# Exact solutions and Birkhoff’s theorem in Jackiw–Teitelboim Gravity

Davood Momeni<EMAIL_ADDRESS>Department of Physics, College of Science,
Sultan Qaboos University, P.O. Box 36,
Al-Khodh 123, Muscat, Sultanate of Oman

###### Abstract

Exact static and time dependent solutions for Jackiw–Teitelboim (JT) gravity
revisited in a comprehensive form. The emphasis is on one parameter solutions
for metric and dilaton field in the static Poincare’s coordinates. We study
cosmological toy models where the time-dependent metric asymptotically
coincides to the AdS hyperbolic spacetime in non static patch. In this case,
the full dilaton profile can be expressed in the form of real time Laplace
transform over spatial coordinate $z$ in real Laplace space $s$. The time-
dependent Laplace amplitude $\tilde{\phi}(t,s)$ satisfies a linear ODE and can
be written as a linear combinations of the associated Legendre’s functions
$P_{1}^{\nu(t,s)}(\tanh(t))$ ,$Q_{1}^{\nu(t,s)}(\tanh(t))$ with complex values
of $\nu(t,s)$. The space time representation of the dilaton profile presented
in terms of Bromwich Integral. over $s$. We also addressed Birkhoff’s theorem
and showed that JT theory globally doesn’t admit such static , time-
independent vacuum solutions. The viable deformed JT action examined for
having exact analytic solutions in static and cosmological forms.

##  I Introduction

Since the discovery of AdS/CFT by Juan Maldacena , the conjecture enjoyed
several applications from pure string theory towards realistic models in
condensed matter physics, for example to describe peculiar behavior of high
temperature superconductors or computing holographic entanglement by
Ryu–Takayanagi conjecture. The Maldacena conjecture stated that there is a
duality between theories with gravity on bulk and certain types of qunatum
field theories on flat boundaries at specific regime of coupling, namely large
$N$ limitMaldacena:1997re . this conjecture interpreted as a direct duality
between physics in hyperbolic space time (AdS) and certain types of quantum
field theories with conformal symmetry (CFT) witten . The conjecture
celebrated as AdS/CFT and it explains a deep hidden connection between
classical negative curvature solutions for Einstein general relativity (GR)
and a theory of qunatum fields with many body interactions. Although there are
some rigorous derivations towards a direct proof for the conjecture developed
by several authors in past, it (was) is still important to see how a realistic
many body system can be described precisely using such curved space time
configurations via AdS/CFT conjecture.

The beauty of the above conjecture is that duality doesn’t restrict only to
the qunatum systems which can be described by CFT. For example if the bulk
theory doesn’t have Lorentz symmetry the dual boundary theory can be a non
relativistic qunatum system described by Schrödinger group. Basically, the
bulk can be a non black hole described by a pure AdS geometry without horizon
temperature, the appropriate dual boundary for it can be a zero temperature
CFT or a QFT with mass gap. The important concept here is to understand
duality as a map between asymptotically AdS spaces and QFT with Ultraviolet
(UV) fixed points.

The idea of finding an exact dual bulk theory for some realistic condensed
matter systems (exactly solvable) motivated by Kitaev in his talks where he
suggested a naive duality between two dimensional black hole correlation
functions and a fully random Majorana fermions many body interacting system
Kitaev2015  . The model considered as an extension of the former model
proposed in past by Sachdev and Yi Sachdev . The Kitaev model considered four
fermions random interactions and it was later became clear that the bulk
theory is a two dimensional toy model Polchinski:2016xgd ; Maldacena:2016hyu .
The SYK model considered as an exact solvable model and very luckily it
provides an exact duality between a gauge theory and another gravity model in
bulk. It has been demonstrated that the model emerges conformal symmetry at
low temperature and possibly the dual bulk for model, provides a toy model for
qunatum gravity. It was not so clear how the AdS/CFT will work on two
dimensions since the boundary theory considered as a simple conformal qunatum
mechanics in past. The exact form of duality between two dimensional gravity
in bulk and an ensemble of hermitian random matrix theory (RMT) proposed in
Ref.Saad:2019lba . The authors in Ref. Saad:2019lba  showed that we can
compute the density function corresponds to the SYK density of the state using
the Euclidean path integral over gravity part. Thes work in Ref. Saad:2019lba
considered as a remarkable study about the SYK/gravity duality as a particular
case of the most general AdS/CFT scenarios.

Einstein gravity doesn’t work properly in two dimensions (2d) and the
Einstein-Hilbert action simply reduces to Euler characteristic, consequently
no one can get black hole in 2d from it. The triviality of 2d gravity
motivated many authors to find alternatives to GR which works in 2d, see for
example Bagrets:2016cdf -Iliesiu:2019xuh  . Unless the main question here was
to find a suitable theory to describe black holes and later Hawking radiation,
but searching for lower dimensional gravity theories also helped us to
understand the structure of a possible UV divergence free qunatum theory for
gravity. Many of such models inspired directly from string theory, mainly they
are considered as reductions of the higher dimensional gravity theories under
specific metric parametrizations. In a simplest form , $2d$ gravity can be
constructed using Einstein–Cartan theory, with suitable set of the variables
and by imposing a class of symmetry groups, starting from the BF models. This
process leads to a class of simple gravity models named as JT gravity
Jackiw:1984je ; Teitelboim:1983ux . The model is a family of scalar field
theories where the scalar field (dilaton) $\phi$ minimally coupled to gravity
in two dimensions. In the simplest form, the Euclidean action of the pure JT
gravity with a negative cosmological constant is represented by

$\displaystyle S=-\frac{1}{2}\int_{\Omega}\sqrt{g}d^{2}x\phi(R+2)+S_{bdy}$ (1)

In the above action, $S_{bdy}$ stands for any boundary topological term (or
the usual GHY boundary term), e.g. Euler characteristic, $R$ is the Ricci
scalar curvature of the Euclidean metric tensor $g_{\mu\nu},\mu,\nu=0,1$. To
not dealing with the boundary action, we suppose that $\Omega$ is a space time
region (without) boundary. Very recently a generalized family of models
proposed in Ref.Ecker:2021guy . The authors showed that JT gravity can be
considered as a particular sub class of another family of models. The
associated boundary action theory for the generalized model derived and it has
been demonstrated that the $2d$ holography can be extended beyond the JT
model.

Different aspects of the JT gravity investigated by many authors like
classical features as black hole solutions and thermodynamics and
gauge/gravity aspects of JT gravity , see for example the works done by
Refs.Almheiri:2014cka ; Maldacena:2016upp ; Engelsoy:2016xyb ; Harlow:2018tqv
.

Modified JT gravity proposed by Turiaci and Maxfield Turiaci:2020fjj  replaces
the linear term $2\phi$ with more general form and it has been demonstrated
that the model can be derived from minimal string action under proper metric
decomposition. Thermal radiation in JT gravity studied in Eom:2020dnx  via a
semi classical treatment of JT gravity. Free energy in JT gravity and the role
of critically investigated in Alishahiha:2020jko . Classical defects of AdS
geometry studied in Momeni:2020tyt . The scalar mode propagation in JT gravity
considered in Momeni:2020zkx . Deformations of JT gravity studied using
holography in Mertens:2019tcm , where the authors studied the deformation of
the Schwarzian theory as the dual quantum boundary action.

In GR, the Birkhoff’s theorem restricts the geometry of the vacuum solutions
to be static ( time- independent) and asymptotically flat,for example with
spherical symmetry, the vacuum Einstein field equations provides time-
independent components for the whole metric under study, i.e, $\frac{\partial
g_{\mu\nu}}{\partial t}=0$ if we use the parametric time coordinate $x^{0}=t$.
Birkhoff’s theorem also required the metric be asymptotically flat. As a
result, Schwarzschild metric is the unique, static , asymptotic flat metric
for exterior regions. In cylindrical symmetric distributions of mass the
metric will be replaced by Levi-Civita and with planar metric in the planar
symmetric configurations. By validity of theorem, we can match an interior
solution in GR to the exterior one successfully, however the problem still
doesn’t have accepted solution for rotating metrics for example the metric for
neutral, rotating , time independent geometries named Kerr metric.

The validity of Birkhoff’s theorem in any type of the models for gravity plays
a crucial role. In alternative models to gravity the (in)validity of theorem
examined in several works. In particular, the authors in Ref. Cavaglia:1997hca
studied Birkhoff’s theorem for a type of dilaton gravity with cosmological
constant known as CGHS model. The model constructed as a gauge invariant
theory as a sub class of BF models introduced firstly in Callan:1992rs  as a
different formulation of gravity in $2d$ manifolds.

Although JT gravity doesn’t represent the realistic geometry for any
astrophysical object but it can be considered as an effective action to
describe near horizon geometry for astrophysical black holes at the extremal
limit. Motivated by the above considerations, we would like to address
Birkhoff’s theorem in pure JT gravity.

The plan of this letter is as the following: In Sec. (II) we present general
framework for JT gravity. In Sec. (III) exact static, time-independent
solutions for theory studied in details. In Sec. (IV) we present exact
solutions for static and cosmological patches of the bulk theory. In Sec. (V)
we address the validity of the Birkhoff’s theorem. In Sec. (VI) we briefly
comment on the the integrability of the viable deformed JT gravity bulk
action. We conclude in Sec. (VII).

##  II Toy model and field equations

We adapt a type of the coordinates $x^{\mu}=(t,z)$ to describe Lorentzian
metric on a two dimensional space time manifold $\mathcal{M}^{2}$, using this
coordinate representation the conformal flat metric can be written as the
follows

$\displaystyle ds^{2}=e^{2\sigma}(-dt^{2}+dz^{2})$ (2)

here the conformal factor (or in the cosmological perspective the scale
factor) is considered as a general function of the coordinates
$\sigma=\sigma(t,z)$. For time slice $\Sigma_{t}=\\{t=t_{0}\\}$, the conformal
factor becomes time-independent, consequently it can describe a static metric.
In some cases, the geometry mimics a blackhole with a horizon given at
$z=z_{H}$. Such horizon black holes in the context of the JT gravity studied
in Ref. Witten:2020ert . In that Ref, Witten studied a general class of black
holes in a slightly different coordinate with definite temperature and
entropy. The family of the black holes extended beyond the JT gravity action,
i.e, (1). In our work we are interesting to conformal flat solutions instead
of the blackhole solutions. The Einstein field equations for variation of the
action (1) with respect to the metric $g_{\mu\nu}$ and scalar mode field are
given as

$\displaystyle R+2=0,$ (3) $\displaystyle\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\nabla_{\alpha}\nabla^{\alpha}\phi+g_{\mu\nu}\phi=0.$ (4)

Note that since the model has only two independent modes, one tensor mode
described by the metric function $\sigma$ and the other corresponds to the
spin $0$ scalar graviton $\phi$, we can work with the eq. (3) and trace of the
equation (4), i.e, the system of the PDEs for our toy model is given as the
following

$\displaystyle\nabla_{\alpha}\nabla^{\alpha}\sigma=e^{2\sigma},$ (5)
$\displaystyle\nabla_{\alpha}\nabla^{\alpha}\phi-2\phi=0.$ (6)

We expand the above field equations using the metric ansatz (2), finally we
end up by the following system of the PDEs, must be integrated to obtain the
general metric profile $\sigma(t,z)$ and scalar graviton mode $\phi(t,z)$:

$\displaystyle\sigma^{\prime\prime}-\ddot{\sigma}=e^{2\sigma}$ (7)
$\displaystyle\phi^{\prime\prime}-\ddot{\phi}=2e^{2\sigma}\phi$ (8)

here primes is used for spatial derivative with respect to the $z$ and dot
denotes time derivative. Note that the system is highly nonlinear in spite of
it’s simple form. There is no simple analytic technique to make it either
linear or integrable unless the metric depends only on the single coordinate
either time $t$ or spatial coordinate $z$. In general, when $\sigma(t,z)$, the
scalar mode profile $\phi(t,z)$ can not be understood easily in terms of the
linear operators. Furthermore it seems that a non static scalar field
$\phi(t,z)$ still can preserves the metric as a static geometry, i.e, if
$\phi=\phi(t,z)$, then it is possible to have $\sigma=\sigma(z)$. This
situation leads to a class of periodic solutions for scalar mode and they have
been investigated recently in Momeni:2020zkx . In this work we will focus only
on three classes of exact solutions: first the ones with static metric and
static scalar mode , second class a cosmological time-dependent class of
metrics with $\sigma(t),\phi(t)$ and finally to search for a class of unique
solutions with arbitrary scalar mode but still leads to the static metrics.
This last category provides an indirect proof for the Birkhoff’s theorem in JT
gravity and we will study it in the Sec. (IV) in details. We mention here that
a null coordinate transformation of the system of the equations presented in
the eqs. (31,32) leads to a non linear PDE can be integrated even for a
hyperbolic geometry with arbitrary genus $n$, this class of solutions along a
possible type of $AdS_{2}\rightarrow AdS_{2}$ phase transition investigated
recently by the author in Momeni:2020tyt . In this work we complete our study
about exact solutions in the JT gravity. The focus is on subclass of the
solutions and to study the uniqueness theorem for vacuum solution in JT
theory.

##  III Static solutions for $\sigma=\sigma(z)$

Time-independent metrics are suitable to study black hole properties in a
gravitational theory, including lower dimensional cases like JT gravity. In
the conformal gauge adapted by us, the single EoM given in (31) is fully
integrable for static conformal factor $\sigma=\sigma(z)$, basically if we
assign a type of conjugate momentum $p_{\sigma}=\sigma^{\prime}$, the first
integral of the equation is

$\displaystyle p_{\sigma}^{2}=e^{2\sigma}+2c_{1}$ (9)

here $c_{1}$ is an integration constant, can be fixed using an initial
conditions as $(\sigma_{0},p_{\sigma}(0))$. In the absence of such initial
profile we just kept it as an arbitrary parameter will be fixed later.
Actually what we are sure about the metric is that the AdS boundary value of
the metric $z\to 0$ should be asymptotically as $e^{\sigma}_{z\to 0}\sim
z^{-1}$. The integration of eq. (9) is elementary, gives us the following
metric as an exact static time-independent solution to JT gravity:

$\displaystyle
ds^{2}=c_{1}^{2}\frac{e^{-2c_{1}(z+c_{2})}}{\sinh^{2}(c_{1}(z+c_{2}))}(-dt^{2}+dz^{2})$
(10)

The above metric has shift symmetry, basically a killing vector
$\zeta_{z}=\partial_{z}$ as well as time translation symmetry $\partial_{t}$.
The reason is clear because the original JT gravity action constructed in a
gauge invariant structure, it is a BF model type of action. It allows us to
set $c_{2}=0$ without loss of the generality. Furthermore, it is possible to
redefine the coordinates as $t\to tc_{1},z\to zc_{1}$, the reason backs again
to the gauge invariance of theory, it implies a global gauge translation,under
this coordinate transformation, the metric finally transforms to the following
static form,

$\displaystyle ds^{2}=\frac{e^{-2z}}{\sinh^{2}z}(-dt^{2}+dz^{2})$ (11)

The above ultimate form of metric is time-independent, exact solution to JT
gravity EoM. It is remarkable to notify here that at the AdS boundary region,
when $z\to 0$, the metric reduces to the Poincare’s half-plane,

$\displaystyle ds^{2}=\frac{1}{z^{2}}(-dt^{2}+dz^{2})$ (12)

With metric (12), we have freedom to either consider the scalar dilaton time-
independent $\phi=\phi(z)$ or $\phi=\phi(t,z)$. In Ref. Momeni:2020zkx , we
studied extensively the scalar field profile for case $\phi=\phi(t,z)$. We
showed that the scalar dilaton on the AdS boundary has a Dirac delta-type
$\delta(t)$ of singularity. For $\phi=\phi(z)$, we can integrate (32) as a
second order position-dependent classical _repulsive_ -harmonic oscillator,

$\displaystyle\phi^{\prime\prime}-\Omega(z)^{2}\phi=0$ (13)

here $\Omega(z)=\sqrt{2}\frac{e^{-z}}{|\sinh z|}$ is the frequency. A exact
solution for scalar dilaton found as follows,

$\displaystyle\phi(z)=Ae^{2z/a}F(a,a,1+a^{3}|e^{2z})+Be^{-2z/a}F(-a,-a,1-a^{3},e^{2z}).$
(14)

here $a=\sqrt[4]{2}$ and $F(a,b,c|x)$ is the hypergeometric function. Except
this nice closed form for dilaton field, it is also remarkable to mention here
that the dilaton field decoupled from the gravity sector , i,e, the EoM for
scalar graviton can be derived independently from the following repulsive
oscillating system,

$\displaystyle\mathcal{L}_{\phi}=\frac{1}{2}\Big{(}\phi^{\prime
2}+\Omega(z)^{2}\phi^{2}\Big{)}.$ (15)

It has been clear that such classical systems with the above repulsive
Lagrangian forms are chaotic . It can provide a simple fully classical
description for emergence of the chaotic behavior in the boundary dual of the
JT action, i.e, to better understanding of the chaotic behavior in the SYK
many body random system. We don’t not argue about this chaotic behavior here
since the origin of the chaos in the SYK well investigated by computing Out-
of-Time-Ordered Correlators in 2d CFT whose are boundary dual to two, or four
point functions in the bulk theory.

##  IV Cosmological solutions $\sigma=\sigma(t)$

In analogue to the time-independent solution obtained in the previous section,
there is another possibility. Let the metric be time-dependent,probably the
scalar dilaton can be either pure time-dependent as $\phi=\phi(t)$ or
$\phi(t,z)$. Although the metric with time-dependent conformal factor
$\sigma(t)$ can not be realized as a realistic cosmological model for
Universe, but it is still possible to think about it as the near horizon
geometry of a type of extremal astrophysical black holes, probably the one we
observed recently. The field equation (31) for time-dependent conformal
geometry can be integrated easily as we did in the previous section. Note that
if one consider a type of weak rotation $z\to it$, a rotation of z-axis to
coincide to the t-axis,the exact solution for metric can be obtained in the
following form:

$\displaystyle
ds^{2}=\frac{\tilde{c}_{1}}{\cosh^{2}(\sqrt{\tilde{c}_{1}}|\tilde{c}_{2}+t|)}(-dt^{2}+dz^{2})$
(16)

here we performed another gauge transformation $t\to t+\tilde{c}_{2}$ in
analogous to the shift symmetry for static geometries. The metric makes sense
only if $\tilde{c}>0$, after a scaling (conformal) transformation
$t\to\sqrt{\tilde{c}_{1}}t,z\to\sqrt{\tilde{c}_{1}}z$ we find the following
exact time-dependent solution,

$\displaystyle ds^{2}=\cosh^{-2}(t)(-dt^{2}+dz^{2})$ (17)

Basically it is just the AdS metric at the boundary if one consider $t\to
z\to\infty$. It defines half-plane geometry. The scalar profile can be
obtained in general by solving the following linear PDE:

$\displaystyle\phi^{\prime\prime}-\ddot{\phi}-2\cosh^{-2}(t)\phi=0.$ (18)

Note that here $z\in[0,\infty)$, a possible integral transformation could be a
Laplace transformation ,

$\displaystyle\tilde{\phi}(t,z)=\int_{0}^{\infty}\phi(t,z)e^{-sz}dz$ (19)

By substituting it in the scalar field PDE, we obtain the following ODE for
Laplace amplitude, $\tilde{\phi}(t,s)$

$\displaystyle\ddot{\tilde{\phi}}(t,s)+\omega^{2}(t,s)\tilde{\phi}(t,s)=0$
(20)

here $\omega^{2}(t,s)=2\cosh^{-2}(t)-s^{2}+s\phi^{\prime}(t,0)+\phi(t,0)$ is
time-dependent natural frequency of the oscillator. With the boundary data
$\phi(t,0),\phi^{\prime}(t,0)$ it is possible to integrate the above ODE,then
the dilaton profile can be obtained using the inverse Laplace transform as a
contour integral in the complex $s$ plane (Bromwich Integral). The system in
fully integrable. A simple classical analogous to the above dilaton system
could be propagation of the mechanical wave in a non homogeneous media.
Suppose that the scalar dilaton at the AdS boundary as
$\phi(t,\epsilon)\propto\epsilon^{\Delta}$ here $\Delta$ is the usual
conformal dimension. Consequently,
$\phi^{\prime}(t,\epsilon)\propto\epsilon^{\Delta-1}$ if $\Delta>1$ then the
ODE for Laplace amplitude can be solved by a pair of associate Legendre
function of complex order
$\nu(t,s)=i\sqrt{-s^{2}+s\phi^{\prime}(t,0)+\phi(t,0)}$, here we assumed that
the dilaton and its first derivative at the AdS boundary remain time-
independent,

$\displaystyle\tilde{\phi}(t,s)=c_{3}(s)P_{1}^{\nu(t,s)}(\tanh(t))+c_{4}(s)Q_{1}^{\nu(t,s)}(\tanh(t)).$
(21)

where $c_{3}(s),c_{4}(s)$ are constants and can be fixed by the initial
condition as $\tilde{\phi}(0,s),\dot{\tilde{\phi}}(0,s)$. The inverse Laplace
transform to give us the dilaton profile is complicated but can be written
formally as the following Bromwich Integral:

$\displaystyle\phi(t,z)=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}e^{sz}ds\Big{(}c_{3}(s)P_{1}^{\nu(t,s)}(\tanh(t))+c_{4}(s)Q_{1}^{\nu(t,s)}(\tanh(t))\Big{)}$
(22)

The exact dilaton profile requires to know the precises values for the
Legendre’s coefficients $c_{3,4}(s)$. There is no ”simple” technique to relate
those coefficients to the initial data for $\phi(0,z),\dot{\phi}(0,z)$. The
formal solution presented here can be used to explore more details about the
boundary value of dilaton field, $\phi_{bdy}=\phi(t,0)$. Such analysis
requires to know the asymptotic values for associate Legendre’s functions at
the regime of $z\to 0$, such series explanations along other useful asymptotic
expressions reported in DLMF . Using those asymptotic expansions, it is
possible to find exact Schwarzsian boundary action in theory.

##  V Comments on Birkhoff’s theorem

Now we discuss the validity of Birkhoff’s theorem in JT gravity. Our aim is to
know in the ”vacuum” whether the metric will remain generally time-independent
or not. Since JT gravity doesn’t require any matter contents in theory, we
have to be very careful about the meaning and realization of the vacuum state.
If the model been quantized, then it was possible to consider vacuum through
zero particle (vacuum to vacuum) propagator. Here we don’t study the qunatum
effects which they require path integral formalism. Our problem is to study
classical geometries without any qunatum effect inside it. Note that since
general relativity in terms of the Einstein theory is trivial in two
dimensions, any type of the energy-momentum tensor vanishes, in our case the
only form of the energy-momentum tensor $T_{\mu\nu}$ is the one defines EoM
for metric, i.e, equation (32). To recover the usual meaning for vacuum, we
need to set $T_{\mu\nu}\equiv 0$. The only possible way is to take the dilaton
as $\phi=0$. Note that in general any other type of dilaton field with
uniform, constant profile $\phi=\phi_{0}$ also works for us. But that requires
to set the cosmological constant zero, in JT gravity we have to kept it as
much as we can. The only metric equation here is (31). If $\phi\equiv 0$, does
it always imply that the following metric is exact solution for metric field
equation or not?. The metric under study is as follows:

$\displaystyle ds^{2}=A(Z)(-dT^{2}+dT^{2})$ (23)

The consistency relation between (23) and (2) requires that there exist always
a set of the Birkhoff’s coordinates $T,Z$ in a such manner that ,

$\displaystyle e^{\sigma(t,z)}dt=A(Z)^{1/2}dT,$ (24) $\displaystyle
e^{\sigma(t,z)}dz=A(Z)^{1/2}dZ$ (25)

Note that in general $T=T(t,z),Z=Z(t,z)$. Lets consider simple case where
$e^{2\sigma(t,z)}=f(t)h(z)$, it is easy to show that in this case a coordinate
transformation from $t\to\int\sqrt{f(t)}dt,z\to\int\sqrt{h(z)}dz$ makes the
above system integrable. This is the only case where we hope to keep
Birkhoff’s theorem valid. But such ansatz must pass the (31), lets us see
whether that PDE has any analytical separable solution or not. If the above
factorization works, then we should have $\sigma(t,z)=\frac{1}{2}(\ln f(t)+\ln
h(z))$, then
$\sigma^{\prime\prime}=\frac{1}{2}(\frac{h^{\prime}}{h})^{\prime},\ddot{\sigma}=\frac{1}{2}\partial_{t}(\frac{\dot{f}}{f})$
the consistency requires we have one of the following constraints :

$\displaystyle h(z)=h_{0}$ (26) $\displaystyle f(t)=f_{0}.$ (27)

The Birkhoff’s theorem remains valid either if $h(z)=h_{0}$ or $f(t)=f_{0}$,
because for example for first case the metric reads as follows

$\displaystyle ds^{2}=h_{0}f(t)(-dt^{2}+dz^{2}),$ (28)

using a conformal transformations as $t\to iz,z\to z$, and scaling $h_{0}=1$,
the metric transforms to the following static form:

$\displaystyle ds^{2}=f(z)(dz^{2}-dt^{2})$ (29)

which is time-independent. If we opt the other choice as $f(t)=f_{0}$ , we
will end up by the same metric in previous case. Note that these metrics
correspond to particular solutions of the system of the equations. The PDE for
$\sigma(t,z)$ have many solutions since it is nonlinear. Only one separable
family of solutions hold the Birkhoff’s theorem. In general, the Birkhoff’s
theorem doesn’t hold in such two-dimensional models for gravity.

The above straightforward addressing of the Birkhoff’s theorem in the
conformal gauge is valid. But it is also important to show that the vacuum
solution can be static even if we use a non conformal general time-dependent
metric as we usually write in the standard GR. Any 2d-metric can be
transformed to the conformal form using a proper infinitesimal transformations
of the coordinates. The reason is that JT theory doesn’t change because is
basically a dimensional reduction of the original GR in higher dimensions
under a specific metric decomposition. As we know such higher dimensional
theories are manifestly diffeomorphism invariant. Consequently we expect a
trace of this symmetry in the reduced lower dimensional theory as well. There
is no need to rewrite or repeat the validity or invalidity of theorem for this
specific gauge. As we already mentioned, the pure JT gravity doesn’t hold the
theorem.

##  VI Deformed JT gravity as an integrable system

The deep dual relation between JT gravity a type of RMT recently generalized
to the scalar potentials differ from the linear regime. As a novel extension
of the JT gravity proposal, recently in Ref. Maxfield:2020ale  the authors
studied the emergence of deformed JT gravity bulk models from $AdS_{3}$
gravity and its RMT dual description. The model motivated by Witten
Witten:2020wvy  in the extended form the bulk point of the view. The
remarkable observation about this deformed model is deformed JT gravity still
enjoys a RMT dual in the gauge/gravity picture. This duality well investigated
by Witten in the second paper Ref. Witten:2020ert . The deformed JT gravity
bulk theory proposed by simple adding a self-coupling potential term $U(\phi)$
of the scalar dilaton field. The model for first order interaction term
reduced to pure JT gravity. Such modified JT gravity as a viable deformation
of the JT gravity bulk action is the unique deformation to the bulk with a
well understood boundary description. Following the above references, we write
the action for deformed (dJT) in the form

$\displaystyle S=-\frac{1}{2}\int_{\Omega}d^{2}x\sqrt{g}\Big{(}\phi
R+U(\phi)\Big{)}\,.$ (30)

Note that pure JT gravity action recovered with $U(\phi)=2\phi$. Our interest
in this work is to find similar exact family of solutions with an arbitrary
potential. The metric gauge which we consider here is the one which we used in
(2). Using the equations of the motion derived by varying of the action (30)
with respect metric function $\sigma$ and scalar dilaton $\phi$ we obtain the
following set of the EoMs,

$\displaystyle\sigma^{\prime\prime}-\ddot{\sigma}=\frac{U^{\prime}(\phi)}{2}e^{2\sigma}$
(31) $\displaystyle\phi^{\prime\prime}-\ddot{\phi}=U(\phi)e^{2\sigma}$ (32)

The interesting case for vacuum when $\phi=0$ recovers that for a class of
self interactions $U(\phi)$, when $U(\phi)=\sum_{n=0}^{\infty}c_{n}\phi^{n}$,
the possible metric is a solution to the following PDE,

$\displaystyle\sigma^{\prime\prime}-\ddot{\sigma}=\frac{c_{1}}{2}e^{2\sigma}$
(33)

Thanks to gauge invariant of theory, we observe that the model reduces to the
following PDE for gauge transformed ”potential” function
$\tilde{\sigma}=\sigma+\log|\frac{c_{1}}{2}|$,

$\displaystyle\tilde{\sigma}^{\prime\prime}-\ddot{\tilde{\sigma}}=e^{2\tilde{\sigma}}$
(34)

which is the same equations as eq. (5). We do not repeat the analysis we did
in Sec. (II). There is still static and non static solutions for this non
linear 2d wave equation. One can realize the vacuum geometry either as static
or cosmological AdS. The existence of such vacuum is very crucial and it
supports the ideas about qunatum gravity via 2d gravity. We believe the above
PDE is fully integrable unless one can show that the solutions violate one of
the conditions which we mentioned in the above, for example they can’t be
represented as gauge transformations of each other. The existence of a general
metric $\tilde{\sigma}(t,z)$ depends strictly on the initial data
$\tilde{\sigma}(0,z),\dot{\tilde{\sigma}}(0,z)$. The above PDE can be realized
as a 2d wave equation in a media with source. Since the origin of the source
term is also written from the metric field $J=|e^{\tilde{\sigma}}|$, one can
interpret the above equation as a non linear Klein-Gordon equation for the
auxiliary wave function $\Psi=e^{\tilde{\sigma}}$. It is possible to provide
soliton solutions similar to Lorentz invariant version of the Gross–Pitaevskii
equation, where the Hamiltonian replaced by the relativistic version of it.
Such soliton solutions can be used to describes the ground state for qunatum
gravity in 2d.

##  VII Summary

JT gravity is the simplest gauge invariant theory to describe gravity in 2d
considered as a possible bulk dual for SYK theory. We revisited exact
solutions for scalar and tensor modes using integral transformation in both
static and time-dependent patches of the hyperbolic space time. The static
metric is a gauge invariant hyperbolic space. The scalar profile for dilaton
expressed in terms of the hypergeometric functions and describes a repulsive
harmonic oscillator with position dependent frequency. Such systems are
potentially chaotic at classical level. In the cosmological patch, we solved
the metric equation of the motion after a proper coordinate transformation.
The metric represented as a time-dependent version of the static metric. The
dilaton field satisfies an inhomogeneous linear wave equation in the compact
space time. Using Laplace transformation and by giving initial data , one is
able to find scalar profile using Bromwich integral, where the kernel
expressed in terms of the associated Legendre functions with complex order. As
an attempt to compare results with GR, we prove generally, the Birkhoff’s
theorem is no longer hold in JT gravity. We comment on possible exact
integrability of the viable deformation of the JT gravity for an arbitrary
smooth self interaction potential term in the action beyond the linear
potential. A naive investigations shows that for a class of potential
functions with $U^{\prime}(0)\neq 0$ , there are static gauge transformed
solution in comparison to the original JT gravity. Our results represent a
step forward towards fully understanding more aspects of this nice lower
dimensional toy model for gravity in two dimensions.

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