\omega}}\dot{N}}{N^{5}}$
(52)

The equations for $N^{\varphi}$ are

${\dot{N^{\varphi}}}^{\prime}=-\frac{3\dot{\bar{\omega}}{N^{\varphi}}^{\prime}}{\omega}$
(53)
${{N^{\varphi}}^{\prime}}^{2}=-\frac{2N}{r^{3}}\Bigl{(}2N^{\prime}+3\frac{N\bar{\omega}^{\prime}}{\bar{\omega}}\Bigr{)}$
(54)
${N^{\varphi}}^{\prime\prime}=-{3N^{\varphi}}^{\prime}\Bigl{(}\frac{\bar{\omega}^{\prime}}{\bar{\omega}}+\frac{1}{r}\Bigr{)}$
(55)

with solution

$N^{\varphi}=F_{2}(t)+\int\frac{1}{r^{3}\bar{\omega}^{3}}dr$ (56)

with $F_{2}(t)$ an arbitrary function in t. A typical solution is also given
by Eq(40)!

Further, we have the constraint

$N^{\prime}=-\frac{3N\bar{\omega}^{\prime}}{2\bar{\omega}}+\frac{e_{2}}{N\bar{\omega}^{6}r^{3}}$
(57)

with $e_{2}$ a constant.

Figure 2: Left: typical plot of $\bar{g}_{tt}$ for the 4D induced spacetime.
Right: typical plot of $\omega^{4/3}\bar{\omega}^{2}\bar{g}_{tt}$ . We observe
that the ergo sphere fades away as time increases

.

In figure 2 we plotted $g_{tt}$ for $\bar{g}_{\mu\nu}$ and $g_{\mu\nu}$.

Again, the dilaton equation is superfluous and the trace of Eq.(24) again
zero. The conservation equations now become

$\bar{\nabla}^{\mu}{\cal
E}_{\mu\nu}=\bar{\nabla}^{\mu}\Bigl{[}\frac{1}{\bar{\omega}^{2}}\Bigl{(}-\Lambda\kappa^{2}{{}^{(4)}{\bar{g}_{\mu\nu}}}\bar{\omega}^{4}+{{}^{(4)}}{T_{\mu\nu}}^{(\bar{\omega})}\Bigr{)}\Bigr{]}$
(58)

which yields differential equations for $\ddot{N^{\prime}}$ and $\dot{N}$ as
boundary conditions at the brane. It can be described as the non-local
conservation equation. In the high energy case close to the horizon, one must
include the ${\cal S}_{\mu\nu}$ term. So the divergence of ${\cal E}_{\mu\nu}$
is constrained. In the non-conformal case, Eq.(58) contains on the right hand
side also the quadratic correction ${\cal S}_{\mu\nu}$ of the matter fields on
the brane. The effective field equations, Eq.(44), are then not a closed
system. One needs the Bianchi equations. In fact, ${\cal E}_{\mu\nu}$ encodes
corrections from the 5D graviton effects and are for the brane observer non-
local. In our model under consideration, we have only the
$T_{\mu\nu}^{(\omega)}$ term and no source terms (apart from the 5D
$\Lambda_{5}$). But it still source the KK modes. The dilaton $\omega$ plays
the role of a ”scalar field”. But we don’t need the 5D equations themselves,
because the solution for $N$ is the same! It is only the $\omega^{4/3}$ which
represents the 5D contribution. There is no exchange of energy-momentum
between the bulk and brane. If one applies the model to a FLRW model[4], then
the evolution equations are very complicated. Then inhomogeneous and
anisotropic effects from the 4D matter radiation distribution on the brane are
source for the 5D Weyl tensor, ${\cal E}_{\mu\nu}$ then causes non-local back-
reaction on the brane. One needs an approximation scheme in order to find the
missing evolution equation for ${\cal E}_{\mu\nu}$.

###  4.3  The 4D and 5D equations written in dimension $n$

One can write the differential equations for $n=5,4$ as

$\ddot{\omega}=-N^{4}\omega^{\prime\prime}+\frac{n}{\omega(n-2)}\Bigl{(}N^{4}\omega^{\prime
2}+\dot{\omega}^{2}\Bigr{)}$ (59)
$\displaystyle\ddot{N}=\frac{3\dot{N}^{2}}{N}-N^{4}\Bigl{(}N^{\prime\prime}+\frac{3N^{\prime}}{r}+\frac{N^{\prime
2}}{N}\Bigr{)}$ (60)
$\displaystyle-\frac{n-1}{(n-3)\omega}\Bigl{[}N^{5}\Bigl{(}\omega^{\prime\prime}+\frac{\omega^{\prime}}{r}+\frac{n}{2-n}\frac{{\omega^{\prime}}^{2}}{\omega}\Bigl{)}+N^{4}\omega^{\prime}N^{\prime}+\dot{\omega}\dot{N}\Bigr{]}$
(61)

with solution

$\omega=\Bigl{(}\frac{a_{1}}{(r+a_{2})t+a_{3}r+a_{4}}\Bigr{)}^{\frac{1}{2}n-1},N^{2}=\frac{1}{5r^{2}}\frac{10a_{2}^{3}r^{2}+20a_{2}^{2}r^{3}+15a_{2}r^{4}+4r^{5}+C_{1}}{C_{2}(a_{3}+t)^{4}+C_{3}}\qquad$
(62)

with $a_{i}$ some constants. The constraint becomes

$\bar{\omega}^{\prime\prime}=-\frac{2n}{n-2}\frac{\Lambda
l\kappa^{\frac{4}{(n-2}}\xi^{\frac{n-2}{{4(n-1)}}}\bar{\omega}^{\frac{n+2}{n-2}}}{N^{2}}-\frac{\omega^{\prime}N^{\prime}}{N}-\frac{\omega^{\prime}}{2r}+\frac{4}{n-2}\frac{\dot{\bar{\omega}}^{2}}{\bar{\omega}N^{4}}-\frac{\dot{\bar{\omega}}\dot{N}}{N^{5}}$
(63)

Note that the differential equations are constraint by an equation for
$\omega^{\prime\prime}$, which contains the $\Lambda$-term. We have now a
closed system of the 4D and 5D equations.

###  4.4  Contribution from the bulk

If one omits the contribution from the bulk, i. e., ${\cal E}_{\mu\nu}$, one
obtains the solution ($d_{i}$ constants)

$\displaystyle\omega=\frac{1}{(t+d_{3})r+d_{2}t+d_{2}d_{3}},\quad
N^{2}=\frac{1}{4r^{2}}\frac{6d_{2}^{2}r^{2}+8d_{2}r^{3}+3r^{4}+D_{1}}{D_{2}(t+d_{3})^{3}+D_{3}}.$
(64)

The solution for $N^{\varphi}$ becomes

$N^{\varphi}=\frac{(t+d_{3})^{2}}{2r^{2}}\Bigl{(}2r^{2}\ln(r)-d_{2}(d_{2}+4r)\Bigr{)}.$
(65)

If we compare this solution with Eq.(51), we observe that the highest power of
the radial coordinate in $N^{2}$ is now $r^{2}$ in stead of $r^{3}$. Further,
for $D_{1}=0$, $N^{2}$ has, for any $d_{2}$, no real roots, so it has naked
singularities. So the bulk contribution generates at least one horizon (see
next section).

##  5 Location of the horizon’s, Hawking radiation and antipodal
identification

Let us inspect more closely the behavior of the induced spacetime on the
brane, found in the preceding section. It turns out, by inspection of
$\bar{g}_{\mu\nu}$, that the spacetime possesses horizons (coordinate
singularities, i. e., 0, 1 or 2) and ergo-sphere (Killing horizon). $N^{2}$
becomes singular for $t=t_{H}=-b_{3}+\sqrt[4]{-\frac{C_{3}}{C_{2}}}$. However,
$\bar{g}_{\mu\nu}$ can be made regular everywhere and singular free by
suitable choices of the parameters $b_{i},c_{i}$ and $C_{i}$. For $C_{1}=0$,
$\bar{g}_{\mu\nu}$ has one real zero $r_{H}=\sim|1.606b_{2}|$ and two complex
zero’s $\sim(0.178\pm 0.638I)b_{2}$. In figure 3 we plotted $N_{1}(r)^{2}$ for
several values of $C_{1}$ and $b_{2}$.

Figure 3: Four possible plots of $N_{1}(r)^{2}$.

However, there is no restriction on the angular velocity of the horizon: the
equation for $N^{\varphi}$ decouples. Further, one must remark that we don’t
deal here with the 3D BTZ (with the obligatory cosmological constant). There
are no gravitational degrees of freedom in 3D Einstein gravity. Still this 3D
model has interesting applications, such has the connection with the
asymptotic $AdS_{3}$. It possesses local defects, black holes and asymptotic
symmetries[31]. In our uplifted model we don’t need a cosmological constant.
If one omits the $dz^{2}$ in Eq.(46), one needs a different approach[28] We
shall see that the properties of our $\bar{g}_{\mu\nu}$ can be compared with
the Kerr black hole[8].

###  5.1 The ”classical” picture.

In order to investigate the features of the time-dependent gravitational field
of our rotating object and the resulting production of Hawking radiation, one
should take into account the complementarity of different observers. The
annihilation operators of particles at late time can be expressed as
superpositions of annihilation and creation operators of particles at early
times (Bogolubov transformation). The created particles form a thermal
spectrum determined by the surface gravity, i. e., the gravitational
acceleration at the horizon. There will be also a back-reaction on the
spacetime. This emission is caused by the formation of an event horizon.
Finally the black hole will evaporate. During the evaporation, the surface
gravity and temperature increases.

The treatment of the rotating black hole thermodynamics can be found in
Wald[32]. The mass loss can be expressed as

$\delta M=\frac{\kappa}{8\pi}\delta A+\Omega_{H}\delta J_{H}.$ (66)

with $\kappa$ the surface gravity of the black hole (see section 5.5), $A$ the
area of the event horizon, $J_{H}$ the angular momentum and

$\Omega_{H}=\frac{d\varphi}{dt}=-\frac{g_{t\varphi}}{g_{\varphi\varphi}}=N^{\varphi}$
(67)

the coordinate angular velocity (the angular velocity of null generators of
the event horizon). $N^{\varphi}$ given by Eq.(40), evaluated at $r=r_{H}$. It
blows-up $\sim t^{3}$. Massless quanta created by the black hole are governed
by null geodesics that begin far outside the collapsing body at early times,
move inwards and become outgoing (radially) null geodesics and escape from the
collapsing body. They reach future null infinity at arbitrarily late times.

To describe the spectrum of the Hawking particles, one usually defines
geodesics[8] $x^{\mu}(\lambda)$, with $\lambda$ an affine parameter. The
geodesic is defined by $\nabla^{\lambda}(dx^{\alpha}/d\lambda)$, with the
covariant derivative along the geodesic. In null coordinates $(u,v)$ one has
for the affine separation between two outgoing null geodesics that escape from
the collapsing body just before the formation of the event horizon,
$u(v)=-\frac{1}{r_{H}}\log\frac{(v_{0}-v)}{K}$, with $K$ a constant.

In the Boyer-Lindquist coordinates, an incoming null geodesic take an infinite
coordinate time to reach the event horizon and it will winds an infinite
number of times around the z-axes.

###  5.2 Changing coordinates

If we define the coordinates, $dr^{*}\equiv\frac{1}{N_{1}(r)^{2}}dr$ and
$dt^{*}\equiv N_{2}(t)^{2}dt$, then our induced spacetime of Eq.(46) can be
written as

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}\frac{N_{1}^{2}}{N_{2}^{2}}\Bigl{(}-dt{{}^{*}}^{2}+dr{{}^{*}}^{2}\Bigr{)}+dz^{2}+r^{2}(d\varphi+\frac{N^{\varphi}}{N_{2}^{2}}dt^{*})^{2}\Bigr{]},$
(68)

with

$\displaystyle
N_{1}^{2}=\frac{10b_{2}^{3}r^{2}+20b_{2}^{2}r^{3}+15b_{2}r^{4}+4r^{5}+C_{1}}{5r^{2}},N_{2}^{2}=\frac{1}{C_{2}(t+b_{3})^{4}+C_{3}}$
(69)

and

$\displaystyle
r^{*}=\frac{1}{4}\sum_{r^{H}_{i}}\frac{r^{H}_{i}\log(r-r^{H}_{i})}{(r^{H}_{i}+b_{2})^{3}},\qquad
t^{*}=\frac{1}{4C_{2}}\sum_{t^{H}_{i}}\frac{\log(t-t^{H}_{i})}{(t^{H}_{i}+b_{3})^{3}}.$
(70)

The sum it taken over the roots of
$(10b_{2}^{3}r^{2}+20b_{2}^{2}r^{3}+15b_{2}r^{4}+4r^{5}+C_{1})$ and
$C_{2}(t+b_{3})^{4}+C_{3}$, i. e., $r^{H}_{i}$ and $t^{H}_{i}$. This
coordinate system is comparable with a Kruskal-Szekeres coordinate system.
Further, one can define the azimuthal angular coordinate
$d\varphi^{*}\equiv(d\varphi+\frac{N^{\varphi}}{N_{2}^{2}}dt^{*})$, which can
be used when an incoming null geodesic falls into the event horizon.
$\varphi^{*}$ is the azimuthal angle in a coordinate system rotating about the
z-axis relative to the Boyer-Lindquist coordinates.

###  5.3 Penrose diagram

If we introduce outgoing and ingoing radial null coordinates,
$u=t^{*}-r^{*},v=t^{*}+r^{*}$, then $u=cst.,$ and $v=cst.,$ describe the
outgoing and ingoing null curves. We then have

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}\frac{N_{1}^{2}}{N_{2}^{2}}dudv+dz^{2}+r^{2}{d\varphi^{*}}^{2}\Bigr{]}.$
(71)

Next we apply the usual Penrose compactification,

$\displaystyle u=\tan U,\qquad v=\tan V.$ (72)

We then obtain

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}\frac{N_{1}^{2}}{N_{2}^{2}}\frac{dUdV}{\cos^{2}U\cos^{2}V}+dz^{2}+r^{2}{d\varphi^{*}}^{2}\Bigr{]}.$
(73)

Further, from Eq.(70) we have

$\displaystyle r^{*}=\frac{1}{2}(v-u)=\frac{1}{2}(\tan V-\tan
U)=\frac{1}{4}\sum_{r^{H}_{i}}\frac{r^{H}_{i}\log(r-r^{H}_{i})}{(r^{H}_{i}+b_{2})^{3}}$
(74) $\displaystyle t^{*}=\frac{1}{2}(v+u)=\frac{1}{2}(\tan V+\tan
U)=\frac{1}{4C_{2}}\sum_{t^{H}_{i}}\frac{\log(t-t^{H}_{i})}{(t^{H}_{i}+b_{3})^{3}}.$
(75)

The Penrose diagram bears strong resemblance with the 3D BTZ black hole
solution. However, there are differences. First of all, we have no
cosmological constant and we have have a physical 4D spacetime. Further, the
factor $\frac{N_{1}^{2}}{N_{2}^{2}}$ is quite different.
$\frac{N_{1}^{2}}{N_{2}^{2}}$ can have no roots, so no horizons. Even for
$r=0$, $\bar{g}_{\mu\nu}$ can be regular and is conformally flat ($z,\varphi$
suppressed). This can also be seen, when one solves from Eq.(75) (for the case
of one horizon $r_{H}\sim 1.606b_{2}$).

$\displaystyle r=r_{H}+exp\Bigl{[}2(v-u)(r_{H}+b_{2})^{3}/rH\Bigr{]},$ (76)
$\displaystyle t=t_{H}+exp\Bigl{[}2C_{2}(v+u)(t_{H}+b_{3})^{3}\Bigr{]},$ (77)

So $\bar{g}_{\mu\nu}$ can be used to locate the light cones.

We can write the spacetime $\bar{g}_{\mu\nu}$ in the form

$\displaystyle\bar{ds}^{2}=(C_{2}(t+b_{3})^{4}+C_{3})\Bigl{[}\frac{10b_{2}^{3}r^{2}+20b_{2}^{2}r^{3}+15b_{2}r^{4}+4r^{5}+C_{1}}{5r^{2}}dudv\Bigr{]}+dz^{2}+r^{2}d{\varphi^{*}}^{2},$
(78)

or

$\displaystyle\bar{d}s^{2}=\Theta^{2}(U,V)dUdV+dz^{2}+r^{2}d{\varphi^{*}}^{2}$
(79) $\displaystyle\Theta^{2}=C_{2}\frac{e^{6(\tan V-\tan
U)(r_{H}+b_{2})^{3}/rH+8C_{2}(\tan V+\tan
U)(t_{H}+b_{3})^{3}}}{\cos^{2}U\cos^{2}V}$ (80)

Finally, we transform to coordinates $T$, $T=U+V,R=U-V$, in order to plot the
Penrose diagram of figure 4. We took $C_{1}=C_{3}=0$ and one horizon $r_{H}$.
We then have

$\displaystyle u=\tan
U=\log\frac{(t-t_{H})^{\beta}}{(r-r_{H})^{\alpha}},\qquad v=\tan
V=\log(t-t_{H})^{\beta}(r-r_{H})^{\alpha}.$ (81)

with

$\alpha=\frac{r_{H}}{4(r_{H}+b_{2})^{3}},\qquad\beta=\frac{1}{4C_{2}(t_{H}+b_{3})^{3}}$
(82)

Figure 4: Kruskal diagram for $\bar{g}_{\mu\nu}$ in $(T,R)$-coordinates.

Note that we have the ”scale-factor” $\Theta^{2}$ in front of
$(-dT^{2}+dR^{2})$, which causes no harm in construction the diagram. For
$t=t_{H}$, $\Theta\rightarrow\infty$! For $t\rightarrow\infty$,
$\Theta\rightarrow\infty$ OR $r\rightarrow r_{H}$ Note that although $r=0$ has
disappeared as singular point, it is in the line element Eq.(78) for
$C_{1}\neq 0$.

If one makes the antipodal identification $U\rightarrow-U,V\rightarrow-V$, one
obtains $\alpha\rightarrow-\alpha$ and $\beta\rightarrow-\beta$. This is only
possible if $(b_{2},b_{3})\rightarrow(-b_{2},-b_{3})$ and $r_{H}$ and $-r_{H}$
are identified!

Following Strauss, et. al.[38], we can use a slightly different coordinate
system, in order to study quantum effects more clearly (again 1 horizon and
$C_{1}=C_{3}=0$).

Figure 5: Kruskal diagram for $\bar{g}_{\mu\nu}$ in $(\tilde{U},\tilde{V})$\-
coordinates. The antipodal map between region I and II is quite clear here. If
one approaches the horizon from the outside and passes the horizon, one
approaches from the ”otherside” the horizon.

We define

$\displaystyle U=e^{\kappa(r^{*}-t^{*})},\quad V=e^{\kappa(r^{*}+t^{*})}\qquad
r>r_{H}$ (83) $\displaystyle U=-e^{\kappa(r^{*}-t^{*})},\quad
V=-e^{\kappa(r^{*}+t^{*})}\qquad r<r_{H}$ (84)

Further,

$\tilde{U}=\tanh U,\qquad\tilde{V}=\tanh V$ (85)

The spacetime can then be written as

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}H(\tilde{U},\tilde{V})d\tilde{U}d\tilde{V}+dz^{2}+r^{2}d\varphi^{*2}\Bigr{]}$
(86)

with

$H=\frac{N_{1}^{2}}{N_{2}^{2}}\frac{1}{\kappa^{2}\operatorname{arctanh}\tilde{U}\operatorname{arctanh}\tilde{V}(1-\tilde{U}^{2})(1-\tilde{V}^{2})}$
(87)

We can write $r$ and $t$ as

$r=r_{H}+\Bigl{(}\operatorname{arctanh}\tilde{U}\operatorname{arctanh}\tilde{V}\Bigl{)}^{\frac{1}{2\kappa\alpha}},\quad
t=t_{H}+\Bigl{(}\frac{\operatorname{arctanh}\tilde{V}}{\operatorname{arctanh}\tilde{U}}\Bigl{)}^{\frac{1}{2\kappa\beta}}$
(88)

In figure 5 we plotted the Penrose diagram in
$(\tilde{U},\tilde{V})$-coordinates. $\tilde{g}_{\mu\nu}$ is regular
everywhere and conformally flat. The ”scale-term” H is consistent with the the
features of the Penrose diagram.

###  5.4 The role of the dilaton field.

Let us now return to
$g_{\mu\nu}=\omega^{4/3}\bar{\omega}^{2}\bar{g}_{\mu\nu}$. In the CDG setting,
the evaporation is expressed by the complementarity transformation of $\omega$
between the in-going and outside observer. Only the outside observer will
notice the outgoing Hawking radiation. The local observer notice only the in-
going radiation[15].

In order to apply quantum mechanics in the vicinity of horizon555crossing the
horizon in this antipodal mapping, also interchanges creation and annihilation
operators and reverse time. So there will be an entanglement between positive
energy particle with the positive energy antiparticles at the antipodes.
Moreover, one can deal with pure quantum states in stead of thermodynamical
mixed states. This resolves the information problem[33]., one can apply an
antipodal identification, a boundary condition in order to remove the inside
of the horizon. In this approach one connects the regions I and II in the
Kruskal diagram of figure 5 in order to solve the unitarity and the
entanglement problems[33]. This procedure shed some inside how particles
transmit the information they carry, across the horizon. The horizon changes
from a 2-cylinder into a projected one. Every point $(r,t)$ corresponds with
two points $(u,v),(-u,-v)$. If one identifies
$(-u,-v,z,\varphi)\equiv(u,v,-z,\varphi+\pi)$ (i.e.,
$t^{*}\rightarrow-t^{*},r^{*}\rightarrow-r^{*},z\rightarrow-z,\varphi\rightarrow\varphi+\pi$),
then region I and III correspond to the same black hole, so there is no
inside.

For $b_{4}=b_{3}b_{2}$, we have now

$\displaystyle
ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}(C_{2}(t+b_{3})^{4}+C_{3})\frac{(10b_{2}^{3}r^{2}+20b_{2}^{2}r^{3}+15b_{2}r^{4}+4r^{5}+C_{1})}{5r^{2}}\cdot$
(89)
$\displaystyle\qquad\qquad\qquad\Bigl{(}-dt{{}^{*}}^{2}+dr{{}^{*}}^{2}\Bigr{)}+dz^{2}+r^{2}(d\varphi+\frac{N^{\varphi}}{N_{2}^{2}}dt^{*})^{2}\Bigr{]}.$
(90)

Different observers use different $\omega$ (disagreement about the vacuum
state ), i.e., the scale, by adjusting the constants. The local observer will
not notify the new boundary condition of the antipodal map. Only the outside
observer will become aware of the mapping. We see that

$\omega^{4/3}\bar{\omega}^{2}=\frac{1}{(r+c_{2})^{2}(t+c_{3})^{2})(r+b_{2})^{2}(t+b_{3})^{2}},$
(91)

approaches zero for coordinate time $t\rightarrow\infty$, so $g_{\mu\nu}$
shrinks to zero. We observe that the time derivative of Eq.(91) is non-zero,
so we have a non-zero $T^{\omega}$ (see Eq.(25)). This is the energy-momentum
tensor of the Hawking particles. By suitable choices of the constants one can
make it asymptotically regular.

###  5.5 Surface gravity

In order to calculate the surface gravity (needed for the Hawking radiation
and the notion of the quasi-local mass)), one must realize that we are dealing
here with a non-stationary black hole. For stationary Kerr black holes, one
defines the Killing vector
$\chi^{\mu}=\xi_{(t)}^{\mu}+\Omega_{H}\xi_{(\varphi)}^{\mu}$, normal to the
Killing horizon666 The Killing vector is a non-affinely parameterized geodesic
on the horizon. $\Omega_{H}$ is the angular velocity of the horizon, given by
Eq.(67) and
$\xi_{(t)}^{\mu}\xi_{(t)\mu}=g_{tt},\xi_{(\varphi)}^{\mu}\xi_{(\varphi)_{\mu}}=g_{\varphi\varphi}$.
The surface gravity is then defined as

$\kappa^{2}=-\frac{1}{2}\nabla^{\mu}\chi^{\nu}\nabla_{\mu}\chi_{\nu},$ (92)

evaluated at the horizon $r=r_{H}$. An equivalent form is
$\chi^{\mu}\nabla_{\mu}\chi^{\nu}=2\kappa\chi^{\nu}$. We say that $\chi^{\mu}$
is geodesic on the horizon and $\kappa$ measures the extent to which
$\chi^{\mu}$ fails to be affinely parameterized.

In our non-stationary case, the calculations complicates, because we have an
evolving horizon[34, 32]777An approved method is the use of Eddington-
Finkelstein coordinates. An other possible way out is to apply the
transformation $t\rightarrow iz,z\rightarrow it$ and follow the method of
Medved, et al.[37].

We need now null generators of the local non-Killing horizon and a suitable
normalization (parameterization) of the surface gravity. This is related to
non-affinity of null geodesics on the horizon. For a Killing vector, it is
easy, because the Killing equation $\nabla_{\mu}l_{\nu}+\nabla_{\nu}l_{\mu}=0$
possesses a gauge freedom, i. e., a constant factor which can be used to re-
scale $l_{\mu}$ and make $l^{\mu}l_{\mu}=-1$ at spacelike infinity. The Lie
derivative then tells us how a spacetime tensor changes when one takes an
infinitesimal step along along the vector field. For a Killing field
$\chi^{\mu}$ we have ${\cal L}_{\chi}g_{\mu\nu}=0$.

Let us define outgoing and ingoing null normals for $\bar{g}_{\mu\nu}$,
$\bar{l}^{\mu}$ and $\bar{m}^{\mu}$
($\bar{l}^{\mu}\bar{l}_{\mu}=\bar{m}^{\mu}\bar{m}_{\mu}=0$, no Killing
fields), such that $\bar{l}^{\mu}\bar{m}_{\mu}=-1$:

$\bar{l}^{\mu}=(1,N\sqrt{N^{2}-r^{2}{N^{\varphi}}^{2}},0,0),\quad\bar{m}^{\mu}=\Bigl{(}-\frac{1}{2r^{2}{N^{\varphi}}^{2}-2N^{2}},-\frac{N}{2\sqrt{N^{2}-r^{2}{N^{\varphi}}^{2}}},0,0\Bigr{)}$
(93)

The surface gravity is then[36, 35]

$\kappa=-\bar{m}^{\nu}\bar{l}^{\mu}\bar{\nabla}_{\mu}\bar{l}_{\nu}.$ (94)

One can evaluate Eq.(94) at the horizon, because we have exact expressions for
$N$ and $N^{\varphi}$. It turns out for suitable parameters, that at
$t=-b_{3}$, $\kappa\rightarrow\infty$.

The value of $\bar{l}^{\mu}\bar{m}^{\nu}\bar{\nabla}_{\nu}\bar{m}_{\mu}$
depends on the normalization for $l^{\mu}$. One should take

$\bar{l}^{\mu}\bar{m}^{\nu}\bar{\nabla}_{\nu}\bar{m}_{\mu}=0,$ (95)

which yields an equation for $\partial_{t}N^{\varphi}$

$\partial_{t}N^{\varphi}=\frac{-1}{r^{2}N^{\varphi}N}(r^{2}N^{\varphi}-2N^{2})\partial_{t}N.$
(96)

This determines the function of $F_{2}(t)$ in Eq(57). This choice is also
necessary, in order to obtain the correct expression for the Lie derivative
${\cal L}_{\bar{l}}\bar{g}_{\mu\nu}$

The surface gravity then becomes

$\kappa=2N\Bigl{(}\partial_{r}(\sqrt{N^{2}-r^{2}{N^{\varphi}}^{2}})+\partial_{t}\Bigl{(}\frac{1}{N}\Bigr{)}\Bigr{)}=2N\Bigl{(}\partial_{t}\sqrt{\bar{g}_{rr}}-\partial_{r}\sqrt{\bar{g}_{tt}}\Bigr{)}$
(97)

This is consistent with the metric definition of $\kappa$ for the time
independent case.

###  5.6 The choice of $\Omega$

If $v^{\mu}$ is a vector tangent to a geodesic with affine parameter
$\lambda$, then

$\partial_{\lambda}(v^{\mu}l_{\mu})=v^{\nu}\nabla_{\nu}(l_{\mu}v^{\mu})=v^{\nu}v^{\mu}\nabla_{\nu}l_{\mu}=\frac{1}{4}v^{\nu}v_{\nu}\nabla^{\mu}l_{\mu},$
(98)

which vanishes for $v^{\nu}v_{\nu}=0$. So $l^{\mu}v_{\mu}$ is conserved along
null geodesics. Remember that we still have the freedom of Eq.(22). One can
verify that $v^{\nu}\nabla_{\nu}v^{\mu}$ transforms as

$v^{\nu}\nabla_{\nu}v^{\mu}\rightarrow
v^{\nu}\nabla_{\nu}v^{\mu}+\frac{1}{\Omega}(2v^{\mu}v^{\nu}\nabla_{\nu}\Omega-v^{\nu}v_{\nu}\nabla^{\mu}\Omega).$
(99)

For $v^{\mu}v_{\mu}=0$ we obtain

$v^{\nu}\nabla_{\nu}v^{\mu}\rightarrow
v^{\nu}\nabla_{\nu}v^{\mu}+\frac{2}{\Omega}v^{\mu}v^{\nu}\nabla_{\mu}\Omega.$
(100)

Now $v^{\mu}$ is a tangent vector to a geodesic if
$v^{\nu}\nabla_{\nu}v^{\mu}=\alpha v^{\mu}$. If $\alpha=0$, we have a affinely
parameterized geodesic. Conformal transformations preserve null geodesics. So
conformal transformations preserve affinely parameterized geodesics, if we
take $\frac{2}{\Omega}v^{\nu}\nabla_{\nu}\Omega=0$, or

$\partial_{t}\Omega=\pm N\sqrt{N^{2}-r^{2}N^{\varphi 2}}\partial_{r}\Omega,$
(101)

which is consistent with the null condition for spacetime
$\tilde{g}_{\mu\nu}$. It makes the surface gravity also conformal invariant
(note that $\nabla_{\mu}\Omega^{2}$ is non-singular at the horizon).

We conjecture that this constraint can be used for the outside observer. Can
one construct from $\bar{l}$ and $\bar{m}$ a conformal Killing vector? (note
that we have ${\cal L}_{\bar{l}}\bar{g}_{\mu\nu}\neq 0$ and ${\cal
L}_{\bar{m}}\bar{g}_{\mu\nu}\neq 0$) If so, then one can construct a true
Killing field for the conformally related metric $\Omega^{2}\bar{g}_{\mu\nu}$
and a conformal invariant surface gravity of a conformal Killing horizon.
Remember, we have the conformal Killing equation for a conformal Killing
vector (not null)

$\bar{\nabla}_{\mu}\bar{l}_{\nu}+\bar{\nabla}_{\nu}\bar{l}_{\mu}=\frac{1}{4}\bar{\nabla}^{\sigma}\bar{l}_{\sigma}\bar{g}_{\mu\nu}.$
(102)

The Lie derivative transforms as

${\cal L}_{\bar{l}}\bar{g}_{\mu\nu}\rightarrow{\cal
L}_{\bar{l}}\log\Omega^{2}\bar{g}_{\mu\nu}.$ (103)

Suppose that $\hat{g}_{\mu\nu}=\Omega^{2}\bar{g}_{\mu\nu}$. A spacetime with a
conformal killing field $\bar{l}$ is conformal to a spacetime for which
$\bar{l}$ is a true Killing field. That is, if ${\cal
L}_{\bar{l}}\hat{g}_{\mu\nu}=2k\hat{g}_{\mu\nu}$, then ${\cal
L}_{\bar{l}}\hat{g}_{\mu\nu}=0$, if

${\cal L}_{\bar{l}}\Omega^{2}+2k\Omega^{2}=0.$ (104)

The solution is given along integral curves of $\bar{l}$ by $\log\Omega=-\int
kdv$, with $\bar{l}^{\mu}\nabla_{\mu}v=1$.

The black hole complementarity is now formulated as follows: both observers
uses different ways to fix the conformal gauge $\Omega$. They disagree about
the vacuum state. In our simplified model, with exact solutions for $\omega$
and $\bar{g}_{\mu\nu}$, we conjecture that these gauges can be found by
constructing conformal killing fields and a conformal invariant surface
gravity.

###  5.7 Flux density

We can also calculate the flux density $\bar{G}_{tr}$.

$\frac{2(t+b_{3})^{3}}{r\Bigl{[}(t+b_{3})^{4}+\frac{C_{3}}{C_{2}}\Bigr{]}}.$
(105)

Then the flux across the horizon $r=r_{H}$ in an amount of proper time can be
found by integrating the expression

$\int_{t_{init}}^{t_{final}}\Bigl{[}4\pi
r^{2}\frac{2N(t+b_{3})^{3}}{r\Bigl{[}(t+b_{3})^{4}+\frac{C_{3}}{C_{2}}\Bigr{]}}dt,$
(106)

which becomes infinite at $t_{H}$.

##  6 Numerical solutions

We can also solve the PDE’s numerically. In figure 6 we plotted a typical
solution for the 5D case.

Figure 6: Example of a regular numerical vacuum solution in case of the 5D
spacetime of section 4.1. We have only gravitational waves.
$c_{1}=1;c_{2}=0.15;c_{3}=-0.2;c_{4}=1$. We observe that an ergo-sphere
($g_{tt}=0$) is dynamically formed at some $r$ and $t$. For the boundary
condition for N, we used Eq(41). Note that the angular momentum is depicted
without an arbitrary function in t (see Eq(39)). It must be chosen such that
$N^{\varphi}$ doesn’t change sign. We also see that $\omega$ shrinks gradually
to zero as the $t\rightarrow\infty$ (see also the exact solution).

The numerical solution of the 4D counterpart case doesn’t differ
significantly. With the numerical ”back-up” solution, one can investigate how
the constants of the exact solution are related to the boundary conditions and
the $\Lambda\omega^{4}$ term.

##  7 Conclusions

We investigated the conformal invariance of the Einstein-Hilbert action on a
dynamical warped 5D black hole spacetime. By solving both Einstein and dilaton
equations, we find an exact solution for the dilaton field and the metric
components of the ”un-physical” spacetime $\bar{g}_{\mu\nu}$, where the real
spacetime is written as
${}^{(5)}{g_{\mu\nu}}=\omega^{4/3}{{}^{(5)}{\tilde{g}_{\mu\nu}}}$ and
${}^{(4)}{\tilde{g}_{\mu\nu}}=\bar{\omega}^{2}{{}^{(4)}\bar{g}_{\mu\nu}}$. In
our model, $\omega$ can be seen as the contribution of gravitational waves
from the bulk (KK-modes), while $\bar{\omega}$ is the brane component. There
is no exchange of energy-momentum between the bulk and brane in our vacuum
model.

It is conjectures that the different conformal gauge freedom, $\Omega$, the
in-going and outside observers possesses, can be calculated by demanding a
conformal invariant surface gravity. This means that the complementarity is
expressed by the different notion of the vacuum state. They disagree about the
interior of the black hole. The solution guarantees regularity of the action
when $\omega\rightarrow 0$. We don’t need a Weyl term in the action (generates
negative metric states). In stead, we have a contribution from the bulk, i.
e., the electric part of the 5D Weyl tensor. It is remarkable that the 5D
field equations and the effective 4D equations can be written for general
dimension $n$, with $n=4,5$. The energy-momentum tensor of the time-dependent
dilaton, determining the Hawking radiation, can be calculated exactly. By
suitable choice of the parameters, the spacetime $\bar{g}_{\mu\nu}$ can be
regular and singular free.

In context of quantization procedures, counter terms in an effective action
will cause problems, only in the bulk spacetime of the ”large” extra dimension
and not for the brane spacetime. When the extra dimensional volume is
significantly above the Planck scale, then the true fundamental scale can be
much less than the effective scale $10^{19}$ GeV. This means that no UV cut-
off is necessary on the brane. This exact solution, nonetheless without mass
terms, can be used to tackle the deep-seated problem of the black hole
complementarity: the infalling and outside observer experience different
$\omega$ by the choice of $\Omega$.

The solution fits also very well in the antipodal mapping, when crossing the
horizon. The Penrose diagram for $\bar{g}_{\mu\nu}$, in suitable Kruskal
coordinates, shows the features of the antipodal map of region I on region II:
the inside of the black hole is removed. The in-going observer, when crossing
the horizon, turns up at ”the other side” of the horizon.

The next task is to incorporate mass into our model and investigate the
dilaton-scalar field interaction. The conformal invariance will then
spontaneously be broken. It is conjectured that the parameters of the model
will be calculable in the final version. This is currently under investigation
by the author.

##  8 References

## References

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risk of getting a singular covariance matrix. With the replication, the GP
variance vanishes for all EI maximizers, even those outside $\mathcal{A}$,
removing any further EI from these $\boldsymbol{\alpha}$’s. The
$\boldsymbol{\alpha}$ space is better explored with benefits on the objective
value (26.26 against 27.13 here).

Figure 39: Optimization with EI maximization in the covering hyper-rectangle
of $\mathcal{A}_{N}$ without (left) or with (right) replication strategy.

The EI strategy which consists in maximizing via the $X$ space of CAD
parameters avoids the $\boldsymbol{\alpha}$ manifold issues. However, it does
not perform well, because of the higher dimensional space where the criterion
is maximized. An additional drawback for efficient maximization is that
$\nabla$EI is not known analytically in this case.

In this catenoid example, the additive GP and the GP in the space of (all) 7
principal components achieve comparable results, both in terms of best value,
and of function evaluations to attain the targets. Indeed, the true dimension
(7) is relatively low, and we have noticed that the 5, 6 or even 7 first
eigenshapes often got classified as active for the additive GP.

#### Heart rectangle

We now consider Example 4 and the minimization of $f_{4}(\mathbf{x})$ that
expresses the distance from a shape to a rectangle deformed as an heart.

As before, different metamodeling and EI maximization options are benchmarked.
They include: the standard approach of doing the process in the space of CAD
parameters (in dimension $d=40$); the optimization in the space of 2, 4, 16 or
40 first principal components, where 100% of the shapes variability is
recovered with 40 eigencomponents as seen in Table 6. Supervised eigenshape
selection methods (Section 3.2) are also used: the GP built over
$\boldsymbol{\alpha}^{a}$ only, and the additive model over
$\boldsymbol{\alpha}^{a}$ and $\boldsymbol{\alpha}^{\overline{a}}$. For the
latter, the 4 EI maximization options of Section 4.1 are compared. In light of
the above optimization results on the catenoid, the three EI maximization
strategies are carried out in the covering hyper-rectangle of
$\mathcal{A}_{N}$ (as opposed to restricted to the neighborhood of
$\mathcal{A}_{N}$), and EI maximizers which do not belong to $\mathcal{A}$ are
nonetheless used for the GP update. Henceforth, the with replication strategy
becomes the new default in all algorithms carrying out EI maximizations in
$\boldsymbol{\alpha}$’s and it will no longer be specified in the algorithms
names.

The statistics on the solutions proposed by the algorithms are reported in
Table 17. They consist in the best objective function values averaged over 10
runs with different initial designs, with standard deviations given in
brackets. The average and standard deviation of the number of function
evaluations to reach certain levels is also given, to compare the ability of
the algorithms to quickly attain near-optimal values. When at least one run
failed in attaining the target, it is replaced by a rough estimator of the
empirical runtime. The algorithms start with a space-filling DoE of 20
individuals and are run for 80 supplementary iterations. In the case of the
CAD parameters GP($X$)-EI($X$) and of
GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$), since
$d=40>20$, the initial DoE contains 50 designs and the algorithm is run for 50
iterations.

Method Best value Time to 0.5 Time to 1 Time to 3 GP($X$)-EI($X$) 1.18 (0.45)
$\times$ 166.9 [4] 42.1 (26.5)
GP($\boldsymbol{\alpha}_{1:2}$)-EI($\boldsymbol{\alpha}_{1:2}$) 9.21 (0.80)
$\times$ $\times$ $\times$
GP($\boldsymbol{\alpha}_{1:4}$)-EI($\boldsymbol{\alpha}_{1:4}$) 0.33 (0.07)
48.8 (21.8) 21.8 (2.2) 21.0 (0.0)
GP($\boldsymbol{\alpha}_{1:16}$)-EI($\boldsymbol{\alpha}_{1:16}$) 0.59 (0.15)
197.8 [3] 50 (15.4) 35.0 (9.7)
GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$) 2.95 (0.97)
$\times$ $\times$ 194.4 [5]
GP($\boldsymbol{\alpha}^{a}$)-EI($\boldsymbol{\alpha}^{a}$) 0.32 (0.09) 33.7
(9.4) 24.5 (3.7) 21.8 (1.3)
AddGP($\boldsymbol{\alpha}^{a}$+$\boldsymbol{\alpha}^{\overline{a}}$)-EI($X$)
0.54 (0.19) 199.4 [4] 40.2 (12.3) 30.2 (10.5)
AddGP($\boldsymbol{\alpha}^{a}$+$\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
0.37 (0.08) 49.0 (21.4) 26.1 (5.6) 22.2 (1.9)
AddGP($\boldsymbol{\alpha}^{a}$+$\boldsymbol{\alpha}^{\overline{a}}$)-EI($\boldsymbol{\alpha}^{a}$)
0.37 (0.09) 33.3 (14.6) 22.7 (2.6) 21.4 (0.7)
AddGP($\boldsymbol{\alpha}^{a}$+$\boldsymbol{\alpha}^{\overline{a}}$)-EI($\boldsymbol{\alpha}$)
0.60 (0.26) 106.7 [6] 41.2 [9] 21.5 (0.5)

Table 17: Minimum objective function values found and number of function
evaluations required to attain a fixed target (average over 10 runs, standard
deviations in brackets) for different metamodels and optimization strategies,
rectangular heart problem (Example 4). The red figures correspond the
empirical runtime, with the number of runs which attained the target in
brackets, and ’$\times$’ signifies that no run was able to attain it within
the limited budget. All algorithms performing an EI search in
$\boldsymbol{\alpha}$’s do it with replication, the henceforth default.

In this test case, as shown in Figure 22, the 2 first eigenshapes modify the
shape’s position, to which $f_{4}$ is insensitive. Poor results are therefore
obtained by GP($\boldsymbol{\alpha}_{1:2}$)-EI($\boldsymbol{\alpha}_{1:2}$)
even though $\mathbf{v}^{1}$ and $\mathbf{v}^{2}$ account for 80% of shape
reconstruction, highlighting the benefits of the determination of active
eigenshapes. In a first order approximation, $\mathbf{v}^{3}$ and
$\mathbf{v}^{4}$ are the most influential eigenshapes with regard to $f_{4}$,
which measures the nodal difference between $\Omega_{\mathbf{x}}$ and the
target $\Omega_{\mathbf{t}}$.
GP($\boldsymbol{\alpha}_{1:4}$)-EI($\boldsymbol{\alpha}_{1:4}$) exhibits very
good results, as well as
GP($\boldsymbol{\alpha}^{a}$)-EI($\boldsymbol{\alpha}^{a}$), which mainly
selects $\mathbf{v}^{3}$ and $\mathbf{v}^{4}$ ($\mathbf{v}^{1}$,
$\mathbf{v}^{2}$ and other eigenshapes are sometimes selected too). Even
though the shape reconstruction is enhanced,
GP($\boldsymbol{\alpha}_{1:16}$)-EI($\boldsymbol{\alpha}_{1:16}$) and
GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$) have poor
results because of the increase in dimension which is not accompanied by
additional information, as already pointed out during the comparison of the
predictive capability of these GPs for small budgets, see Table 12.
GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$) performed
better than GP($X$)-EI($X$) in Table 12, yet its optimization performance is
decreased. This is certainly due to the initial DoE: both DoEs are space-
filling in their respective input space ($X$ or the hyper-rectangle of
$\boldsymbol{\alpha}\in\mathcal{A}$ containing $\mathcal{A}_{N}$). However,
there is a significant difference between the minima in these DoEs: the
average minimum over the 10 runs was 2.57 for GP($X$)-EI($X$) (hence better
than the eventual average best value for
GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$)), and 9.22
for GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$). While
GPs built over the entire $\boldsymbol{\alpha}$ space (e.g. the additive one)
suffer from the same drawback, the selection of variables identifies the
dimensions to focus on to rapidly decrease the objective function. This remark
applies only to the rectangular heart test case and one may wonder what level
of generality it contains. Contrarily to the previous example where building
the GP in the space of all (informative) eigenshapes led to the best results,
this strategy
(GP($\boldsymbol{\alpha}_{1:40}$)-EI($\boldsymbol{\alpha}_{1:40}$)) performs
weakly here because of the higher dimension.

The variants of the additive GP perform well too but they are slightly
outperformed by
GP($\boldsymbol{\alpha}_{1:4}$)-EI($\boldsymbol{\alpha}_{1:4}$). As the
objective function mainly depends on $\mathbf{v}^{3}$ and $\mathbf{v}^{4}$,
always classified as active, strategies that do not put too much emphasis or
that neglect $\boldsymbol{\alpha}^{\overline{a}}$ (namely,
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
and
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI($\boldsymbol{\alpha}^{a}$))
perform the best. This explains the good performance of
GP($\boldsymbol{\alpha}_{1:4}$)-EI($\boldsymbol{\alpha}_{1:4}$), which
disregards $\alpha_{5},\dotsc,\alpha_{40}$. The maximization of the EI with
respect to the full $\boldsymbol{\alpha}$ is hindered by the high dimension.
Again, the performance decreases when the EI is maximized via the $X$ space.
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
and GP($\boldsymbol{\alpha}_{1:4}$)-EI($\boldsymbol{\alpha}_{1:4}$) need more
iterations to attain good values (smaller than 0.5) than
GP($\boldsymbol{\alpha}^{a}$)-EI($\boldsymbol{\alpha}^{a}$) and
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI($\boldsymbol{\alpha}^{a}$)
which are early starters. This might be due to the additional though less
critical components ($\overline{\alpha}$ or $\alpha_{1},\alpha_{2}$,
respectively) considered by these methods.

#### NACA 22 optimization

In this last test case, we compare two of the aforementioned algorithms by
optimizing the lift coefficient and the drag coefficient of a NACA 22 airfoil
($f_{7L}$ and $f_{7D}$). The simulation is made with a computational fluids
dynamic code that solves the Reynolds Averaged Navier-Stokes (RANS) equations
with $k-\varepsilon$ turbulence model. Since a single call to the simulator
(one calculation of $f_{7}$) takes about 20 minutes on a standard personal
computer, only two runs are compared for each objective. The first algorithm
is the classical Bayesian optimizer where the GP is built in CAD parameter
space, GP($X$)-EI($X$). In the second algorithm,
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed,
the GP is built in the $\mathcal{V}$ basis of eigenshapes, while prioritizing
the active dimensions, $\boldsymbol{\alpha}^{a}$, via the additive GP and the
EI random embedding method with the replication option, see Section 4.2. The
optimization in the eigenshape basis starts with a DoE of $n=10$ designs and
is run for $p=90$ additional iterations while, because there are 22 $x_{i}$’s,
the optimization in the CAD parameters space starts using $n=50$ designs and
is run for $p=50$ iterations.

Figure 40 shows the optimization runs of both algorithms for the minimization
of the NACA 22’s drag (top) and lift (bottom), and Figure 41 the resulting
airfoils.

Figure 40:  Top row: drag optimization of the NACA 22 airfoil in the reduced
eigenbasis with
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
(left) or carried out in the CAD parameters space with GP($X$)-EI($X$)
(center). Low drag airfoils are found with
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
while the classical method still evaluates the airfoils of the initial design
of experiments (right). Bottom row: lift optimization of the NACA 22 airfoil
in the reduced eigenbasis with
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
(left) or carried out in the CAD parameters space with GP($X$)-EI($X$)
(center). High lift airfoils are found while the classical method still
evaluates the airfoils of the initial design of experiments (right), i.e.,
lower objective functions are obtained faster.

Drag optimization

Lift optimization

GP($X$)-EI($X$)

AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed

GP($X$)-EI($X$)

AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed

Figure 41: Airfoils found by the compared optimization algorithms. Top: drag
minimization, bottom: lift maximization. Left: optimization with the
GP($X$)-EI($X$) algorithm, right: optimization with the
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
algorithm.

In this application, the main advantage of the
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
(Figure 40, top left and bottom left) over the standard Bayesian optimizer
(top center and bottom center) is that it enables an early search for low
drag, respectively high lift airfoils, at a time when the standard approach is
still computing its initial DoE. Indeed, the classical method needs much more
function evaluations for building the initial surrogate model (black dots)
because the inputs live in a space of higher dimension. The approach
introduced in this paper would further gain in relevance in problems with more
than $d=22$ CAD parameters, where it would almost be impossible to build a
large enough initial design of experiments (whose size is typically of the
order of $10\times$ dimension [28]).

It is observed in Figure 41 that smoother airfoils are obtained with
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
(right column), because it uses a shape coordinate system instead of treating
the $L_{i}$’s (i.e., $x_{i}$’s with local influences on the airfoil, see
Figure 27) separately, as is done by GP($X$)-EI($X$) (left column). When the
optimization aims at minimizing the drag, the
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
airfoil (top right) is smoother than the GP($X$)-EI($X$) one (top left). And
when the objective is to maximize the lift, the camber of the
AddGP($\boldsymbol{\alpha}^{a}+\boldsymbol{\alpha}^{\overline{a}}$)-EI embed
airfoil (bottom right) is increased in comparison with the design yielded by
GP($X$)-EI($X$) (bottom left).

##  5 Conclusions

In this paper we have proposed a new methodology to apply Bayesian
optimization techniques to parametric shapes and other problems where a pre-
existing set of relevant points and a fast auxiliary mapping exist. Instead of
working directly with the CAD parameters, which are too numerous for an
efficient optimization and may not be the best representation of the
underlying shape, we unveil the lower dimensional manifold of shapes through
the auxiliary mapping and PCA. The dimensions of this manifold that contribute
the most to the variation of the output are identified through an $L^{1}$
penalized likelihood and then used for building an additive Gaussian Process
with a zonal anisotropy on the selected variables and isotropy on the other
variables. This GP is then utilized for Bayesian optimization.

The construction of the reduced space of variables opens the way to several
strategies for the maximization of the acquisition criterion, in particular
the restriction or not to the manifold and the replication. The different
variants for the construction of the surrogate model and for the EI
maximization have been compared on 7 examples, 6 of them being analytical and
easily reproducible, the last one being a realistic airfoil design.

Even though specific variants are more or less adapted to features of specific
test problems, the supervised dimension reduction approach and the
construction of an additive GP between active and inactive components have
given the most reliable results.

Regarding the EI maximization our experiments highlight the efficiency of the
random embedding in the space of inactive variables in addition to the
detailed optimization of the active variables. It is a trade-off between
optimizing the active variables only, and optimizing all variables. Benefits
have been observed for not restricting this inner maximization to the current
approximation of $\mathcal{A}$ as well as for the virtual replication of
points outside $\mathcal{A}$ when $\boldsymbol{\alpha}\notin\mathcal{A}$ is
promoted by the EI.

Further research should consider shapes made of multiple elements such as the
one in Example 3. This is of practical importance and it brings a new
theoretical feature, the presence of symmetries in $\Phi$. The knowledge about
symmetries has to be propagated to the eigenshape space to enhance the
surrogate model.

## Acknowledgments

This research was partly funded by a CIFRE grant (convention #2016/0690)
established between the ANRT and the Groupe PSA for the doctoral work of David
Gaudrie.

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