2022 [1,2]Reinoud Jan Slagter [1]Asfyon, Astronomisch Fysisch Onderzoek
Nederland , Bussum, The Netherlands 2]former: University of Amsterdam,
Department of Theoretical Physics, The Netherlands

# Conformal Dilaton-Higgs Gravity on Warped Spacetimes: Black Hole Paradoxes
Revisited

<EMAIL_ADDRESS><EMAIL_ADDRESS>*  [

###### Abstract

We investigate on a Randall-Sundrum warped spacetime, a Kerr-like black hole
in the conformal dilaton-Higgs $(\omega,\Phi)$ gravity model. We applied the
antipodal boundary condition on the Klein surface using the
$\mathds{Z}_{2}$-symmetry in the "large" (bulk) extra dimension. It turns out
that the pseudo-Riemannian 5D manifold can be written as an effective 4D
Riemannian brane spacetime, $\mathds{R}^{2}_{+}\times\mathds{R}^{1}\times
S^{1}$, where $\mathds{R}^{2}_{+}$ is conformally flat. The solution in valid
on both manifolds. So the solution can equally well described by an instanton
solution. An advantage is that antipodicity can be maintained without a "cut-
and-past" method or to rely on quantum cloning, when treating the scattering
description of the evaporation process of the Hawking radiation. We need the
windingnumber as quantum number. Moreover, the equations are invariant under
time reversal. The problem of finding the matching condition of the near-
horizon approximation and the far-away Regge-Wheeler approximation, can
possibly be solved by splitting the spacetime in a dilaton field times an "un-
physical" spacetime, which is conformally flat. In the case of a constant
gauge field, we find that the conform invariant mass term
$\sim\Phi^{2}\omega^{2}$ in the Lagrangian follows directly from the
superfluous dilaton equation by suitable choice of the scale of the extra
dimension. Finally, we bring forward the relation between the embedded Klein
surface in $\mathds{R}^{4}$ and the quantum mechanical information paradox.

###### keywords:

Conformal invariance, Dilaton-Higgs fields, Warped spacetimes, Antipodal
boundary condition, Klein surface, Instanton, Black hole paradoxes, Minimal
surfaces

##  1 Introduction

One of the most profound unsolved problems in theoretical physics is the
discrepancy between quantum mechanics (QM) and general relativity theory
(GRT). One should like to obtain a consistent theory of quantum gravity, a
major goal of theoretical physics (the "holy grail") that would reconcile QM
and Einstein’s GRT. It is conjectured that the problem how to handle quantum-
gravity effects, will be found near the horizon of black holes.

Back holes are in quite another perspective, incredibly unique and fascinating
objects. The black hole can take part in the weird and wonderful behavior of
QM. Particles can be in the quantum superposition of multiple states. They
will be entangled until observers will act on the system (¨collapse of the
wavefunction¨). So new Planck scale physics should be necessary to address the
information paradox, complementarity and to avoid the firewall.

During the last decades, several authors proposed different approaches in
order to attack this epic problem. Pioneer work was delivered by
Hawkinghawking1975 , who investigated particle creation near the horizon of a
black hole. He found that black holes radiate as black bodies with a thermal
spectrum. So the black hole would be in a mixed state. However, this seems to
suggest violation of unitary evolution, because all information connected with
the in-falling particle (¨in-state¨), appears to have lost after the
evaporation of the black hole. In quantum field theory (QFT) one usually deals
with pure quantum states which evolve unitarily. A possible solution was
addressed by Almheiri, et al.,alm2013 , in order to solve the conflict between
complementarity and firewalls. The in-falling observer would burn up at the
horizon´s firewall. However, this viewpoint conflicts GRT. Maldacena, et
al,maldacena2013  proposed the ¨ER = EPR¨ hypothesis, which is based on the
suggestion that connectivity in spacetime is equivalent to quantum
entanglement. They assert that quantum entanglement of distant black holes is
equivalent to the existence of an Einstein-Rosen bridge connecting them.
However, one ignores the gravitational interaction between the two regions I
and II in the Penrose diagram. Also the emergence of string theory, holography
and gauge-gravity duality could shed light on this problem. However, the
information retrieval process is not quite clearmathur2009 . Recent
investigation by Foo, et al.foo2021 , suggests that even so-called “spacetime
superpositions” could exist, i.e., that quantum superpositions of different
spacetimes not related by a global coordinate transformation could be
possible222after an proposal of Bekenstein long time agobekenstein1974 ..

In general, one can state that in all these models the dynamical back reaction
is ignored. For example, the evaporation process will have an impact on the
near-horizon spacetime. There will be a strong interaction between, for
example, a scalar field and the gravitation degrees of freedombetz2021 .

The departure from Hawking‘s calculations, leads to quantum mechanical
scattering algebra, realized by in- and out-going wave functions near the
horizonthooft2015 . Later, it was argued that one can reformulate this
departure, by considering soft graviton exchange between matter fields near
the horizon333See the clear overview of Betzios, et al.betz2021 .. Away from
the horizon, one can then introduce an effective Regge-Wheeler gravitational
potential. However, in all these models, the calculations on the quasi-normal
modes were done on a fixed Schwarzschild background geometry, which is nor
adequate for black hole evaporation process: the potential is time dependent.
One should like to obtain a complete scattering description for the outside
observer, which is consistent with the local observer444Remember, an freely
in-falling observer perceive spacetime as Minkowski, so will not notice the
horizon..

This complementarity issue, which means the discrepancy between the local
observer and the far away observer, can be reformulated in conformal dilaton
gravity (CDG) by introducing a dilaton field $\omega$ and by writing
$g_{\mu\nu}=\omega^{\frac{4}{d-2}}\tilde{g}_{\mu\nu}$thooft2015c . Different
observers experience different notions of $\omega$, i.e., the scale of
spacetime. Moreover, $\omega$ (and a scalar field $\Phi$) and
$\tilde{g}_{\mu\nu}$ are invariant under
$\tilde{g}_{\mu\nu}\rightarrow\Omega^{\frac{4}{d-2}}\tilde{g}_{\mu\nu},\omega\rightarrow\Omega^{-\frac{d-2}{2}}\omega,\Phi\rightarrow\Phi^{-\frac{d-2}{2}}\omega$.
$\Omega$ can be used, for example, to make Ricci scalar for the local observer
zero. Further, one observer sees Hawking radiation as matter, while on other
one as part of his vacuum. The in-falling observer will not notice any change
in the mass of the black hole. The out side observer, on the other hand,
observes a gradually shrinking mass. It is conjectured that this conformal
description will be necessary in order understand the black hole paradoxes.

The approach we will address in this manuscript, relies on the antipodal
boundary condition on a warped 5D Spacetime. This idea was introduces decades
ago by Schrödingerschrod1957 . It is sometimes called the elliptic
interpretation of spacetime. Interesting investigation on this mapping where
done by Sanchez, et al.san1987 ; san1988 ; fol1987  on a AdS spacetime. The
model was intensively extended by´t Hooftthooft2016 ; thooft2018b ; thooft2019
in a slightly different setting, by considering a new treatment of the
gravitational back-reaction. In brief, antipodal points on the horizon
represent the same world-point or event. The spacetime inside the horizon is
removed such that the edges are glued together by identifying the antipodes.
In the language of a Penrose diagram, "region II" is the antipode of "region
I".

In order to avoid wormhole constructions or demanding "an other universe" in
the Penrose diagram, it is essential that the asymptotic domain of the
manifold maps one-to-one onto the ordinary spacetime in order to preserve the
metric. In fact, one deals with one black hole. ´t Hooft performed
calculations on the unitary evolution matrix by using the antipodal boundary
condition, i.e., the transverse spherical coordinates $(\theta,\varphi)$ at
region II, represent the antipode of region I. So there are no fixed points.
In fact, there is no "hidden" sector in the Penrose diagram. The resulting
Hartle-Hawking vacuum state remains a pure state in stead of a
thermodynamically mixed state for the outside observer. The Hawking particles
are emerging from I, are maximally entangled with particles emerging from II.
This approach has the potential to solve the information paradox and a
firewall is not necessary. The antipodal boundary condition555Also called the
”cut-and-past” procedure or firewall transformation., minimizes the number of
unconventional assumptions or to rely on full string theory and ¨fuzzyballs".
It will require that the local laws of physics are invariant across the black
hole boundarythooft2021 . Usually, one considers the Hawking particles as
excitations of low energy particles that fulfill the standard model and
applies perturbative quantum gravity on a fixed background metric (i.e., the
gravitons). The distant observer notices the Hilbert space of low energy
particles at given time. One then maps the states at later times, 1 on 1 on
earlier times. One basically bypasses the Planck-area. It manifests itself
only by the higher modes of the spherical harmonics (in the angle variables).
An unitary S matrix can then be constructed if one applies a cut-off for the
higher modes of the harmonics. In a recent study,´t Hooftthooft2022  tries to
avoid the antipodal boundary condition, in order to overcome some
inconsistencies, into a quantum clone description, in spherical symmetric
case.

In our 5D warped spacetime, however, this change is not necessary, due to the
cylindrical symmetry of the Kerr-type spacetime we considered. It is strongly
conjectured that the center of galaxies harbors a spinning Kerr black holes
and not a Schwarzschild black hole. So there is a preferred direction, i.e.,
the $z$-axis. If one omits the $dz^{2}$, one obtains the well known exact
Ban̆ados-Teitelboim-Zanelli (BTZ) black hole solution in $(2+1)$-dimensional
spacetimebanadoz1992 ; slagter2019b .

In the recently found vacuum solutionslagter2022c , no further inconsistency
was encountered, because we had only to deal with the gravitational freedom,
i.e., the dilaton field $\omega$, which was solved exactly666Note that, due to
Birkhoff’s theorem, spherical symmetric objects will not emit gravitational
radiation..

In the present case, the scalar field will be written as $\Phi=\eta
X(t,r)e^{in\varphi}Y_{m}(\varphi)$ and with the gauge field
$A\sim\frac{n}{e}$. In order find a consistent description of the antipodal
map, the warped fifth dimension is mandatory, in order to apply the non-
orientable Klein surface.

In our model we will also try to avoid in the dynamical evolution of the
fields (scalar, dilaton and metric components) on the Kerr spacetime, the
late-times and early-times behavior, by writing the PDE´s in Kruskal
coordinates777We will work in polar coordinates, because the Kerr spacetime is
axisymmetric. In general, the field variables will then be written as
$F(t,r,z,\varphi,y)=F(U,V)H(z,\varphi)K(y)$, with $H$ cylindrical harmonics,
$(U,V)$ Kruskal coordinates and $y$ the RS bulk dimension. Note: the
axisymmetry is also needed when the scalar-gauge field, considered as vortex,
is incorporated.

We know that local QFT and GRT are invariant under CPT transformations. In the
model we consider here, we advocate that antipodal map preserves also CPT. The
basic argument being that CPT is an exact symmetry of nature, i. e., of QFT
around flat background. It has the potential to be elevated to a form of
discrete gauge symmetry in a full theory of quantum gravity. One says that
region II is a CPT-transformed quantum copy of region I.

There are some other approaches to this invariance. The conformal group
implies Poincaré invariance and so PT invariancemannheim2013 . The PT symmetry
would be a necessary and sufficient condition for unitary time evolution
whereas hermiticity is only a sufficient condition. Dropping the hermiticity
in favor of PT symmetry is not generally accepted (see also the discussion by
Betzios, et al.bet2016 ; bet2017  and references therein).

There are several reasons for considering a warped spacetime of Randall and
Sundrum (RS)ran1999a ; ran1999b . The recently found exact solution of the
Kerr-like spacetime without a scalar field, was found in the CDG model on a
warped RS spacetimeslagter2018 ; slagter2021 ; slagter2022b ; slagter2022c .
These so-called brane-world models provide a simplification of the full string
model. In the later string models, the extra dimensions are compactified or
fold in on themselves in many ways, meaning that there are to many possible
solutions to be able to make a clear prediction. In the RS model, only gravity
can propagate into the bulk, while all other fields resides on the brane.
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 problemmann2005 .
In a former studyslagterpan2016  we applied this model on a Friedmann-
Lemaître-Robertson-alker (FLRW) spacetime. There is a contribution from the
projected 5D Weyl tensor on the effective 4D brane, which 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 $M_{Pl}$. There are some other
arguments which advocate for the 5D model. First, the warped model could
possibly solve the hierarchy problem. Secondly, the description of the
antipodal boundary condition by means of the Möbius strip in the 4D model, can
be extended by considering the Klein surface, which can be embedded in
$\mathds{R}^{4}$. Quite recently, Maldacena, et al.maldacena2021 , found an
interesting wormhole solution on the RS model. It is remarkable that the found
exact spacetime solution $\tilde{g}_{\mu\nu}$ together with the dilaton
solution, is the same for the 5D and effective 4D Einstein equations. In a
covariant approach of the model, it is mandatory to solve these equations
together in order to obtain consistencyshirom2000 ; shirom2003 . Moreover, it
turns out that the solution can be maintained, when one switches to the
Riemannian 5D spacetime (Wick rotation). This instanton solution can then be
embedded in $\mathds{R}^{4}$, i.e., our effective brane in the RS model. This
becomes clear when one complexifies the space as
$\mathds{C}^{1}\times\mathds{C}^{1}$. By the Hopf fibration, one then makes
the connection with our real space.

As already mentioned, the BTZ solution shares some features with the 4D Kerr
solution and gained new interest because it can be used to study quantum
gravity issues. The model needs a cosmological constant and is asymptotically
AdS. The relation with the dynamical "uplifted"" BTZ solution888In this case
the $dz^{2}$ term is maintained. It is remarkable that $\Lambda$ must then be
taken zero. was presented by Slagterslagter2019b . After the discovery of the
AdS/CFT correspondence, the BTZ solution gained new interest and became a tool
to understand black hole entropy. It is not yet clear whether pure 3D Einstein
gravity make sense quantum mechanically without string theory embedding.
Adding a scalar field will make the model more realistic. The improvement of
the treatment of the gravitational interaction of the propagating modes on a
Schwarzschild background was done by Betzios, et al.betz2021 . They
distinguish two regions. Far from the horizon the propagating fields evolve
semi-classical under the Regge-Wheeler potential. In the near-horizon region,
where the Regge-Wheeler potential is taken zero, they apply a gravitational
scattering that captures non-perturbative soft graviton exchange. An
appropriate matching of the two dynamical systems is then necessary in order
to obtain a unitary scattering matrix.

It is worth making some remarks on the recent work of Gaddam, et.
al.gaddam2020 ; gaddam2021 ; gaddam2022 ; gaddam2022b , on the information
paradox. They calculated the soft graviton exchange between the in- and out-
going quanta, using the eikonal phase approach, in order to find the
gravitation back reaction near the horizon. It is possible that the so-called
multiple-scale wave approximation could be better appliedchoq1969 ;
slagter2000 , in order to keep track of the different orders of approximation.
The expansion parameter will then be the ratio of the background scale and the
fast varying scale. The expansion in harmonics will follow directly from the
several orders of perturbation equations.

In this manuscript, we try to make no distinction between the two regions. By
considering the conformal invariant model with the dilaton and scalar field,
we obtain two (quantum) interacting fields, which differ only in the effective
potential. They contain also the metric component, which determines the near
horizon behavior. In the vacuum situation, an exact time dependent solution
was found. In the new model, one can be guided by this solution, in order to
obtain a numerical solution. Obstructions encountered in the antipodal
boundary condition can be avoided in the 5D warped version: the antipodal map
can be constructed on a Klein surfaceslagter2022c . It is interesting to
compare the dynamical behavior of the scale function $\omega$999which is,
after all, part of the spacetime. with the theory of gravitational echoes and
so-called scrambling timeabedi2017 .

The outline of this manuscript is the following. In section 2 we describe the
new CDH model and the related, earlier found, vacuum solution. In section 3 we
present the connection of the Klein surface with antipodicity. In section 4 we
outline some topological aspects. In section 5 we return to the information
paradox in context with the Klein surface and in section 6 we briefly
summarize the application of the high-frequency approximation.

##  2 The model

It is conjectured that a conformal invariant theory of gravity promises
interesting results when applied to the quantum-gravity area. In particular,
the conformal dilaton gravity model (CDG)alvarez2014 ; codello2013 ;
thooft2015b ; oda2015 . We shall see that it is a route to construct a
topologically regular theory of gravity.

A theory is called conformally invariant at the classical level, if its action
is invariant under the conformal group of translations, dilatations LT and
special conformal transformationsfelsager1998 . This is a local symmetry, if
the metric is dynamically, as will be considered here101010It is a global
symmetry, when one considers the spacetime fixed.. Let us consider the
conformal invariant Lagrangian

$\displaystyle S=\int
d^{d}x\sqrt{-\tilde{g}}\Bigl{[}-\frac{1}{2}\xi(\Phi\Phi^{*}+\omega^{2})\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}\Bigl{(}{\cal
D}_{\mu}\Phi({\cal
D}_{\mu}\Phi)^{*}+\partial_{\mu}\omega\partial_{\nu}\omega\Bigr{)}$ (1)
$\displaystyle-\frac{1}{4}F_{\alpha\beta}F^{\mu\nu}-V(\Phi,\omega)-\Lambda\kappa^{\frac{4}{d-2}}\xi^{\frac{d}{d-2}}\omega^{\frac{2d}{d-2}}\Bigr{]},$
(2)

which is invariant under

$\tilde{g}_{\mu\nu}\rightarrow\Omega^{\frac{4}{d-2}}\tilde{g}_{\mu\nu},\quad\omega\rightarrow\Omega^{-\frac{d-2}{2}}\omega,\quad\Phi\rightarrow\Omega^{-\frac{d-2}{2}}\Phi$
(3)

for suitable choice of $V$ (for example the simplest one, $V=0$). Further,
$\xi=(d-2)/4(d-1)$. The covariant derivative is taken with respect to the "un-
physical" $\tilde{g}_{\mu\nu}$, which is defined in the CDG model, initiated
by ’t Hooftthooft2015b , by writing

$g_{\mu\nu}=\omega^{4/(d-2)}\tilde{g}_{\mu\nu}.$ (4)

$\omega$ represents a dilaton field, which represents the scale dependency.
The gauge-covariant derivative is ${\cal
D}_{\mu}\Phi=\tilde{\nabla}_{\mu}\Phi+ieA_{\mu}\Phi$ and $F_{\mu\nu}$ the
Abelian field strength. Further, we redefined the dilaton,
$\omega^{2}\rightarrow-\frac{6\omega^{2}}{\kappa^{2}}$, in order to ensure
that the field $\omega$ has the same unitarity and positivity properties as
the scalar field. The potential $V(\Phi,\omega)$ is still unspecified. A
massive term in $V(\Phi,\omega)$ will break, in general, the tracelessness of
the energy momentum tensor and therefore breaks the conformal invariance. One
considers then the symmetry as exact and spontaneously broken, just as the BEH
mechanism. $\omega\rightarrow 0$ describes the small distance limit. No
singularity should occur in this limit. We parameterize the scalar field and
gauge field as

$A_{\mu}=\Bigl{[}0,0,0,\frac{1}{e}(P(t,r)-n)\Bigr{]},\quad\Phi=\eta
X(t,r)e^{in\varphi}.$ (5)

The metric $\tilde{g}_{\mu\nu}$ is actually a metatensor. All the scale
dependencies are contained in the dilaton and will be handled on equal footing
with the scalar field and can be extended to scales close to the Planck scale.
We consider here the Kerr-like spacetime on a warped 5D spacetime with
$\mathds{Z}_{2}$-symmetry slagter2021 ; slagter2022c

$ds^{2}=\omega(t,r,y)^{4/3}\Bigl{[}-N(t,r)^{2}dt^{2}+\frac{1}{N(t,r)^{2}}dr^{2}+dz^{2}+r^{2}(d\varphi+N^{\varphi}(t,r)dt)^{2}+d\mathpzc{y}^{2}\Bigr{]},$
(6)

where $\mathpzc{y}$ is the extra dimension (not to confuse with the Cartesian
y111111One can also use the Eddington-Finkelstein coordinates
$(U,r,z,\varphi,\mathds{y})$.). Here $\omega$ was called a "warp factor" in
the formulation of RS 5D warped spacetime with one large extra dimension and
negative bulk tension. It turns out that one can write
$\omega(t,r,\mathpzc{y})=\omega_{1}(t,r)\omega_{2}(\mathpzc{y})$, with
$\omega_{2}(\mathpzc{y})=y_{0}$=constant (the length scale of the extra
dimension). The Standard Model (SM) fields are confined to the 4D brane, while
gravity acts also in the fifth dimension. It possesses
$\mathds{Z}_{2}$-symmetry, which means that when one approaches the brane from
one side and go through it, one emerges into the bulk that looks the same, but
with the normal reversed. Since the pioneering publication of RS, many
investigations were done in several related domains. In particular, Shiromizu
et.al. shirom2003 , extended the RS model to a fully covariant curvature
formalism. See also the work of Maartens maartens2010 . It this extended
model, an effective Einstein equation is found on the brane, with on the right
hand side a contribution from the 5D Weyl tensor which carries information of
the gravitational field outside the brane. So the brane world observer may be
subject to influences from the bulk.

One then obtains field equations for the 5D spacetime together with the
effective 4D Einstein equations (we took an empty bulk) slagter2021 ;
slagter2022c

${{}^{(5)}}{G_{\mu\nu}}=-\Lambda_{5}{{}^{(5)}g_{\mu\nu}},$ (7)
${{}^{(4)}G_{\mu\nu}}=-\Lambda_{eff}{{}^{(4)}g_{\mu\nu}}+\kappa_{4}^{2}{{}^{(4)}T_{\mu\nu}}+\kappa_{5}^{4}{\cal
S}_{\mu\nu}-{\cal E}_{\mu\nu},$ (8)

where we have written

${{}^{(5)}g_{\mu\nu}}={{}^{(4)}g_{\mu\nu}}+n_{\mu}n_{\nu},$ (9)

with $n^{\mu}=[0,0,0,0,\frac{1}{\sqrt{y_{0}}}]$ the unit normal to the brane.
Here ${{}^{(4)}T_{\mu\nu}}$ is the energy-momentum tensor on the brane, which
also contains the contribution from the dilaton

${{}^{(4)}T_{\mu\nu}}=T^{(\omega)}_{\mu\nu}+T^{(scalar)}_{\mu\nu}$ (10)

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

${\cal
E}_{\mu\nu}={{}^{(5)}}C^{\alpha}_{\beta\rho\sigma}n_{\alpha}n^{\rho}{{}^{(4)}}g_{\mu}^{\beta}{{}^{(4)}}g_{\nu}^{\sigma},$
(11)

represents the projection of the bulk Weyl tensor orthogonal to $n^{\mu}$. The
effective gravitational field equations on the brane are not closed. One must
solve at the same time the 5D gravitational field in the bulk. For an empty
bulk, there is no exchange of energy-momentum between bulk and brane. The
interaction is purely gravitational. From the 4D conservation equations
(contracted Bianchi identities), one then obtains $\nabla^{\mu}{\cal
E}_{\mu\nu}=\kappa_{5}^{4}{\cal S}_{\mu\nu}$. It tells us that (1+3)-spacetime
variations in the matter-radiation can source KK modes. Now we replace again

${{}^{(4)}\tilde{g}_{\mu\nu}}\rightarrow\bar{\omega}^{2}{{}^{(4)}\tilde{g}_{\mu\nu}}.$
(12)

Variation of the action Eq.(2) for a general $V$121212We omit for the time
being the cosmological constant term $\sim\Lambda\omega^{4}$. with respect to
$\tilde{g}_{\mu\nu}$, $\Psi$ and $\omega$ yields ($d=4,5$)

$\displaystyle\xi\bar{\omega}\tilde{R}-\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\bar{\omega}-\frac{\partial
V}{\partial\bar{\omega}}=0,$ (13)
$\displaystyle\xi\Phi\tilde{R}-\tilde{g}^{\mu\nu}{\cal D}_{\mu}{\cal
D}_{\nu}\Phi-\frac{\partial V}{\partial\Phi^{*}}=0,$ (14)

and

$\displaystyle(\bar{\omega}^{2}+\Phi\Phi^{*})\tilde{G}_{\mu\nu}=T^{(\Phi)}_{\mu\nu}+T_{\mu\nu}^{(\omega)}-\tilde{g}_{\mu\nu}V-(\bar{\omega}^{2}+\Phi\Phi^{*}){\cal
E}_{\mu\nu}\delta^{4}_{d},$ (15)

with

$\displaystyle
T_{\mu\nu}^{(\omega)}=\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\bar{\omega}^{2}-\tilde{g}_{\mu\nu}\tilde{\nabla}^{2}\bar{\omega}^{2}+\frac{1}{\xi}\Bigl{(}\frac{1}{2}\tilde{g}_{\alpha\beta}\tilde{g}_{\mu\nu}-\tilde{g}_{\mu\alpha}\tilde{g}_{\nu\beta}\Bigr{)}\partial^{\alpha}\bar{\omega}\partial^{\beta}\bar{\omega},$
(16)

and

$T^{(\Phi)}_{\mu\nu}={\cal D}_{\mu}{\cal
D}_{\nu}\Phi\Phi^{*}-\tilde{g}_{\mu\nu}{\cal D}^{\alpha}{\cal
D}_{\alpha}\Phi\Phi^{*}+\frac{1}{\xi}\Bigl{(}\frac{1}{2}\tilde{g}_{\alpha\beta}\tilde{g}_{\mu\nu}-\tilde{g}_{\mu\alpha}\tilde{g}_{\nu\beta}\Bigr{)}{\cal
D}^{\alpha}\Phi{\cal D}^{\beta}\Phi^{*}.$ (17)

Note that on the right hand side of the 4D effective Einstein equations now
appears for $d=4$, the contribution from the bulk in the correct form. The
only unknown functions are the metric components, the dilaton and scalar
fields. $\omega$ and $\Phi$ can be treated on equal footing as normal quantum
fields on the small scale. The dilaton equation Eq.(13) is superfluous in the
vacuum caseslagter2022c . In the general case with a scalar field, there will
be an interaction between $\omega$ and $\Phi$ and some constraint on $V$.

###  2.1 The dilaton-Higgs model in conformal invariant gravity

####  2.1.1 case: $A_{\mu}(t,r)=0$

Firstly, we can switch off the gauge potential, $A_{\mu}=0$. From the Einstein
field equations and the scalar equation one can isolate the PDE’s for $N$,
$\omega$ and $\Phi$:

$\displaystyle\ddot{N}=-N^{4}\Bigl{(}N^{\prime\prime}+3\frac{N^{\prime}}{r}+\frac{N^{\prime
2}}{N}\Bigr{)}+3\frac{\dot{N}^{2}}{N}$ (18)
$\displaystyle\frac{1}{(\eta^{2}X^{2}+\omega^{2})}\Bigl{[}6N^{5}(\eta^{2}X^{\prime
2}+\omega^{2})-6(\eta^{2}\dot{X}^{2}+\dot{\omega}^{2})-\frac{3}{2r}N^{5}(\eta^{2}XX^{\prime}+\omega\omega^{\prime})$
(19) $\displaystyle+\frac{9}{2r^{2}}\eta^{2}n^{2}N^{2}X^{2}-6N^{3}\Bigl{(}\eta
X\frac{\partial V}{\partial X}+\omega\frac{\partial
V}{\partial\omega}+V\Bigr{)}\Bigr{]}$ (20)
$\displaystyle\ddot{X}=N^{4}\Bigl{(}X^{\prime\prime}+\frac{X^{\prime}}{r}+2\frac{X^{\prime}N^{\prime}}{N}\Bigr{)}+2\frac{\dot{X}\dot{N}}{N}+\frac{2X}{(\eta^{2}X^{2}+\omega^{2})}\Bigl{[}N^{4}(\eta^{2}X^{\prime
2}+\omega^{\prime 2})$ (21)
$\displaystyle-(\eta^{2}\dot{X}^{2}+\dot{\omega}^{2})+N^{2}(V-\omega\frac{\partial
V}{\partial\omega})-\frac{N^{2}}{2X}(\eta^{2}X^{2}-\omega^{2})\frac{\partial
V}{\partial X}\Bigr{]}$ (22)
$\displaystyle\ddot{\omega}=N^{4}\Bigl{(}\omega^{\prime\prime}+\frac{\omega^{\prime}}{r}+2\frac{\omega^{\prime}N^{\prime}}{N}\Bigr{)}+2\frac{\dot{\omega}\dot{N}}{N}+\frac{2\omega}{(\eta^{2}X^{2}+\omega^{2})}\Bigl{[}N^{4}(\eta^{2}X^{\prime
2}+\omega^{\prime 2})$ (23)
$\displaystyle-(\eta^{2}\dot{X}^{2}+\dot{\omega}^{2})+N^{2}(V-\eta
X\frac{\partial V}{\partial
X})-\frac{N^{2}}{2\omega}(\eta^{2}X^{2}-\omega^{2})\frac{\partial
V}{\partial\omega}\Bigr{]}.$ (24)

The superfluous equation for $\omega$ delivers a constraint equation,

$\displaystyle{N^{\varphi}}^{2}=\frac{1}{\eta^{2}n^{2}X^{2}}\Bigl{[}\frac{\eta^{2}n^{2}X^{2}}{r^{2}}+\eta
X\frac{\partial V}{\partial X}+\omega\frac{\partial
V}{\partial\omega}+V\Bigr{]},$ (25)

which is consistent with the requirement of the tracelessness of the energy
momentum tensor131313The tracelessness of the energy-momentum tensor can be
seen as part of the field equations.. This constraint is used in order to
decouple the equations for $N^{\varphi}$.

We observe that the equations for the dilaton and scalar equations are
identical, except the potential terms, as expected. They are of the type of a
Klein-Gordon PDE.

Further, the equations for $X$ and $\omega$ are invariant under
$t\rightarrow-t$ and $r\rightarrow-r$. So the antipodal pulse wave fulfills
the same equation. From the equations for $N^{\varphi}$, it follows that the
most realistic solution is $N^{\varphi}=0$. The potential becomes

$V=\alpha_{1}X^{\alpha_{2}}\omega^{\frac{2}{3}-\eta\alpha_{2}}-\frac{3n^{2}\eta^{2}}{2r^{2}(3\eta-1)}(X^{2}+\alpha_{3}X^{\frac{2}{3\eta}}),$
(26)

with $\alpha_{i}$ constants. Another solution is given by

$N^{\varphi}=F_{1}(t)-\frac{3}{2}\int
F_{2}(r)\log(\eta^{2}X^{2}+\omega^{2})dr,$ (27)

with $F_{1}$ and $F_{2}$ arbitrary functions. It turns out that this solution
is of less importance.

####  2.1.2 case: $A(t,r)=cst$.

A more interesting situation is obtained by taking a constant gauge potential
$A_{\varphi}=-\frac{n}{e}$, i.e., $P(t,r)=0$ in Eq.(5). The equations then
becomes

$\displaystyle\ddot{N}=-N^{4}\Bigl{(}N^{\prime\prime}+3\frac{N^{\prime}}{r}+\frac{N^{\prime
2}}{N}\Bigr{)}+3\frac{\dot{N}^{2}}{N}+\frac{2N}{(\eta^{2}X^{2}+\omega^{2})}\Bigl{[}y_{0}N^{2}V$
(28) $\displaystyle+3N^{4}(\eta^{2}X^{\prime 2}+\omega^{\prime
2})-3(\eta^{2}\dot{X}^{2}+\dot{\omega}^{2})-\frac{3}{4r}N^{4}(\eta^{2}XX^{\prime}+\omega\omega^{\prime})\Bigr{]}$
(29)
$\displaystyle\ddot{X}=N^{4}\Bigl{(}X^{\prime\prime}+\frac{X^{\prime}}{r}+2\frac{X^{\prime}N^{\prime}}{N}\Bigr{)}+2\frac{\dot{X}\dot{N}}{N}+\frac{N^{2}}{\eta}\frac{\partial
V}{\partial X}$ (30)
$\displaystyle+\frac{2X}{(\eta^{2}X^{2}+\omega^{2})}\Bigl{[}N^{4}(\eta^{2}X^{\prime
2}+\omega^{\prime
2})-(\eta^{2}\dot{X}^{2}+\dot{\omega}^{2})+\frac{y_{0}}{3}N^{2}V\Bigr{]}$ (31)
$\displaystyle\ddot{\omega}=N^{4}\Bigl{(}\omega^{\prime\prime}+\frac{\omega^{\prime}}{r}+2\frac{\omega^{\prime}N^{\prime}}{N}\Bigr{)}+2\frac{\dot{\omega}\dot{N}}{N}+y_{0}N^{2}\frac{\partial
V}{\partial\omega}$ (32)
$\displaystyle+\frac{2\omega}{(\eta^{2}X^{2}+\omega^{2})}\Bigl{[}N^{4}(\eta^{2}X^{\prime
2}+\omega^{\prime
2})-(\eta^{2}\dot{X}^{2}+\dot{\omega}^{2})+\frac{y_{0}}{3}N^{2}V\Bigr{]}.$
(33)

From the superfluous $\omega$ equation we obtain that the potential must be
taken

$V(X,\omega)=\beta_{1}X^{\beta_{2}}\omega^{\frac{2}{3}-\frac{\eta\beta_{2}}{|y_{0}|}},$
(34)

which is consistent with the tracelessness of the energy-momentum
tensor141414Remember that $\omega$ contains the gravitational constant
$\kappa$ by the redefinition. . Observe that the scale of the extra dimension
enters the equations by $y_{0}$. It is remarkable that we can obtain in the
Lagrangian a quartic conformal invariant matter coupling

$M\sim X^{2}\omega^{2},$ (35)

(Eq.(34)) for suitable combination of the parameters, i.e.,

$|y_{0}|=\frac{3}{2}\eta.$ (36)

If we inspect the equations for the scalar and dilaton field (for $N=1$), we
observe that the total potential is

$V_{X}=\frac{\alpha\Bigl{(}\eta
X^{2}(3\eta\beta+2y_{0})+3\beta\omega^{2}\Bigr{)}}{3\eta(\eta^{2}X^{2}+\omega^{2})}X^{\beta-1}\omega^{\frac{(2y_{0}-3\beta\eta)}{3y_{0}}},$
(37)
$V_{\omega}=\frac{\alpha\Bigl{(}\eta^{2}X^{2}(2y_{0}-3\eta\beta)+\omega^{2}(4y_{0}-3\beta\eta)\Bigr{)}}{3(\eta^{2}X^{2}+\omega^{2})}X^{\beta}\omega^{-\frac{(y_{0}+3\beta\eta)}{3y_{0}}}.$
(38)

In fact, one can still integrate the functional integral over $\omega$
exactly. So non-conformal matter does not effect the conformal invariance of
the effective action after integrating over $\omega$ (see for example the
discussion by ´t Hooftthooft2015b ).

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

$\displaystyle\dot{N^{\varphi}}^{\prime}=-3{N^{\varphi}}^{\prime}\frac{(\eta^{2}X\dot{X}+\omega\dot{\omega})}{(\eta^{2}X^{2}+\omega^{2})},$
(39)
$\displaystyle{N^{\varphi}}^{\prime\prime}=-3{N^{\varphi}}^{\prime}\Bigl{[}\frac{1}{r}+\frac{(\eta^{2}XX^{\prime}+\omega\omega^{\prime})}{(\eta^{2}X^{2}+\omega^{2})}\Bigr{]}.$
(40)

The general solution is

$N^{\varphi}=f_{1}\int\frac{1}{r^{3}(\eta^{2}X^{2}+\omega^{2})^{3/2}}dr,$ (41)

with $f_{1}$ a constant. The scalar and dilaton equations differ only by the
potential term. This difference is manifest in the numerical solution,
presented in the next section. Note that $n$ has disappeared from the PDE’s.
We have absorbed the windingnumber (or vortex number) $n$ in the constant
gauge field $|A_{\varphi}|=\frac{n}{e}$151515The scalar-gauge system possesses
a quantized magnetic flux $\sim\frac{n}{e}$, which equals the first Chern
number of $A$ on $\mathds{R}^{2}_{+}$. This has an important consequence in
the expansion of the scalar field in cylindrical harmonics. See section 4.

Figure 1: stable solution of $X,\omega$ and $N$ for the case of a constant
gauge field. The potential is
$V=\beta_{1}X^{\beta_{2}}\omega^{\frac{2}{3}-\frac{\eta\beta_{2}}{y_{0}}}$. We
took for the initial values of the scalar and dilaton field a cosine and sine
function respectively. For $N$ we took the vacuum solutions of the Appendix A.
Further, we applied Neumann boundary conditions. Note that $N$ develops a
singularity and approaches to a constant value for increasing time. The
solution depends critically on the parameters of the potential, for example,
the scale $y_{0}$ of the extra dimension.

####  2.1.3 The numerical solution

It is clear that we can have breather modes of our scalar-dilaton system. In
figure 1 and 2 we plotted a typical solution in the case of a constant gauge
field.

Figure 2: As figure 1, with now as initial value for $\omega$ a $\tanh$
function. We observe that the polynomial behavior in the metric component $N$
is induced by the dilaton. We also plotted the total metric component
$\omega^{4}N^{2}$

##  3 On the double cover of the Klein surface, Riemann surfaces and
antipodicity

###  3.1 Motivation for the Klein surface

For the $S^{2}$ we could apply the stereographic projection
$S^{2}\rightarrow\mathds{R}P^{2}\rightarrow\mathds{C}P^{1}$. In order to make
the map one-to-one, one applies the $\mathds{Z}_{2}$ symmetry identification:
the two antipodal planes are identified, $z\rightarrow-\frac{1}{\bar{z}}$ (see
figure 3). In our situation, we uplift the projection to
$S^{3}\subset\mathds{R}^{4}$ and use cylindrical coordinates $(r,\varphi)$.

Figure 3: Left: the stereographic projected
$S^{2}\rightarrow\mathds{R}P^{2}\rightarrow\mathds{C}P^{1}$. In order to make
the map one-to-one, one applies the $\mathds{Z}_{2}$ symmetry identification:
the two antipodal planes are identified, $z\rightarrow-\frac{1}{\bar{z}}$.
Right: stereographic projection of
$S^{3}\subset\mathds{R}^{4}\simeq\mathds{C}\times\mathds{C}$ with also
antipodal identification, representing the projection of an embedded Klein
bottle.

Figure 4: Artist impression of the trajectory of a scalar (Hawking-) particle
on the Klein surface. $P$ and $\bar{P}$ are the antipodes. Again, we have a
$\mathds{Z}_{2}$ symmetry identification (see figure 3), now for the both
sides of the brane: $g_{\mu\nu}({\bf x},\mathpzc{y})=g_{\mu\nu}({\bf
x},\mathpzc{-y})$. Note that we apply a slightly different antipodal mapping.
We are dealing here with cylindrical symmetric Kerr-like spacetime, where we
expand the scalar field in cylindrical harmonics $\Phi_{n,m}=\eta
X(r,t)e^{in\varphi}Y_{m}(\Omega)$, in stead of spherical ones. Further, we
replace the "cut-and-paste" method by the antipodicity in the bulk space. We
can bypass the notion of "quantum clones" proposed by ´t Hooft
recentlythooft2022 .

By means of the Hopf fibration, one then makes the connection with real
spacetime. For details, we refer to Slagterslagter2022c  and Urbantkeur2003 .
Another important characteristic of the non-orientable Klein surface is the
fact that meromorphic functions remain constantalling1971 . Our solution for
$N$ is meromorphic161616Remember that our solution for $N$ of section 4.2 (or
see Appendix A) could be written asslagter2022c  as a meromorphic polynomial
$\frac{P({\bf z})}{Q({\bf z})}$, with $Deg(P)=5$. . Further, the Klein surface
is homeomorphic to the connected sum of two projective planes.

In the past, intensive research was done on the fundamental group structure of
compact surfaces in $m$ dimensions. It is clear that, for example, the torus
possesses the structure of two infinite cyclic groups. For the projective
plane, the cyclic group of order 2. For the Klein bottle, the fundamental
group can be presented by two generators $a,b$ and $baba^{-1}$massey1971 .

We constructed our model in the 5D RS warped spacetime. So the question is how
we can imagine the evolution of the Hawking particles created near the
horizon, say point $P(U,V,z,\varphi,\mathpzc{y})$ in figure 4. The in-going
particle can travel on the Klein surface, in order to leave in the antipode
$\bar{P}(-U,-V,-z,\varphi+\pi,-\mathpzc{y})$ the horizon
($J:P\rightarrow\bar{P}$ is an inversion in 5D).

The real physics still takes place on the brane. Effectively, we replaced the
"cut-and-past" method in $\mathds{R}^{3}$ by the imaginary transport in
$\mathds{R}^{4}$ (via the Hopf fibration of the 3-sphere in $\mathds{R}^{4}$.

We know that complete embedded surfaces in $\mathds{R}^{3}$ must be
orientable, otherwise we have self-intersection, so the genus changes. Non-
orientable surfaces immersed in $\mathds{R}^{3}$ does not have global Gauss
maps, so no well defined mean curvature. So the application of the Klein
surface embedded in $\mathds{R}^{4}$ is quite interesting in our context.
First, we complexify our hyper-surfaceslagter2022c . Let us write our
coordinates $(r,z,\mathpzc{y},\varphi^{*})$ as171717with
$x=r\sin\varphi,y=r\cos\varphi$,

${\cal V}=z+i\mathpzc{y}=Re^{i\varphi_{1}},\qquad{\cal
W}=x+iy=re^{i\varphi_{2}},$ (42)

where the antipodal map is now ${\cal V}\rightarrow-{\cal
V}\equiv-R^{2}/\bar{\cal V},{\cal W}\rightarrow-{\cal W}\equiv-r^{2}/\bar{\cal
W}$.181818 because $e^{i(\varphi+\pi)}=-e^{i\varphi}$.  Further, ${\cal
V}\bar{\cal V}=z^{2}+\mathpzc{y}^{2}=R^{2},{\cal W}\bar{\cal
W}=x^{2}+y^{2}=r^{2}$. And after inversion

$\displaystyle z=\frac{1}{2}({\cal V}+\bar{\cal
V}),\qquad\mathpzc{y}=\frac{1}{2i}({\cal V}-\bar{\cal V}),\hskip 14.22636pt$
(43) $\displaystyle x=\frac{1}{2}({\cal W}+\bar{\cal W}),\qquad
y=\frac{1}{2i}(({\cal W}-\bar{\cal W}),$ (44)
$\displaystyle\varphi_{1}=i\log\sqrt{\frac{\bar{\cal W}}{{\cal
W}}}\quad\varphi_{2}=i\log\sqrt{\frac{\bar{\cal V}}{{\cal V}}}.$ (45)

We can write

$dz^{2}+d\mathpzc{y}^{2}+dx^{2}+dy^{2}=d{\cal V}d\bar{\cal V}+d{\cal
W}d\bar{\cal W}.$ (46)

We now have $|{\cal V}|^{2}+|{\cal
W}|^{2}=x^{2}+y^{2}+z^{2}+\mathpzc{y}^{2}=r^{2}+R^{2}$. We identified
$\mathds{C}^{1}\times\mathds{C}^{1}$ with $\mathds{R}^{4}$ and so contains
$S^{3}$, given by $|{\cal V}|^{2}+|{\cal W}|^{2}=const.$ Every line through
the origin, represented by $({\cal V},{\cal W})$ intersects the sphere
$S^{3}$, for example $(\lambda{\cal V},{\lambda\cal W})$ with
$\lambda=\frac{1}{\sqrt{|{\cal V}|^{2}+|{\cal W}|^{2}}}$. Thus the homogeneous
coordinates can be restricted to $|{\cal V}|^{2}+|{\cal W}|^{2}=1.$ The point
$({\cal V},{\cal W})\in S^{3}\subset\mathds{C}^{1}\times\mathds{C}^{1}$ with
$|{\cal V}|^{2}+|{\cal W}|^{2}=1$, becomes by the complexification, a point of
$S^{2}$, so with the single complex coordinate ${\cal Z}=\frac{{\cal V}}{{\cal
W}}$. We have now a map $H:S^{3}\rightarrow S^{2}$, which is continuous. One
calls this a Hopf map. For each point of $S^{2}$, the coordinate $({\cal
V},{\cal W})$ is non unique, because it can be replaced by $(\lambda{\cal
V},\lambda{\cal W})$, such that $|\lambda|^{2}=1,\lambda\in S^{1}$. SWe will
now write $\mathds{C}_{1}$ for $S^{2}/\\{\infty\\}$ and $\mathds{C}_{2}$ for
$S^{2}/\\{0\\}$ and admitting coordinates ${\cal Z}$ and ${\cal
Z}^{\prime}=\frac{1}{{\cal Z}}$ respectively191919Let $G$ be the group of
self-homeomorphisms of the product space $S^{2}\times S^{2}$, generated by
interchanging the two coordinates of any point and by the antipodal map on
either factor. $G$ is then isomorphic to the dihedral group. It contains 3
subgroups, for example
$K=\\{I,(x,y)\rightarrow(-x,y),(x,y)\rightarrow(x,-y),(x,y)\rightarrow(-x,-y)\\}$.
It acts freely on $S^{2}\times S^{2}$. Then $(S^{2}\times
S^{2})/K=\mathds{R}P^{2}\times\mathds{RP}^{2}$. The most interesting feature
is the fact that the 2-fold symmetric product of $\mathds{R}P^{2}$,
$SP^{2}(\mathds{R}P^{2})=\mathds{R}P^{4}$..

If we write

${\bf H}=\begin{pmatrix}z+i\mathpzc{y}&x+iy\\\
-x+iy&z-i\mathpzc{y}\end{pmatrix}=\begin{pmatrix}\cal{V}&\cal{W}\\\
-\cal{\bar{W}}&\cal{\bar{V}}\end{pmatrix}$ (47)

with $det({\bf H})=|{\cal V}|^{2}+|{\cal W}|^{2}$ and $\mathds{R}^{4}\cong{\bf
H}$, we can describe the mapping also with normalized quaternions
$\mathds{A}=[a,{\bf A}]=[\cos\alpha,\sin\alpha{\bf n}]$ as binary
rotationsaltman1986 . This is clarifying, when we consider the covariance
covering groups, such as the $SO(3)$ and $SU(2)$. A rotation is actually
represented by two antipodal points on the sphere (a kind of double cover) as
a point on the hypersphere in 4D. Each rotation represented as a point on the
hypersphere is matched by its antipode on that hypersphere. The quaternion
represents a point in 4D. Constrained to unit magnitude, yields then a 3D
space, i.e., the surface of a hypersphere. The group of unit quaternions
$S^{3}\subset{\bf H}$ is isomorphic with $SU(2)$. We identify $S^{3}$ and
$SU(2)$ with the isomorphism. The relation with the Möbius group $G$ is then
easily made (see Tothtoth2002  or Slagterslagter2022c ) and so with the
alternating group ${\cal A}_{5}$, isomorphic with the binary symmetry group of
the icosahedron (see for exampletoth2002 ). Because the icosahedron can be
circumscribed by 5 tetrahedra, they form an orbit of order 5 symmetry
rotations of the icosahedron. These symmetries are subgroups of the
icosahedron symmetry group. The connection with our quintic solution was made
in a former studyslagter2022c , by the observation that the vertices of the
icosahedron, stereographically projected to $\mathds{C}$, are

$0,\infty,\gamma^{j}(\gamma+\gamma^{4}),\gamma^{j}(\gamma^{2}+\gamma^{3}),\quad
j=0,..4,\quad\gamma=e^{\frac{2\pi i}{5}}.$ (48)

Shortly stated, for the icosahedral Möbius group, by suitable orientation of
the axes, we can write the linear fractional transformations as
$\zeta\rightarrow\gamma^{m}\zeta(m=0..4)$. The binary icosahedral group
$G^{*}$ associated with the Möbius group $G\subset{\cal
M}_{0}(\mathds{C})$202020Remember, $SO(3)\cong{\cal
M}_{0}(\mathds{C})=SU(2)/\\{\pm I\\}$..

Let us now try to make the connection with state vectors in $\mathds{C}^{2}$.
On a complex Hilbert space ${\cal H}$, taken as the space $\mathds{C}^{2}$ of
pairs $\lvert{\cal U}\rangle=({\cal V},{\cal W})$ and equipped with scalar
product, one can define a state vector by the set of multiples
$\lambda\lvert{\cal U}\rangle$, with $|\lambda|=1$ and normalization condition
$\langle{\cal U}|{\cal U}\rangle=1$. Further, we have the matrix

$\displaystyle\rho:=\lvert{\cal U}\rangle\langle{\cal
U}\rvert,\quad\rho_{ij}=\begin{pmatrix}\cal{V}\cal{\bar{V}}&\cal{V}\cal{\bar{W}}\\\
\cal{W}\cal{\bar{V}}&\cal{W}\cal{\bar{W}}\end{pmatrix},$ (49)

with $\rho^{\dagger}=\rho$ and ${\bf Tr}(\rho)=1$. We define now the vector
$\vec{{\bf V}}=(V_{1},V_{2},V_{3})\in\mathds{R}^{3}$, with
$\rho_{11}=\frac{1}{2}(1-V_{3}),\rho_{22}=\frac{1}{2}(1-V_{3})$ and
$\rho_{21}=\bar{\rho}_{12}=\frac{1}{2}(V_{1}+iV_{2})$. We can write
$\rho=\frac{1}{2}(\mathds{I}+\vec{{\bf V}}.{\bf\sigma})$, with $\sigma$ the
Pauli matrices. Furter, $1-{\bf V}^{2}\geq 0$. We define the vector $\vec{{\bf
X}}=(x,y,z,\mathpzc{y})$ and take $\lambda=e^{i\alpha}$ a phase factor. We
then write ${\cal W}_{i}\rightarrow e^{i\alpha}{\cal W}_{i},{\cal
V}_{i}\rightarrow e^{i\alpha}{\cal V}_{i}$. So $\vec{{\bf
X}}\rightarrow\cos(\alpha)\vec{{\bf X}}+\sin(\alpha)J\vec{{\bf X}}$, with
$J\vec{{\bf X}}=(-X_{2},X_{1},-X_{4},X_{3})=(-y,x,-\mathpzc{y},z)$ a
normalized combination of two orthogonal unit vectors of $\mathds{R}^{4}$ for
all $\alpha$. In fact, we obtain a Hopf fibration of $S^{3}$. This fibering is
the stereographic projection on $X_{4}=\mathpzc{y}=0$ of $S^{3}$, an analogy
of the stereographic projection of one dimension lower. See figure 3.
Remember, we identified the antipodal points, so we obtain a Klein bottle
$\mathds{K}=\\{({\cal V},{\cal W})\mathchar 24635\relax\;|{\cal V}|^{2}-|{\cal
W}|^{2}=\cos(\theta)\\}\in S^{3}\subset\mathds{C}^{2}$ in $\mathds{R}^{3}$
(see Eq.(51)). However, it is embedded in our $\mathds{R}^{4}$. If we project
from $(0,0,0,\pm\mathpzc{y}_{0})$, the running point $\vec{{\bf X}}$ on
$S^{3}$ is now related to its stereographic image $\vec{{\bf
x}}=(x_{1},x_{2},x_{3})$ by $X_{4}=\pm\frac{{\bf x}^{2}-1}{{\bf x}^{2}+1}$,
$X_{i}=\frac{2x_{i}}{{\bf x}^{2}+1}$. For normalization of $\lvert{\cal
Z}\rangle$, we first define $\varphi=\varphi_{2}-\varphi_{1}$ and write

$\displaystyle{\cal
W}=\cos(\frac{\theta}{2})e^{i\varphi_{1}}=\cos(\frac{\theta}{2})e^{-i\varphi/2}e^{i(\varphi_{1}+\varphi_{2})/2}$
(50) $\displaystyle{\cal
V}=\sin(\frac{\theta}{2})e^{i\varphi_{2}}=\sin(\frac{\theta}{2})e^{i\varphi/2}e^{i(\varphi_{1}+\varphi_{2})/2}.$
(51)

Varying $\alpha$ will only change $(\varphi_{1}+\varphi_{2})/2$ and not
$\varphi$ and $\theta$. Further we have $V_{3}=\cos(\theta)$ and
$V_{1}+iV_{2}=\sin(\theta)e^{i\varphi}$, ${\cal Z}=\frac{{\cal V}}{{\cal
W}}=\tan(\theta/2)e^{i\varphi}$, with the antipodal identification
$\varphi\rightarrow\varphi+\pi$.

One can compare this visualization by the orientable counterpart situation of
the torus. For details, see Tothtoth2002  (section 1.4) or Urbantkeur1990

So if we consider a Hawking particle falling in, it travels for a while in
$\mathds{R}^{4}$ from $\mathpzc{y}_{0}\rightarrow-\mathpzc{y}_{0}$ arriving at
the antipodal point in $\mathds{R}^{3}$.

###  3.2 The elapse time in the bulk

One could wonder if one can estimate the elapse propertime of a particle when
it moves in the bulk space and back to the brane212121Another interesting
method could be delivered by the equation of the directrix of the Klein
bottle. From the integral curve from the PDE’s, $\frac{\partial r}{\partial
t_{e}}$, one could find, in principle, the elapse timemassey1971 .. Our
horizon $r_{H}$ is determined by $a_{i}$ (see Eq.(95)). We assume that
$y_{0}\sim 10^{-5}m$maartens2010  and let us take $r_{H}\sim 10^{7}$m. See,
for example, Maldacenamaldacena2021 222222There is a difference between
Maldacena’s method and our model: we don’t need a matter field in the bulk..
The proper traversal time, $t_{e}$, of the Klein surface will be of the order
of $r_{H}$. Further, the trajectory on the Klein surface in the bulk, $l_{b}$,
will be of the order $\frac{t}{t_{e}}$, with $t$ the asymptotic time and
$t_{e}$ the proper time. If we adapt the approximation of the wormhole model
on the RS spacetime of Maldacena, we could obtain an elapse time of the order
$10^{-2}$ s, while $t$ would be must larger.

It would be of interest if one could measure this elapse time in the Hawking
radiation. One could obtain an estimate of $y_{0}$.

###  3.3 The scalar equation on the single-sided black hole

It is obvious that strong gravitation interactions will take place near the
horizon. The scattering process by in- and out-going wave functions can be
handled by different approximations. One can make a distinction between the
near horizon behavior and the further away approximation by means of the
Regge-Wheeler potentialbetzios2021 . An effective gravitational potential for
the propagating modes on a Schwarzschild background then enters the Klein-
Gordon equation.

We will consider here the antipodal boundary condition of the single-sided
black hole. This idea was already studied by Schrödingerschrod1957 . It was
used by Gibbonsgib1986  in connection with CPT invariance and quantum
mechanics on the identified space. The method was also used by Sanchez, et
al.san1987  in connection with the quantum Fock space. We applied the mapping
on the RS spacetimeslagter2022c . We already observed that the scale of the
extra dimension $y_{0}$ entered the effective 4D field equations. Further, the
dynamical evolution of the 4D Einstein equations were modified by the
projection of the 5D Weyl tensor on the brane and carried the information of
the gravitation field outside the brane (the so-called "Kaluza-Klein"modes).

We will use the expansion of the scalar field in cylindrical harmonics232323We
work in the axially symmetric spacetime. Moreover, our scalar gauge field is
also rotational symmetric, a fundamental property of the vortex, which
contains the winding number n, which is a topological parameter.

$\Phi_{n,m}=\eta X(r,t)e^{in\varphi}{\bf Y}_{m},$ (52)

where $X(r,t)$ fulfills the PDE’s of section 2. Further, we have
$e^{in(\varphi+\pi)}=(-1)^{n}e^{in\varphi}$. For ${\bf Y}_{m}$ we can choose

${\bf Y}_{m}=\sum_{m}\Bigl{(}A_{m}\cos(m\varphi)+B_{m}\sin(m\varphi)\Bigr{)}.$
(53)

We can in this way always obtain a factor $(-1)^{(n+m)}$ in front of the
right-hand side in Eq.(52). Further, the field equations are invariant under
$t\rightarrow-t$ (see 4.2 and the Appendix A). Note that for antipodicity, we
need the map: $e^{i\varphi}\rightarrow e^{i(\varphi+\pi)}$.

The reader must remind himself that in our conformal model, the scale is
determined by the dilaton $\omega$. In the vacuum situation (see Appendix A)
an exact time-dependent solution was found. Now we conjecture that in the non-
vacuum situation, more information can be gathered by our model about the
scattering process when approaching smaller scales. The distinction one
usually makes between the near horizon region and the Regge-Wheeler region,
can be replaced by the dynamical interaction of the scalar waves and the
dilaton field. The distinction was necessary because gravity comes into play.
Further, the small scale behavior translates itself in the dilaton and the
conformally flat $\tilde{g}_{\mu\nu}$. Remember now that the dilaton field
(part of the gravitation metric) describes the complementarity between the
local and outside observer and can be used (by $\Omega$, see Eq.(3)) to define
the vacuum state the local observer experiences (see section 2.1). It is
conjectured (also from the numerical solutions) that the spacetime becomes
singular-free when $t\rightarrow\infty$, except for $r=0$, which is no problem
because there is no inside of the Klein surface.

###  3.4 Comparison with AdS

The elliptic interpretation was historically first studied in cosmology, i.e.,
in the deSitter spacetime. Let us consider the inversion in
$\mathds{R}^{5},J:{\bf X}^{a}\rightarrow-{\bf X}^{a},(a=1...5)$, and the
hypersurface $\frac{1}{H^{2}}=-t^{2}+x^{2}+y^{2}+z^{2}+\mathpzc{y}^{2}$, which
is the deSitter spacetime in a 5D spacetime242424Note that we consider in our
model the topology of the Klein surface.

$ds^{2}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}+d\mathpzc{y}^{2},$ (54)

with $H^{-1}$ the radius of the hypersphere and $t$ the proper time along a
set of geodesics. The symmetries of this pseudo sphere is the 10 parameter
Poincaré group in Minkowski spacetime. One writes the pseude sphere
($\eta=-H^{-1}exp(-Ht)$)

$ds_{dS}^{2}=-dt^{2}+e^{2Ht}(dx^{2}+dy^{2}+dz^{2})=\frac{1}{H^{2}\eta^{2}}(-d\eta^{2}+dx^{2}+dy^{2}+dz^{2}).$
(55)

The group invariance of the deSitter spacetime can now be used to apply a
high-frequency behavior of the scalar field in order to define the prefered
vacuum state as measured by the observers on the geodesics. From the scalar
equation (like Eq.(14)), one easily find

$\partial_{t}^{2}\phi+3H\partial_{t}\phi-e^{-2Ht}\sum_{i}\partial_{i}^{2}\phi+M^{2}\phi=0,$
(56)

with $M^{2}=m^{2}+12\xi H^{2}$. We expand the field operator $\phi$ now as

$\phi=\sum_{k}\Bigl{\\{}A_{\vec{k}}f_{\vec{k}}(x)+A^{\dagger}_{\vec{k}}f^{*}_{\vec{k}}(x)\Bigr{\\}},\quad
f_{\vec{k}}=\frac{1}{\sqrt{2Ve^{3Ht}}}2^{i\vec{k}.\vec{x}}h_{k},$ (57)

with $k^{i}=2\pi n^{i}/L,n^{i}$ an integer, $k=|\vec{k}|$ and $L$ the
dimension of the 3-space ($V\sim L^{3}$). The term $Ve^{3Ht}$ represents here
the physical volume of the cube on which one applies the periodic boundary
conditions. Further $h_{k}$ satisfies a Bessel equation

$v^{2}\frac{d^{2}}{dv^{2}}h_{k}+v\frac{d}{dv}{h_{k}}+(v^{2}-\nu^{2})h_{k}=0,\quad\nu=\sqrt{\frac{9}{4}-\frac{M^{2}}{H^{2}}},\quad
v=\frac{ke^{-Ht}}{H}.$ (58)

One can also use $u\equiv-v$ as independent variable. $v$ is the ratio of the
momentum of a particle to the expansion $H$. In this context, an observer on a
geodesic in the deSitter spacetime will be surrounded by isotropic thermal
radiation ( with wavelength $\lambda$) coming from de geodesic’s Hubble
horizon with temperature $T_{H}=H/(2\pi k_{B})$. So $v$ represents also the
ratio of the proper radius $H^{-1}$ to $\lambda/(2\pi)$. For small wavelength,
compared with the radius of curvature and slowly changing of
$a(t)=e^{Ht}$252525we write in general,
$ds^{2}=a(t)^{2}(-dt^{2}+dx^{2}+dy^{2}+dz^{2})$., we assume that the positive
frequency solutions $f_{\vec{k}}$ possesses an adiabatic form, i.e., the WKB-
solution. We can write

$f_{\vec{k}}({\bf
x})=\frac{1}{\sqrt{2Va^{3}\omega_{k}}}e^{i(\vec{k}.\vec{x}-\int\omega_{k}dt)},\quad\omega=k/a(t).$
(59)

Further, $h_{k}$ has the asymptotic form

$h_{k}\sim\frac{1}{\sqrt{\omega}}e^{-i\int\omega_{k}dt}.$ (60)

For large $k,h_{k}\sim\frac{1}{\sqrt{Hv}}e^{iv}$. The general solution of the
Bessel equation, Eq.(58), is

$h_{k}=\sqrt{\frac{\pi}{2H}}\Bigl{\\{}E(k)H_{\nu}^{(2)}(v)+F(k)H_{\nu}^{(1)}(v)\Bigr{\\}},$
(61)

with $H_{\nu}$ Hankel functions. When $k\rightarrow\infty$, we have $E(k)\sim
0,F(k)\sim 1$. From the deSitter symmetries, one finally obtains the creation
and annihilation operators

$f_{\vec{k}}=\sqrt{\frac{\pi}{4HV}}e^{-3/2Ht}H_{\nu}^{(1)}(k\frac{k}{H}e^{-HT})e^{i\vec{k}.\vec{x}},$
(62)

and hence the vacuum state262626Also called the Bunch-Davies vacuum, valid
also for the conformally coupled massless scalar. However, in our case, we
could add the mass term of Eq(35). For $\xi=1/6$ and $m=0$, we obtain
$H_{\frac{1}{2}}^{(1)}=-i\sqrt{\frac{2}{\pi v}}e^{iv}$. For
$u=-v,H_{\nu}^{(1)}(v)$ is replaced by $H_{\nu}^{(2)}(u)$. Note that in
calculating the expectation values such as $<\phi(\vec{x},t)^{2}>$, there
appears infrared or ultra violet singularities in this two point function, at
least for $\xi=0$.

In the antipodal situation of the deSitter spacetime, the expectation value
becomes (symmetric and antisymmetric)fol1987

$<\phi_{JA/S}(\vec{x},t)^{2}>=<\phi(\vec{x},t)^{2}>\pm<\bar{\phi}(\vec{x},t)^{2}>,$
(63)

with

$<\bar{\phi}(\vec{x},t)^{2}>=\frac{1}{16\pi\cos(\pi\nu)}\\{m^{2}+12(\xi-1/6)H^{2}\\}.$
(64)

What remains is the problem with the Fock vacuum. This is essential, because
one can always add an antipodal source (see discussion by Gibbonsgib1986 ).

We now return to our situation. In the next sections we investigate some
topological aspects.

##  4 Topological aspects: relation with the instanton and minimal embedding
in $\mathds{K}$

###  4.1 Some history

It is well known that on Minkowski spacetime $\mathds{R}^{d+1}$, the action of
the static Yang-Mills-Higgs configuration $({\bf A},\Phi)$ equals the
Euclidean action on $\mathds{R}^{d}$. The field equations of the Euclidean
action, for example, in $\mathds{R}^{4}$ in this model, allow soliton
solutions, which are so time-independent finite energy solutions to the
variational equations for the action density on the $\mathds{R}^{5}$ Minkowski
spacetime. For the $d=4$ pure Yang-Mills case, they are called instantons. It
turns out that their "curvatures" are self-dual272727If one allows a Higgs
field, then for $d=2$ one calls the configuration a vortex and for $d=3$ a
monopole..

In the "transcendental" book of Jaffe and Taubesjaf1981  one finds a nice
introduction of these issues. We can now make the connection with our model.

###  4.2 Connection with the warped spacetime solution

Let us write our (4+1) dimensional spacetime on the 5 dimensional Riemannian
manifold:

$ds^{2}=\omega(\tau,r,y)^{4/3}\Bigl{[}N(\tau,r)^{2}d\tau^{2}+\frac{1}{N(\tau,r)^{2}}dr^{2}+dz^{2}+r^{2}(d\varphi+N^{\varphi}(\tau,r)d\tau)^{2}+d\mathpzc{y}^{2}\Bigr{]}.$
(65)

One easily obtains the field equations in $d$ ($d=4,5$; compare with Eq.(90)
and Eq.(92) in the pseudo-Riemannian spacetime)

$\ddot{\omega}=N^{4}\omega^{\prime\prime}-\frac{d}{\omega(d-2)}(N^{4}\omega^{\prime
2}-\dot{\omega}^{2}),$ (66)
$\displaystyle\ddot{N}=\frac{3\dot{N}^{2}}{N}+N^{4}(N^{\prime\prime}+\frac{3N^{\prime}}{r}+\frac{N^{\prime
2}}{N})$ (67)
$\displaystyle+\frac{(d-1)}{(d-3)\omega}\Bigl{[}N^{5}\Bigl{(}\omega^{\prime\prime}+\frac{\omega^{\prime}}{r}+\frac{d}{2-d}\frac{\omega^{\prime
2}}{\omega}\Bigr{)}+N^{4}(\omega^{\prime}N^{\prime}-\dot{\omega}\dot{N}\Bigr{]},$
(68)

with the dot representing now $\partial_{\tau}$. Note that we are dealing here
now with hyperbolic PDE’s in stead of elliptic ones.

The exact solution is again given by the solution of Eq.(93) and Eq.(95) (in
the pseudo-Riemannian spacetime)282828The replacement $t\rightarrow i\tau$ has
no influence on the solution.. We repeat that our original pseudo-Riemannian
5D spacetime delivers an effective 4D Riemannian spacetime on the brane

$ds^{2}=\frac{N_{1}^{2}}{N_{2}^{2}}(dt^{*2}+dr^{*2})+dz^{2}+r^{2}d\varphi^{*2},$
(69)

where $r$ can be expressed in $r^{*}$ (Appendix B). The topology is
$\mathds{R}^{2}_{+}\times\mathds{R}^{1}\times S^{1}$. The $\varphi^{*}$ causes
no problems, because the $N^{\varphi}$ component decouples from the equations
for $N$ and $\omega$. $\mathds{R}^{2}_{+}$ is still conformally flat.

It is legitimate to conjecture that in the non-vacuum case, the scalar field
will also be the same on the Riemannian spacetime. This could be used in order
to solve the issue of construction of the quantum wave functions as elements
of the Fock space. One of the entangled particles travels on the Klein surface
on the hypersurface embedded in the 5D spacetime and remains a pure state. In
the next sections, we will enlighten this issue.

###  4.3 Some other aspects of the Klein bottle

In our model, we need the Klein surface $\mathds{K}$, the compact sum of two
projective planes. Our conjecture is, that our solution can be seen as a
dynamical solution embedded in our $(4+1)$ spacetime. More precisely, if
${\cal N}$ is the sub manifold with metric $g$, i.e., the metric on our
effective 4D spacetime, then ${\cal N}$ is diffeomorphic to the hyperbolic 5D
spacetime: $\chi:\mathds{R}^{5}\rightarrow{\cal N}$. The Riemannian manifold
$({\cal N},g)$ must be conformally flat, which is the case here.

Now physicists are interested in the topology of moduli spaces of self-dual
connections on vector bundles over Riemannian manifolds. One reason was that
on these spaces the instanton approximation to the Green functions of
Euclidean quantum gravity Yang-Mills theory can be expressed in terms of
integrals over moduli spaces. One needs then the metric and volume form of the
moduli spaces.

From the investigations of Groisser, et.al,gross1987 , we conjecture that we
can consider $\mathds{K}$ as a 4-sphere in our hyperbolic pseudo Riemannian
spacetime. This suspicion is fueled by the solution of section 4.2, and the
work on the orientable counterpart model of Groisser (and references therein).

More precisely, on a Riemannian 5-manifold, one can proof that there exists a
coordinate diffeomorphism $\xi:\mathds{R}^{5}\rightarrow{\cal N}$ for which
the pullback of the metric $\mathpzc{g}$ on ${\cal N}$ is given by
$(\xi^{*}\mathpzc{g})_{ij}=\Psi^{2}({\bf x})\delta_{ij}$. Further, the
Riemannian manifold $({\cal N},\mathpzc{g})$ is conformally flat with finite
radius and volume. The action of $SO(5)$ on $S^{4}$ induces an isometry on
${\cal N}$ whose pullback, via $\xi$, is the usual action $SO(5)$ action on
$\mathds{R}^{5}$. So ${\cal N}$ can isometrically included as the interior of
a compact Riemannian manifold with boundary, say $\bar{\cal N}$, whose
boundary $\partial\bar{\cal N}$ is isometric to the 4-sphere of constant
radius. The embedding $\partial\bar{\cal N}\rightarrow{\cal N}$ is totally
geodesic. The sphere $\partial\bar{\cal N}$ is conformally equivalent to the
original manifold $(S^{4},\mathpzc{g})$ and points on $\partial\bar{\cal N}$
corresponds to instantons which are concentrated at a single point on $S^{4}$.
It is remarkable that the $\Phi({\bf x})$ is determined by a PDE which is
comparable with our scalar equation (${\cal N}$ has no constant curvature).

A remark must be made about the $\mathds{Z}_{2}$ symmetry in the original
description of the antipodal mappingthooft2021 . At the border of region I and
II in the Penrose diagram, the antipodes on a 3-sphere were glued together and
the transverse $(\theta,\varphi)$ part is a projected 2-sphere. In our model,
it is replaced by the projected 3-sphere $(z,\varphi,\mathpzc{y})$ using the
$\mathds{Z}_{2}$ symmetry of the bulk space $(U,V,z,\varphi,\mathpzc{y})$. No
cut and past procedure is necessary.

However, a lot of unsolved issues need to be attacked. In particular, the
firewall transformation description in the warped spacetime.

##  5  Notes on the information paradox

There are some known facts about the information paradox. An outside observer
will register the Hawking radiation as thermal, i.e., in a mixed state, while
a local observer is in doubt about the vacuum state. Further, the evolution of
the wave function of the in-falling particle must be unitary, i.e., it satisfy
the Schrödinger equation
$\lvert\chi(t_{1})\rangle=U(t_{1},t_{2})\lvert\chi(t_{2})\rangle$ and is
bijective. It is believed that during black hole evaporation, information of
the quantum state is preserved. Information loss is inconsistent with
unitarity. So the problem is how to handle the controversy between the pure
quantum state of the in-falling particles and the mixed state property of the
Hawking radiation.

In our model we do not rely on replica wormholes, but instead on the warped
spacetime and Klein surface.

By the antipodal mapping in $\mathds{R}^{4}$, we have also
$e^{in\varphi}\rightarrow e^{in(\varphi+\pi)}$. This can be "visualized" by
considering in figure 3 (right) points on the circle on $S^{3}$ (for example
$\alpha\sim e^{in\varphi}$) for which

$\sin(\theta/2)e^{in\varphi}{\cal V}(\alpha)-\cos(\theta/2){\cal
W}(\alpha)=0.$ (70)

From the real part, we obtain the plane after the stereographic projection,

$x_{3}=\tan(\theta/2)(\cos(\varphi)x_{1}-\sin(\varphi)x_{2}.$ (71)

If we apply $e^{in\varphi}\rightarrow e^{in(\varphi+\pi)}$, the plane is
rotated over $\pi$. The imaginary part delivers the two spheres ${\bf
x}^{2}\mp\tan(\theta/2)(\sin(\varphi)x_{1}+\cos(\varphi)x_{2})=1$.

We found in section 3 that the in-falling Hawking particle travels for a while
on the Klein surface in $\mathds{R}^{4}$. Now consider the state

$\lvert\chi\rangle=\lvert\hat{\chi}\rangle e^{in\varphi}$ (72)

and the density matrix

$\rho_{A}=\sum_{i}p_{i}\lvert\chi_{i}\rangle_{A}{{}_{A}}{\langle\chi_{i}\rvert}$
(73)

as mixture of pure states and weights $p_{i}$ with $\sum p_{i}=1$.
Further,${\bf Tr}\rho_{A}=1$ and $\rho_{A}^{2}=\rho_{A}$. The density matrix
doesn’t contain the phase. So the change of the azimuthal angle $\varphi$ has
no influence on the pure states. This can be compared with the the "pureness"
of $\rho$ in Eq(49) of normalized state vectors on
$S^{2}\subset\mathds{R}^{3}$. In addition to $\vec{{\bf V}}$

$\vec{{\bf V}}=\Bigl{(}{\cal V}{\cal\bar{W}}+{\cal W}{\cal\bar{V}},i({\cal
V}{\cal\bar{W}}-{\cal W}{\cal\bar{V}}),|{\cal V}|^{2}-|{\cal W}|^{2}\Bigr{)},$
(74)

independent of a phase change in $\lvert{\cal U}\rangle=({\cal V},{\cal W})$,
we define

$\vec{{\bf Z}}=\vec{{\bf P}}+i\vec{{\bf Q}}=\Bigl{(}{\cal V}^{2}-{\cal
W}^{2},i({\cal V}^{2}+{\cal W}^{2}),-2{\cal V}{\cal W}\Bigr{)}.$ (75)

$(\vec{{\bf V}},\vec{{\bf P}},\vec{{\bf Q}})$ form a positively oriented
orthonormal triad. Now $\vec{{\bf P}}$ is a unit tangent vector to $S^{2}$ at
$\vec{{\bf V}}$. If $\lvert{\cal U}\rangle\rightarrow e^{i\alpha}\lvert{\cal
U}\rangle$, then $\vec{{\bf V}}$ is independent of this phase change, while
$\vec{{\bf Z}}\rightarrow e^{2i\alpha}\vec{{\bf Z}}$. One can verify that
$\vec{{\bf P}}$ registers phase change $mod(\pi)$ only. When $\lvert{\cal
U}\rangle\rightarrow-\lvert{\cal U}\rangle$, $\vec{{\bf U}}$ still delivers
the same $\vec{{\bf P}}$, while $(\vec{{\bf R}},\vec{{\bf P}})$ determine
$\lvert{\cal U}\rangle$ up to sign. Let us now interpret $|\langle{\cal
U}|{\cal U^{\prime}}\rangle|^{2}$ (independent of the phase) as a
"probability". In language of the geometric picture here,
$\sqrt{1-|\langle{\cal U}|{\cal U^{\prime}}\rangle|}=\frac{1}{2}|\vec{{\bf
V}}-\vec{{\cal{\bf V^{\prime}}}}|$ represents the diameter of $S^{2}$.

This "visualization"is in fact a Hopf fibration of $S^{3}$ (see also section
3.1) In our case, we are dealing with a time depended situation and state
vectors follow the Schrödinger equation

$\frac{d}{dt}\lvert{\cal U}\rangle(t)=-\frac{1}{\hbar}H\lvert{\cal
U}\rangle(t),$ (76)

with $H$ a Hermitian matrix, with $H=H_{0}\mathds{I}+\vec{{\bf
H}}.{\bf\sigma}$. One finally obtains

$\frac{d{\vec{{\bf V}}}}{dt}=\vec{{\bf V}}=\frac{2}{\hbar}\vec{{\bf
H}}\times\vec{{\bf V}},$ (77)

the time evolution in $S^{2}$. From Eq.(77) we have $\vec{{\bf V}}.\vec{{\bf
V}}=0$, so pure states remain pure.

###  5.1 Another argument

There is another strong argument for considering the topology of the Klein
surface. In short, we blow up the 4-manifold to 5D in order to handle the
singularities in the curvature and to apply the antipodal map. One can
mathematically formulate the topology of a 4-manifold using self-dual
connections over de Riemannian $S^{4}$freed1984 . It depends only upon the
conformal class of the Riemannian metric. This self-dual connection can be
interpreted by the conformal map: $\mathds{R}^{4}\rightarrow S^{4}/\\{0\\}$ as
a self-dual connection or "instanton".

In our 5D RS model (with a finite number of singularities), we found the
metric is determined by $N$ and $\omega$ (see appendix A). The solution for
$N$ on the effective 4D spacetime is the same, while the $\omega$ contribution
is different. This is solely due to the contribution of ${\cal E}_{\mu\nu}$
(Eq.(8)). If we switch to Riemannian $(t\rightarrow i\tau$, a Wick rotation),
the solution is for both $N$ and $\omega$ unaltered. Moreover the
$\tilde{g}_{\mu\nu}$ is conformally flat. The embedding of the Klein surface
was done using the extra dimension ${\bf\mathpzc{y}}$ and the effective
spacetime is conformally flat.

Consider now the ball $B^{4}\subset\mathds{R}^{4}$ with the induced metric
$\omega^{4}g_{\mu\nu}$ (Eq.(93))292929Remember that we used twice the dilaton
separation.. So the scale is $\omega^{4}$. Then the ball
$\frac{r}{2\omega}B^{4}\subset\mathds{R}^{4}$ has a curvature
$\leq\frac{const}{|\omega|^{4}}$ We interpret the Riemannian 5D warped
spacetime Eq.(65), an open 5-ball, as an instanton on $\mathds{R}^{4}$.

In the pseudo-Riemannian spacetime of Eq.(6), the boundary is the non-
orientable Klein surface, which we used for the antipodicity in order to
maintain the pure states of the Hawking particles.

The importance of the instanton trick, is the fact that in the Riemannian
space (or easier, in Minkowski space) they play a crucial role in calculating
path integralsfelsager1998 . It turns out that a static solution in $m$ space
dimensions is completely equivalent to an instanton in $m$ space-time
dimensions.

##  6 High-frequency approximation close to the horizon

Our time evolution was governed by the Klein-Gorden-type PDE. Although we
found a time dependent exact and numerical solution, one should like to
describe the quantum effects more precisely. It is a nice exercise to derive
the Schrödinger equation from the Klein-Gordon, using the so-called multiple-
scale (or high-frequency (HF)) method (see appendix E).

However, we are dealing here also with the polar angle dependency. One usually
deals with separability of the wave function in a $(t,r)$ part and spherical
harmonics303030They are needed to describe the gravitational back reaction.
See for example refthooft2019 . . In our case, this is not possible without
extra assumptions. The HF method can offer a solution for this situation (see,
for example, Slagterslagter2022d , section 3.3). In some sense, one can keep
track of the back reaction of the emitted Hawking particles. In the appendix
E.4, we applied the method for the Vaidya spacetime in Eddington-Finkelstein
coordinates

$ds^{2}=-\Bigl{(}1-\frac{2M(u)}{r}\Bigr{)}du^{2}-2dudr+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (78)

which is the Schwarzschild spacetime with $u=t-r-2M\log(r/2M-1)$. This
spacetime is used to demonstrate the loss of mass by gravitational radiation.
From Eq.(166) and Eq.(167) we obtain

$h_{rr}=h_{r\theta}=h_{r\varphi}=0,\qquad h_{\varphi\varphi}=-\sin^{2}\theta
h_{\theta\theta}$ (79)
$\displaystyle\ddot{k}_{rr}=0,\qquad\dot{h}_{\theta\theta}=r\partial_{r}\dot{h}_{\theta\theta},\qquad\dot{h}_{\theta\varphi}=r\partial_{r}\dot{h}_{\theta\varphi}.$
(80)

So one writes

$h_{\theta\theta}=r\alpha(u,\theta,\varphi,\xi),\quad
h_{\theta\varphi}=r\beta(u,\theta,\varphi,\xi),\quad
h_{\varphi\varphi}=-r\alpha\sin^{2}\theta.$ (81)

Further, we have

$\displaystyle\ddot{k}_{r\theta}=\frac{1}{r}\Bigl{(}2\dot{\alpha}\cot\theta+\partial_{\theta}\dot{\alpha}+\frac{1}{\sin^{2}\theta}\partial_{\varphi}\dot{\beta}\Bigr{)},$
(82)
$\displaystyle\ddot{k}_{r\varphi}=\frac{1}{r}\Bigl{(}\dot{\beta}\cot\theta-\partial_{\varphi}\dot{\alpha}+\partial_{\theta}\dot{\beta}\Bigr{)},$
(83)
$\displaystyle\frac{dM}{du}=-\frac{\ddot{k}_{\phi\phi}+\sin^{2}\theta\ddot{k}_{\theta\theta}}{4\sin^{2}\theta}-\frac{1}{2}r\dot{h}_{uu}-\frac{1}{4}\Bigl{(}\dot{\alpha}^{2}+\frac{\dot{\beta}^{2}}{\sin^{2}\theta}\Bigr{)}+\frac{1}{4}\Bigl{(}\ddot{\alpha^{2}}+\frac{\ddot{\beta^{2}}}{\sin^{2}\theta}\Bigr{)}.$
(84)

Not all the components of $h_{\mu\nu}$ and $k_{\mu\nu}$ are physical, so one
needs some extra gauge conditions. Suitable choice of $\alpha$ and $\beta$
(Choquet-Bruhat uses, for example, $\alpha=0,\beta=g(u)h(\xi)\sin\theta$),
leads to a solution to second order which is in general not axially symmetric.
We can integrate these zero order equations with respect to $\xi$. One obtains
then some conditions on the background fields, because terms like
$\int\dot{\alpha}d\xi$ disappear. From Eq.(84), we obtain

$\frac{dM}{du}=-\frac{1}{4\tau}\int_{0}^{\tau}\Bigl{(}\dot{\alpha}^{2}+\frac{\dot{\beta}^{2}}{\sin^{2}\theta}\Bigr{)}d\xi,$
(85)

which is the back-reaction of the high-frequency disturbances on the mass $M$.
$\tau$ is the period of $\dot{h}_{\mu\nu}$. This expression can be substituted
back into Eq.(84). However, in the non-vacuum case, the right-hand side will
also contain contributions from the matter fields. In order to obtain
propagation equations for $h_{\mu\nu}$ and $k_{\mu\nu}$, one proceeds with the
next order equation $R_{\mu\nu}^{(1)}=0$. First of all, Eq.(82), (83) are
consistent with $R_{r\varphi}^{(1)}=0$ and $R_{r\theta}^{(1)}=0$. Further, one
obtains propagation equations for $\alpha$ and $\beta$ and for some second
order perturbations, such as $k_{\varphi\varphi}$. Moreover, the
$(\varphi,\theta)$-dependent part of the PDE’s for $\alpha$ and $\beta$ (say
$A(\theta,\varphi),B(\theta,\varphi)$) can be separated (for the case
$k_{\theta\varphi}\neq 0$):

$\partial_{\varphi}B+2\sin\theta\cos\theta
A+\sin^{2}\theta\partial_{\theta}A=0,$ (86)
$\displaystyle\sin^{2}\theta\partial_{\theta\theta}A+7\sin\theta\cos\theta\partial_{\theta}A+4\cot\theta\partial_{\varphi}B+2(5\cos^{2}\theta-1)A+2\partial_{\theta\varphi}B=0.$
(87)

A non-trivial simple solution is

$A=\frac{\cos\theta(sin\varphi+\cos\varphi)}{sin^{3}\theta},\quad
B=\frac{\sin\varphi-\cos\varphi}{\sin^{2}\theta}+G(\theta),$ (88)

with $G(\theta)$ arbitrary. So the breaking of the spherically and axial
symmetry is evident.

##  7 Conclusion

On a warped 5D Randall-Sundrum Kerr-like black hole spacetime, we investigated
the time evolution of the scalar-gauge field in the conformal invariant
gravity model. We analyzed the dilaton-Higgs $(\omega,\Phi)$ system and
compared the numerical solution with the exact vacuum counterpart model found
in an earlier study. It turns out that the presentation of the model on a
Klein surface, fits very well in the antipodal interpretation of the black
hole paradoxes. We find that the potential $V(\omega,\Phi)$ is determined by
the superfluous dilaton equations. A connection is made with the instanton
solution on the effective 4D Riemannian space. There remain a lot of
questions. First, could one explain that fluctuations in the dilaton generate
the spacetime as noticed by the outside observer? Secondly, the zero’s of the
quitic polynomial $N_{1}(r)$, which determine the singular point of the
spacetime, can be written as (in the complex plane) without a scalar field

$\frac{c+10r^{2}(r-a)^{3}-10r^{3}(r-a)^{2}+r^{4}(r-a)-r^{5}}{5r^{2}}$ (89)

When $r\rightarrow a$, we are left with a singularity at $r=0$ when $a$
becomes very small in the far future $t\rightarrow t_{H}$ (Eq.(95)). Could $a$
be related to the mass of the black hole in its end-time? When the scalar
gauge field is included, one should like to find an exact solution, guided by
the vacuum exact solution and the numerical investigation. Thirdly, one can
apply an approximation scheme in order to obtain the graviton exchange on the
horizon. This is intensive work. It could be possible that the eikonal
approximation must be replaced by the two-timing method, or high-frequency
approximation. Fourth, the construction of the Fock space in the warped
spacetime must be investigated more deeply. It is conjectured that the
$\mathds{Z}_{2}$ symmetry of the bulk space is necessary in order to construct
a consistent description of evaporation of the black hole. Further, the role
of the dilaton in connection with the information paradox, must be
investigated. This "quantum"field $\omega$ determines the vacuum state of the
local observer. These issues are currently under investigation by the author.
Acknowledgments This research was solely done under ASFYON, which is financial
independent. We thanks several fellow researchers for valuable comments.
=======================oOo==================

##  Appendix A Former results without a scalar field

In a former studyslagter2018 ; slagter2021 ; slagter2022c  we found in the
case without a scalar field, that an exact solution exists for $\omega$ and
$N$. The equation for the dilaton was obtained from the Einstein equations and
was of the elliptic type (d=4,5):

$\ddot{\omega}=-N^{4}\omega^{\prime\prime}+\frac{d}{\omega(d-2)}\Bigl{(}N^{4}\omega^{\prime
2}+\dot{\omega}^{2}\bigr{)}.$ (90)

The equation for $N$ is

$\displaystyle\ddot{N}=\frac{3\dot{N}^{2}}{N}-N^{4}\Bigl{(}N^{\prime\prime}+\frac{3N^{\prime}}{r}+\frac{N^{\prime
2}}{N}\Bigr{)}\hskip 133.72786pt$ (91)
$\displaystyle-\frac{d-1}{(d-3)\omega}\Bigl{[}N^{5}\Bigl{(}\omega^{\prime\prime}+\frac{\omega^{\prime}}{r}+\frac{d}{2-d}\frac{{\omega^{\prime}}^{2}}{\omega}\Bigl{)}+N^{4}\omega^{\prime}N^{\prime}+\dot{\omega}\dot{N}\Bigr{]}.$
(92)

The exact solution is

$\omega=\Bigl{(}\frac{a_{1}}{(r+a_{2})t+a_{3}r+a_{2}a_{3}}\Bigr{)}^{\frac{1}{2}d-1},$
(93) $\displaystyle N^{2}\equiv
N_{1}(r)^{2}N_{2}(t)^{2}\qquad\qquad\qquad\qquad\qquad$ (94)
$\displaystyle=\frac{10a_{2}^{3}r^{2}+20a_{2}^{2}r^{3}+15a_{2}r^{4}+4r^{5}+C_{1}}{5C_{2}r^{2}(a_{3}+t)^{4}+C_{3}}=\frac{4\int
r(r+a_{2})^{3}dr}{r^{2}[C_{2}(a_{3}+t)^{4}+C_{3}]},$ (95)

with $a_{i},C_{i}$ some constants. $a_{2}$ is related to the mass. Note that
the solution for $N$ is the same in the bulk and on the brane. The
singularities of the spactime are determined by a quintic. For more details of
the zeros and the remarkable relation with Klein surface, we refer to
Slagterslagter2022c .

It is remarkable that one can write the solution for $N_{1}(r)$ (see second
expression in Eq.(95)) as a result of a first-order differential equation;

$N_{1}^{2}+rN_{1}\frac{\partial N_{1}}{\partial r}=2(r+a_{2})^{3}.$ (96)

Without the bulk contribution, it becomes

$N_{1}^{2}+rN_{1}\frac{\partial N_{1}}{\partial r}=\frac{3}{2}(r+a_{2})^{2}.$
(97)

We also compared the solution with the Baňados-Teitelboim-Zanelli (BTZ) black
hole solution in $(2+1)$-dimensional spacetime (without the $dz^{2}$ term).
The solution could be written as

$N_{1}^{2}=\frac{4}{l^{2}r^{2}}\int r(r+l\sqrt{4GM})(r-l\sqrt{4GM}),$ (98)

solution of the first order differential equation

$N_{1}^{2}+rN_{1}\frac{\partial N_{1}}{\partial
r}=\frac{1}{2l^{2}}(r+l\sqrt{4GM})(r-l\sqrt{4GM}).$ (99)

Here $l$ is the scale at which curvature sets in and is related to the
cosmological constant $\Lambda=-\frac{1}{l^{2}}$. It seems that the solution
on the brane, without the bulk contribution (Eq.(97)), must be related to the
BTZ solution of Eq.(99).

##  Appendix B Kruskal coordinates and the Penrose diagram

We define the coordinates,

$dr^{*}\equiv\frac{1}{N_{1}(r)^{2}}dr\quad dt^{*}\equiv N_{2}(t)^{2}dt.$ (100)

Our induced effective spacetime can be written asslagter2022c

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}\frac{N_{1}^{2}}{N_{2}^{2}}\Bigl{(}-dt{{}^{*}}^{2}+dr{{}^{*}}^{2}\Bigr{)}+dz^{2}+r^{2}(d\varphi+\frac{N^{\varphi}}{N_{2}^{2}}dt^{*})^{2}\Bigr{]},$
(101)

due to the fact that the solution for the metric component $N$ is a quotient
$\frac{N_{1}(r)}{N_{2}(t)}$. Further,

$\displaystyle
r^{*}=\frac{1}{4}\sum_{r^{H}_{i}}\frac{r^{H}_{i}\log(r-r^{H}_{i})}{(r^{H}_{i}+b_{2})^{3}},\qquad
t^{*}=\frac{1}{4C_{2}}\sum_{t^{H}_{i}}\frac{\log(t-t^{H}_{i})}{(t^{H}_{i}+b_{3})^{3}}.$
(102)

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 polynomial in $r$ defining the roots of $N_{1}^{2}$, is a quintic
equation, which has some interesting connection with Klein’s icosahedron
solution. 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. Next, we define the
coordinates strauss2020  (in the case of $C_{1}=C_{3}=0$ and 1 horizon, for
the time being)

$\displaystyle U_{+}=e^{\kappa(r^{*}-t^{*})},\hskip
34.14322ptV_{+}=e^{\kappa(r^{*}+t^{*})}\hskip 28.45274ptr>r_{H}$ (103)
$\displaystyle U_{-}=-e^{\kappa(r^{*}-t^{*})},\hskip
24.18501ptV_{-}=-e^{\kappa(r^{*}+t^{*})}\hskip 17.07182ptr<r_{H},$ (104)

with $\kappa$ a constant. The spacetime becomes

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}\frac{N_{1}^{2}}{N_{2}^{2}}\frac{dUdV}{\kappa^{2}UV}+dz^{2}+r^{2}d\varphi^{*2}\Bigr{]}.$
(105)

The antipodal points $P(X)$ and $P(\bar{X})$ are physically identified. If we
compactify the coordinates,

$\tilde{U}=\tanh U,\qquad\tilde{V}=\tanh V,$ (106)

then the spacetime can be written as

$ds^{2}=\omega^{4/3}\bar{\omega}^{2}\Bigl{[}H(\tilde{U},\tilde{V})d\tilde{U}d\tilde{V}+dz^{2}+r^{2}d\varphi^{*2}\Bigr{]},$
(107)

with

$H=\frac{N_{1}^{2}}{N_{2}^{2}}\frac{1}{\kappa^{2}\operatorname{arctanh}\tilde{U}\operatorname{arctanh}\tilde{V}(1-\tilde{U}^{2})(1-\tilde{V}^{2})}.$
(108)

We can write $r$ and $t$ as

$r=r_{H}+\Bigl{(}\operatorname{arctanh}\tilde{U}\operatorname{arctanh}\tilde{V}\Bigl{)}^{\frac{1}{2\kappa\alpha}},\quad
t=t_{H}+\Bigl{(}\frac{\operatorname{arctanh}\tilde{V}}{\operatorname{arctanh}\tilde{U}}\Bigl{)}^{\frac{1}{2\kappa\beta}},$
(109)

with

$\alpha=\frac{r_{H}}{4(r_{H}+b_{2})^{3}},\qquad\beta=\frac{1}{4C_{2}(t_{H}+b_{3})^{3}}.$
(110)

Further, $N_{1}$ and $N_{2}$ can be expressed in $(U,V)$ (or
$(\tilde{U},\tilde{V})$). One easily checks that for $U=0$ and $V=0$ one
obtains the horizons as depicted in the Penrose diagram of figure B1. Now
$N_{1}$ and $N_{2}$ are polynomials in $r$ and $t$. So for $U=0$ and $V=0$
they approach $r_{H}$ and $t_{H}$ as expected. So the "scale-term" $H$ is
consistent with the features of the Penrose diagram.

Figure 5: Penrose diagram. The antipodal points $P(X)$ and $P(\bar{X})$ are
identified. Particles going in will generate waves approaching the horizon
from the outside and passes the horizon, will reappear from "the other side"
of the black hole. Note that $t^{*}=log(U/V)\sim log(t-t_{H})$.

Note that $ds^{2}$ and $H$ are invariant under
$\tilde{U}\rightarrow-\tilde{U}$ and $\tilde{V}\rightarrow-\tilde{V}$.
$\tilde{g}_{\mu\nu}$ is regular everywhere and conformally flat. Note that the
hyperbolas in the Penrose diagram (see Figure B1) can be described by

$UV=e^{2\kappa r^{*}}=(r-r_{H})^{\frac{\kappa r_{H}}{2(r_{H}+b_{2})^{3}}},$
(111)

where $b_{2}$ contains the mass and $\kappa$ a constant. So when the black
hole is evaporating, the hyperbola moves to the left in region I in the
Penrose diagram. If $r$ is close to $r_{H}$, the hyperbola approaches the
Planckian area.

##  Appendix C  In $(U,V)$-coordinates

In light cone coordinates $u=r-t$ and $v=r+t$, one obtains a set of PDE’s

$\displaystyle\Delta
N=\frac{1}{1+N^{4}}\mathpzc{A},\quad\Delta\omega=\frac{1}{1-N^{4}}\mathpzc{B},\quad\Delta
X=\frac{1}{1-N^{4}}\mathpzc{C},$ (112)

where $\mathpzc{A},\mathpzc{B},\mathpzc{C}$ are expressions in $N,\omega,X$
and their derivatives. Further,
$\Delta\equiv\partial_{u}^{2}+\partial_{v}^{2}$. These PDE’s can be used, in a
numerical setting, in order to follow the Hawking particles in the $(U,V)$
plane. 313131This technical aspect is currently under investigation by the
author. One also can proof that Laplace equations make sense on Klein
surfacesalling1971

##  Appendix D Notations

$d$: spacetime dimension
$n$: topological windingnumber or Chern number
$(t,r,z,\varphi)$: coordinates of the 4D spacetime
$\mathpzc{y}$: fifth dimension in RS model
$y_{0}$: dimension of the bulk spacetime
$\Bigl{(}N(t,r),N^{\varphi}(t,r)\Bigr{)}$: metric components
${\cal V}$: $z+i\mathpzc{y}$
${\cal W}$: $x+iy$
${\cal Z}:\frac{{\cal V}}{{\cal W}}$
$\mathds{R}^{n}$: n-dimensional Euclidean space
$\mathds{C}^{n}$: complex n-dimensional space
$\eta$: vacuum expectation value
$\omega(t,r)$: dilaton field
$\Phi=\eta X(t,r)e^{in\varphi}$: complex scalar (Higgs) field
$A_{\mu}$: gauge field =$(0,0,0,A_{\varphi},0)$
$S^{m}$: m-sphere, $\\{x\in\mathds{R}^{m+1}\mathchar 24635\relax\;|x|=1\\}$
$B^{m}$: m-ball
$\mathds{R}P^{m}$: real projective m-space
$\mathds{C}P^{m}$: complex projected m-space
$\mathds{Z}_{2}$: reflection symmetry, i.e. a symmetry operation that yields
the identity if applied twice
$\mathds{T}$: torus $S^{1}\times S^{1}$
$\mathds{K}:$ Klein surface = compact surface (i.e., triangulation possible)
of the connected sum of two projective planes
$\mathds{R}P^{1}\\#\mathds{R}P^{1}$
$f:S^{2}\rightarrow\mathds{R}P^{1}$: map of 2-sphere onto its quotient space
(or identification space topology)
$(S^{2},f)$: covering space of $\mathds{R}P^{1}$

##  Appendix E The Schrödinger equation and the multiple-scale method

###  E.1  The $\Phi^{4}$-Klein-Gordon equation

Consider the 1+1 dimensional Lagrangian density323232We follow the notation of
Felsagerfelsager1998 . Here one can find more details of the short
presentation here.

${\cal
L}=\frac{1}{2}\Bigl{[}-\partial_{\mu}\Phi\partial^{\mu}\Phi-V(\Phi\Bigr{]},$
(113)

with $V=\frac{1}{4}\beta(\Phi^{2}-\frac{\alpha}{\beta})^{2}$. From Euler-
Lagrange equations we obtain the well-known $\Phi^{4}$ model

$\Phi_{tt}-\Phi_{xx}=-(\beta\Phi^{2}-\alpha)\Phi.$ (114)

Note that $-\frac{1}{2}\alpha\Phi^{2}$ is the mass term in the potential. The
potential possesses two distinct minima, so we have a classical degenerated
vacuum, $\Phi_{0}=\pm\sqrt{\frac{\alpha}{\beta}}$. At infinity, the field is
static and the configuration must have finite energy. This results in boundary
conditions

$\displaystyle\Phi_{\pm\infty}=\lim_{x\rightarrow\pm\infty}\Phi(x,t)=\pm\sqrt{\frac{\alpha}{\beta}},\qquad
U(\Phi_{\pm\infty})=0.$ (115)

###  E.2 The "kink"-sector

The most simple topological soliton solution of the $\Phi^{4}$ model occur in
one space dimension. Solutions interpolate between the two vacua
$\Phi_{\pm}=\sqrt{\frac{\alpha}{\beta}}$. The topological "charge" is then

$Q=\int_{-\infty}^{+\infty}\frac{\partial\Phi}{\partial
x}dx=\Phi(+\infty,t)-\Phi(-\infty,t)=\frac{\Phi_{+}-\Phi_{-}}{2\sqrt{\frac{\alpha}{\beta}}},\quad
Q=\\{-1,0,1\\}.$ (116)

We see that $Q$ only depends on the asymptotic behavior at infinity. Note that
in the related sine-Gordon model, one has an infinite series of discrete
minima, (Q = 0…n) in contrast with the $\Phi^{4}$ model. So in this case Q is
topologically quantized. However, in 2 space dimensions $(r,\varphi)$ in the
axially symmetry, one has the topological quantization via the winding
number333333Note that solitary waves doesn’t exists in 2 space dimensions
(Derrick’s theorem). So one needs the vortex solution (Nielsen-Oleson string).

One can wonder, if the vacuum degeneracy will survive the quantization. The
answer is yes. The soliton solution is quantum-mechanically stable, i.e.,
cannot decay into one of the vacua.

Figure 6: The $\Phi$-kink needs an infinite amount of energy in order to
overcome the ridge of infinite length.

A traveling wave solution can easily be obtained.

If one wants a localized traveling wave solution of the form (solitary wave)

$\Phi(x,t)=f(x-vt)=f(\xi),\qquad\lim_{\xi\rightarrow\pm\infty}f(\xi)=\Phi_{\pm\infty},$
(117)

then one can write the equation as

$\frac{df}{d\xi}=\pm\gamma\sqrt{2U[f]},\qquad\gamma=\frac{1}{\sqrt{1-v^{2}}},$
(118)

or

$\lambda(\xi-\xi_{0})=\pm\int\frac{df}{\sqrt{2U[f]}}.$ (119)

It possesses the "kink" and "anti-kink"-solutions.

###  E.3 The method of multiple-scales

Let us expand

$\Phi(x,t\mathchar
24635\relax\;\epsilon)=\sum_{1}^{\infty}\epsilon^{kp}\Phi^{(k)}(x,t\mathchar
24635\relax\;\epsilon)$ (120)

with $\epsilon$ a small parameter. Introduce "slow" variables

${\rm X}_{k}=\epsilon^{k}x,\qquad{\rm T}_{k}=\epsilon^{k}t,\quad k=1,2,..$
(121)

So we write

$\Phi(x,t\mathchar 24635\relax\;\epsilon)=\tilde{\Phi}(x,t,{\rm X}_{1},{\rm
T}_{1},{\rm X}_{2},{\rm T}_{2},...\mathchar 24635\relax\;\epsilon)$ (122)

and

$\frac{\partial\Phi}{\partial x}=\frac{\partial\tilde{\Phi}}{\partial
x}+\sum_{k=1}\epsilon^{k}\frac{\partial\tilde{\Phi}}{\partial{\rm
X}_{k}},\qquad\frac{\partial\Phi}{\partial
t}=\frac{\partial\tilde{\Phi}}{\partial
t}+\sum_{k=1}\epsilon^{k}\frac{\partial\tilde{\Phi}}{\partial{\rm T}_{k}}$
(123)

So $\epsilon$ measures the ratio of the fast scale to the slow scale.
Substitution into Eq.(114) with also

$\tilde{\Phi}(x,t,{\rm X}_{1},{\rm T}_{1},...\mathchar
24635\relax\;\epsilon)=\sum_{k=1}^{\infty}\epsilon^{kp}\tilde{\Phi}^{(k)}(x,t,{\rm
X}_{1},{\rm T}_{1},...\mathchar 24635\relax\;\epsilon)$ (124)

and equating coefficients of equal powers of $\epsilon$, results in:(tilde
omitted):

1\. term of ${\cal O}(\epsilon^{p})$

$\epsilon^{p}\Bigl{(}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial t^{2}}+\alpha\Bigr{)}\Phi^{(1)}$ (125)

2\. term of ${\cal O}(\epsilon^{p+1})$

$2\epsilon^{p+1}\Bigl{(}\frac{\partial^{2}}{\partial x\partial{\rm
X}_{1}}-\frac{\partial^{2}}{\partial t\partial{\rm T}_{1}}\Bigr{)}\Phi^{(1)}$
(126)

3\. term of ${\cal O}(\epsilon^{p+2})$

$\epsilon^{p+2}\Bigl{(}\frac{\partial^{2}}{\partial{\rm
X}_{1}^{2}}-\frac{\partial^{2}}{\partial{\rm
T}_{1}^{2}}+2\frac{\partial^{2}}{\partial x\partial{\rm
X}_{2}}-2\frac{\partial^{2}}{\partial t\partial{\rm T}_{2}}\Bigr{)}\Phi^{(1)}$
(127)

4\. term of ${\cal O}(\epsilon^{2p})$

$\epsilon^{2p}\Bigl{(}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial t^{2}}+\alpha\Bigr{)}\Phi^{(2)}$ (128)

5\. term of ${\cal O}(\epsilon^{2p+1})$

$2\epsilon^{2p+1}\Bigl{(}\frac{\partial^{2}}{\partial x\partial{\rm
X}_{1}}-\frac{\partial^{2}}{\partial t\partial{\rm T}_{1}}\Bigr{)}\Phi^{(2)}$
(129)

6\. term of ${\cal O}(\epsilon^{2p+2})$

$\epsilon^{2p+2}\Bigl{(}\frac{\partial^{2}}{\partial{\rm
X}_{1}^{2}}-\frac{\partial^{2}}{\partial{\rm
T}_{1}^{2}}+2\frac{\partial^{2}}{\partial x\partial{\rm
X}_{2}}-2\frac{\partial^{2}}{\partial t\partial{\rm T}_{2}}\Bigr{)}\Phi^{(2)}$
(130)

7\. term of ${\cal O}(\epsilon^{3p})$

$\epsilon^{3p}\Bigl{(}(\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial
t^{2}}+\alpha\Bigr{)}\Phi^{(3)}+\epsilon^{3p}\beta(\Phi^{(1)})^{3}$ (131)

So to lowest order we get:

$\boxed{\Bigl{(}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial t^{2}}+\alpha\Bigr{)}\Phi^{(1)}=0}$ (132)

This is the linearized equation. The next term is ${\cal O}(\epsilon^{p+1})$
and/or ${\cal O}(\epsilon^{2p})$. However $2p<(p+1)$ is impossible, because
$p\in{\rm N}$. So for the next term we have $p+1<2p$ or $p+1=2p$. If $p+1=2p$
then $p=1$ and we have (we ignore for the moment the other possibility
$p+1<2p,i.e.,p=2$ or 3 or 4, …) for the next order:

$\Bigl{(}\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial
t^{2}}+\alpha\Bigr{)}\Phi^{(2)}+2\Bigl{(}\frac{\partial^{2}}{\partial
x\partial{\rm X}_{1}}-\frac{\partial^{2}}{\partial t\partial{\rm
T}_{1}}\Bigr{)}\Phi^{(1)}=0$ (133)

and

$\displaystyle\Bigl{(}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial
t^{2}}+\alpha\Bigr{)}\Phi^{(3)}+\Bigl{(}\frac{\partial^{2}}{\partial{\rm
X}_{1}^{2}}-\frac{\partial^{2}}{\partial{\rm
T}_{1}^{2}}+2\frac{\partial^{2}}{\partial x\partial{\rm X}_{2}}$ (134)
$\displaystyle-2\frac{\partial^{2}}{\partial t\partial{\rm
T}_{2}}\Bigr{)}\Phi^{(1)}-\beta(\Phi^{(1)})^{3}+2\Bigl{(}\frac{\partial^{2}}{\partial
x\partial{\rm X}_{1}}-\frac{\partial^{2}}{\partial t\partial{\rm
T}_{1}}\Bigr{)}\Phi^{(2)}=0$ (135)

One can solve in principle Eq.(132)-Eq.(135). From Eq.(132) we have the
solution

$\Phi^{(1)}(x,t,{\rm X}_{1},{\rm T}_{1},{\rm X}_{2},{\rm
T}_{2},...)=A^{(1)}({\rm X}_{1},{\rm T}_{1},{\rm X}_{2},{\rm
T}_{2},...)e^{i\hbar(kx-\nu t)}+CC$ (136)

with $\hbar^{2}(k^{2}-\nu^{2})=\alpha$ (dispersion relation), $k$ the wave
number (number of oscillations per $2\pi$ in space) and $\nu$ the frequency
(number of oscillations per $2\pi$ in time). Substitution in Eq.(133) yields

$\boxed{\Bigl{(}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial
t^{2}}+\alpha\Bigr{)}\Phi^{(2)}=-2i\hbar\Bigl{(}k\frac{\partial
A^{(1)}}{\partial{\rm X}_{1}}+\nu\frac{\partial A^{(1)}}{\partial{\rm
T}_{1}}\Bigr{)}e^{i\hbar(kx-\nu t)}+CC}$ (137)

The solution will contain a secular term of the form

$B({\rm X}_{1},{\rm T}_{1},...)(kx-\nu t)e^{i\hbar(kx-\nu t)}$ (138)

due to the fact of resonance with $e^{i\hbar(k^{\prime}x-\nu^{\prime}t)}$,
which disturbs the asymptotic behavior. So we take

$\Bigl{(}k\frac{\partial A^{(1)}}{\partial{\rm X}_{1}}+\nu\frac{\partial
A^{(1)}}{\partial{\rm T}_{1}}\Bigr{)}=0$ (139)

That means, $A^{(1)}$ will be of the form

$A^{(1)}({\rm X}_{1},{\rm T}_{1},{\rm X}_{2},{\rm T}_{2},...)=A^{(1)}(\xi,{\rm
X}_{2},{\rm T}_{2},...),$ (140)

with

$\xi\equiv{\rm X}_{1}-\frac{k}{\nu}{\rm T}_{1}.$ (141)

So we have as first approximate solution of Eq.(114)

$\displaystyle\Phi^{(1)}(x,t\mathchar 24635\relax\;\epsilon)$ $\displaystyle=$
$\displaystyle A^{(1)}\Bigl{(}{\rm X}_{1}-\frac{k}{\nu}{\rm T}_{1},{\rm
X}_{2},{\rm T}_{2},...\Bigr{)}e^{i\hbar(kx-\omega t)}+CC$ (142)
$\displaystyle=$ $\displaystyle
A^{(1)}\Bigl{(}\epsilon(x-\frac{k}{\nu}t),\epsilon^{2}x,\epsilon^{2}t,...\Bigr{)}e^{i\hbar(kx-\omega
t)}+CC$ (143) $\displaystyle\approx$ $\displaystyle
A^{(1)}\Bigl{(}\epsilon(x-\frac{k}{\nu}t),0,0,...\Bigr{)}e^{i\hbar(kx-\nu
t)}+CC.$ (144)

We recognize the well-known carrier-wave (amplitude modulation)-solution: the
amplitude varies slowly with respect to the wave $e^{i\hbar(kx-\nu t)}$. The
envelope $A^{(1)}$ has a velocity of $\frac{k}{\nu}=\frac{d\nu}{dk}$, i. e.,
the velocity of the wave-packet.

Now we want to find the expression for $A^{(1)}$. From Eq.(137) and Eq.(139)
we obtain

$\Bigl{(}\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial
t^{2}}+\alpha\Bigr{)}\Phi^{(2)}=0,$ (145)

so a solution

$\Phi^{(2)}=A^{(2)}({\rm X}_{1},{\rm T}_{1},{\rm X}_{2},{\rm
T}_{2},...)e^{i\hbar(k^{\prime}x-\nu^{\prime}t)}+CC$ (146)

with $\hbar^{2}(k^{\prime 2}-\nu^{\prime 2})=\alpha$. Substitution in Eq.(135)
yields:

$\displaystyle\Bigl{(}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial
t^{2}}+\alpha\Bigr{)}\Phi^{(3)}=-\Bigl{(}1-\frac{k^{2}}{\nu^{2}}\Bigr{)}\frac{\partial^{2}A^{(1)}}{\partial\xi^{2}}e^{i\hbar(kx-\omega
t)}$ (147) $\displaystyle-2i\hbar\Bigl{(}k\frac{\partial A^{(1)}}{\partial{\rm
X}_{2}}+\nu\frac{\partial A^{(1)}}{\partial{\rm
T}_{2}}\Bigr{)}e^{i\hbar(kx-\nu t)}-2i\hbar\Bigl{(}k^{\prime}\frac{\partial
A^{(2)}}{\partial{\rm X}_{1}}+\nu^{\prime}\frac{\partial A^{(2)}}{\partial{\rm
T}_{1}}\Bigr{)}e^{i\hbar(k^{\prime}x-\nu^{\prime}t)}$ (148)
$\displaystyle+\beta\Bigl{(}(A^{(1)})^{3}e^{3i(kx-\nu
t)}+3(A^{(1)})^{2}\bar{A}^{(1)}e^{i(kx-\nu t)}\Bigr{)}+CC\qquad\qquad$ (149)

Again we skip the secular terms:

$\displaystyle 1.\quad k^{\prime}\frac{\partial A^{(2)}}{\partial{\rm
X}_{1}}+\nu^{\prime}\frac{\partial A^{(2)}}{\partial{\rm T}_{1}}=0$ (150)

so we have

$A^{(2)}({\rm X}_{1},{\rm T}_{1},{\rm X}_{2},{\rm
T}_{2},...)=A^{(2)}(\xi^{\prime}.{\rm X}_{2},{\rm T}_{2},...),$ (151)

with $\xi^{\prime}={\rm X}_{1}-\frac{k^{\prime}}{\nu^{\prime}}{\rm T}_{1}$,
and

$2.\quad\Bigl{(}1-\frac{k^{2}}{\nu^{2}}\Bigr{)}\frac{\partial^{2}A^{(1)}}{\partial\xi^{2}}+2i\hbar\Bigl{(}k\frac{\partial
A^{(1)}}{\partial{\rm X}_{2}}+\nu\frac{\partial A^{(1)}}{\partial{\rm
T}_{2}}\Bigr{)}-3\beta|A^{(1)}|^{2}A^{(1)}=0.$ (152)

If we introduce the moving coordinate system (group-velocity
$\frac{k}{\omega}$) $\xi_{2}={X}_{2}-\frac{k}{\nu}{T}_{2}$ and $T_{2}$, in
order to get rid of $\partial X_{2}$, we obtain for the amplitude

$A^{(1)}(\xi,\xi_{2},{\rm T}_{2},{\rm X}_{3},{\rm T}_{3},...)\approx
A^{(1)}(\epsilon(x-\frac{k}{\nu}t),\epsilon^{2}(x-\frac{k}{\nu}t),\epsilon^{2}t,...)$
(153)

the partial differential equation:

$\boxed{i\hbar\frac{\partial A^{(1)}}{\partial{\rm
T}_{2}}-\frac{\alpha}{2\hbar^{2}\nu^{3}}\frac{\partial^{2}A{(1)}}{\partial\xi_{2}^{2}}-\frac{3\beta}{2\nu}|A^{(1)}|^{2}A^{(1)}=0}.$
(154)

which represents a cubic Schrödinger equation. We used the dispersion relation
$k^{2}-\omega^{2}=\frac{\alpha}{\hbar^{2}}$. A typical solution is obtained
for $\alpha=-2\hbar^{2}\omega^{3}$

$A^{(1)}=C_{1}Exp\Bigl{[}i\Bigl{(}C_{2}\xi_{2}-(\frac{3\beta}{2\omega}C_{1}^{2}+C_{2}^{2})T_{2}+C_{3}\Bigr{)}\Bigr{]}$
(155)

where $C_{i}$ are constants. So we have the first order approximation solution
for $\Phi^{(1)}$

$\boxed{\Phi^{(1)}(x,t\mathchar
24635\relax\;\epsilon)=\Phi^{(1)}(x,t,\xi,\xi_{2},T_{2},...)=C_{1}e^{i(C_{2}\xi_{2}-(\frac{3\beta}{2\omega}C_{1}^{2}+C_{2}^{2})T_{2}+C_{3})}e^{i\hbar(kx-\omega
t)}+CC}$ (156)

This form can be compared with the so-called "amplitude modulation". Now in
our model, $\alpha$ is related to the mass in this $\Phi^{4}$ model. Other
interesting solutions are

$A^{(1)}=2C_{1}\sqrt{\frac{\omega}{3\beta}}\frac{e^{i(C_{1}^{2}T_{2}+C_{2})}}{\cosh(C_{1}\xi_{2}+C_{3})}$
(157)
$A^{(1)}=2C_{1}\sqrt{\frac{\omega}{3\beta}}\frac{e^{i\Bigl{(}C_{2}\xi_{2}+(C_{1}^{2}-C_{2}^{2})T_{2}+C_{1}\Bigr{)}}}{\cosh(C_{1}\xi_{2}-2C_{1}C_{2}T_{2}+C_{3})}$
(158)
$A^{(1)}=\frac{C_{1}}{\sqrt{T_{2}}}e^{i\Bigl{[}\frac{(\xi_{2}+C_{2})^{2}}{4T_{2}}+(\frac{3\beta}{2\omega}C_{1}^{2}\ln
T_{2}+C_{3})\Bigr{]}}$ (159)

We have now two scales, the slow $(\xi_{2},T_{2})$ and the fast $(t,x)$. Most
of the solutions are obtained by the inverse-scattering method, the method of
Zakharov-Shabat. Further, there are conserved quantities,

$\displaystyle\int_{-\infty}^{+\infty}|\Phi|^{2}dx,\quad\int_{-\infty}^{+\infty}i(\bar{\Phi}\partial_{x}\Phi-\Phi\bar{\partial\Phi})dx,\quad\int_{-\infty}^{+\infty}\Bigl{(}|\partial_{x}\Phi|^{2}+\frac{3\beta}{4\omega}|\Phi|^{4}\Bigr{)}$
(160)

It would be interesting to explore the case $p=2$.

###  E.4 Application to the Vaidya spacetime

One expands the relevant fields

$V_{i}=\sum_{n=0}^{\infty}\frac{1}{\nu^{n}}F_{i}^{(n)}({\bf x},\xi),$ (161)

where $\nu$ represents a dimensionless parameter ("frequency"), which will be
large. Further, $\xi=\nu\Theta({\bf x})$, with $\Theta$ a scalar (phase)
function on the manifold. The small parameter $\frac{1}{\nu}$ can also be the
ratio of the characteristic wavelength of the perturbation to the
characteristic dimension of the background. On warped spacetimes it could also
be the ratio of the extra dimension to the background dimension. In the vacuum
case, we expand the metric

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\frac{1}{\nu}h_{\mu\nu}({\bf
x},\xi)+\frac{1}{\nu^{2}}k_{\mu\nu}({\bf x},\xi)+...,$ (162)

where we defined

$\displaystyle\frac{dg_{\mu\nu}}{dx^{\sigma}}=g_{\mu\nu,\sigma}+\nu
l_{\sigma}\dot{g}_{\mu\nu},\quad g_{\mu\nu,\sigma}=\frac{\partial
g_{\mu\nu}}{\partial x^{\sigma}},\quad\dot{g}_{\mu\nu}=\frac{\partial
g_{\mu\nu}}{\partial\xi},$ (163)

with $l_{\mu}=\frac{\partial\Theta}{\partial x^{\mu}}$. One then says that

$V_{i}=\sum_{n=-m}^{\infty}\frac{1}{\nu^{n}}F_{i}^{(n)}({\bf x},\xi)$ (164)

is an approximate wavelike solution of order n of the field equation, if
$F_{i}^{(n)}=0,\forall n$. One can substitute the expansion into the field
equations. The Ricci tensor then expands as

$R_{\mu\nu}\rightarrow\omega
R_{\mu\nu}^{(-1)}+\Bigl{(}\bar{R}_{\mu\nu}+R_{\mu\nu}^{(0)}\Bigr{)}+\frac{1}{\omega}R_{\mu\nu}^{(1)}+...$
(165)

By equating the subsequent orders to zero, we obtain

$R_{\mu\nu}^{(-1)}=0=\frac{1}{2}\bar{g}^{\beta\lambda}(l_{\lambda}l_{\mu}\ddot{h}_{\beta\nu}+l_{\nu}l_{\beta}\ddot{h}_{\mu\lambda}-l_{\lambda}l_{\beta}\ddot{h}_{\mu\nu}-l_{\nu}l_{\mu}\ddot{h}_{\beta\lambda}),$
(166) $\displaystyle R_{\mu\nu}^{(0)}+\bar{R}_{\mu\nu}=0,\qquad
R_{\mu\nu}^{(1)}=0,\qquad....$ (167)

The bar stands for the background. Here we used $l_{\mu}l^{\mu}=0$. The rapid
variation is observed in the direction of $l_{\mu}$. The eikonal condition,
$l_{\mu}l^{\mu}=0$, in linear approximations, is adopted as gauge condition.
In the high-frequency approximation, however, it follows from the highest
order ($n=-1$) equations. In the radiative outgoing Eddington-Finkelstein
coordinates, we have $x^{1}=u=\Theta({\bf x})=t-r$ and $l_{\mu}=(1,0,0,0)$.
=========================oOo==========================

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