aainstitutetext: Institut für Theoretische Physik, Goethe Universität, Max-
von-Laue-Str. 1, 60438 Frankfurt am Main, Germanybbinstitutetext: Theoretical
Physics Department, CERN, CH-1211 Genève 23, Switzerlandccinstitutetext:
Departament de Física Quàntica i Astrofísica and Institut de Ciències del
Cosmos (ICC),
Universitat de Barcelona, Martí i Franquès 1, ES-08028, Barcelona,
Spaineeinstitutetext: Institució Catalana de Recerca i Estudis Avançats
(ICREA), Passeig Lluís Companys 23,
ES-08010, Barcelona, Spain

# Holographic Evolution with Dynamical Boundary Gravity

Christian Ecker b Wilke van der Schee c,d David Mateos c Jorge Casalderrey-
Solana<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Holography has provided valuable insights into the time evolution of strongly
coupled gauge theories in a fixed spacetime. However, this framework is
insufficient if this spacetime is dynamical. We present a scheme to evolve a
four-dimensional, strongly interacting gauge theory coupled to four-
dimensional dynamical gravity in the semiclassical regime. As in previous
work, we use holography to evolve the quantum gauge theory stress tensor,
whereas the four-dimensional metric evolves according to Einstein’s equations
coupled to the expectation value of the stress tensor. The novelty of our
approach is that both the boundary and the bulk spacetimes are constructed
dynamically, one time step at a time. We focus on Friedmann-Lemaître-
Robertson-Walker geometries and evolve far-from-equilibrium initial states
that lead to asymptotically expanding, flat or collapsing Universes.

##  1 Introduction

Holography relates the quantum-mechanical time evolution of a strongly
coupled, four-dimensional gauge theory to that of classical gravity in a five-
dimensional asymptotically anti de Sitter (AAdS) spacetime. The power of this
correspondence is that it allows for the use of classical gravity in five
dimensions to tackle otherwise intractable problems on the gauge theory side.

The spacetime where the gauge theory is formulated is identified with the
boundary of AAdS. We will refer to its four-dimensional metric as the
“boundary metric”, and to the five-dimensional metric in AAdS as the “bulk
metric”. In many applications of holography the boundary metric is taken to be
non-dynamical. For example, this metric is flat in the holographic description
of the quark-gluon plasma Casalderrey-Solana:2011dxg ; Busza:2018rrf  or in
applications to condensed matter systems Zaanen:2015oix ; Hartnoll:2016apf ;
Nastase:2017cxp . Applications with a curved metric include gauge dynamics in
black hole backgrounds Marolf:2013ioa  or in de Sitter (dS) space
Buchel:2002wf ; Maldacena:2012xp ; Fischler:2013fba ; Buchel:2017qwd ;
Buchel:2017lhu ; Buchel:2019pjb ; Buchel:2019qcq ; Casalderrey-Solana:2020vls
. In all these cases the boundary metric influences, but is unaffected by, the
gauge theory dynamics. In other words, the backreaction of the gauge degrees
of freedom on the metric is not included.

Despite its successes, this framework is insufficient if the boundary metric
is dynamical. This limits potential applications of holography to cosmological
defects, phase transitions in the early Universe, neutron star mergers,
inflation, pre- or re-heating, cosmological instabilities, etc. In these
applications one is interested in the semiclassical-gravity regime. This means
that the gauge theory is quantum mechanical but the metric obeys the classical
Einstein equations sourced by the expectation value of the gauge theory stress
tensor:

$R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}=8\pi G\,\langle
T_{\mu\nu}\rangle\,.$ (1)

All quantities in this equation, including Newton’s constant $G$ and a
possible cosmological constant $\Lambda$, refer to the four-dimensional
boundary theory. Hereafter we will refer to the gauge theory stress tensor
simply as “the stress tensor”. Since this is $\mathcal{O}(N^{2})$ in the
large-$N$ limit, we assume that $G$ is $\mathcal{O}(N^{-2})$ in order to have
a finite back-reaction. In the following we work with $N$-independent
quantities defined via the rescalings

$T_{\mu\nu}\to\left(2\pi^{2}/N^{2}\right)T_{\mu\nu}\,,\qquad
G\to\left(N^{2}/2\pi^{2}\right)G\,.$ (2)

The key point in the semiclassical regime is to determine the quantum-
mechanical evolution of the stress tensor, which must be done self-
consistently in the presence of the dynamical metric $g_{\mu\nu}$. We use
holography to determine this evolution (see Fig. 1).

Figure 1: Penrose diagram of our evolution scheme. The diagonal blue lines are
four-dimensional null slices in the bulk. Each point on the vertical black
line is a three-dimensional spatial slice of the boundary spacetime.

The initial state at time $t_{0}$ is defined by the five-dimensional fields on
a bulk null slice, together with the four-dimensional metric on a boundary
spatial slice. These two sets of initial data must satisfy non-trivial
“corner” consistency conditions that we will analyse below (see
Friedrich:1995vb ; Enciso:2014lwa ; Carranza:2018wkp ; Horowitz:2019dym  for
related discussions). For the moment, it suffices to say that the leading term
in the near-boundary fall-off of the bulk metric must coincide with the
boundary metric, whereas the subleading term in this fall-off determines the
expectation value of the stress tensor. To evolve to a time
$t_{1}=t_{0}+\Delta t$, we first use equation (1) to determine the new
boundary metric at $t_{1}$. Because AAdS is not globally hyperbolic, this new
metric provides necessary boundary conditions that allow us to evolve the
five-dimensional bulk equations to determine the new bulk fields at $t_{1}$.
The subleading term of the five-dimensional metric near the boundary then
determines the stress tensor at $t_{1}$.

The semiclassical regime has been previously extensively considered in the
holographic context. An incomplete list of references includes Csaki:1999jh ;
Kehagias:1999vr ; Cline:1999ts ; Csaki:1999mp ; Gubser:1999vj ; Dvali:2000rv ;
Karch:2000ct ; Kiritsis:2005bm ; Apostolopoulos:2008ru ; Compere:2008us ;
Erdmenger:2011sy ; Dong:2011uf ; Banerjee:2012dw ; Fischetti:2014hxa ;
Buchel:2017pto ; Chesler:2020exl ; Ghosh:2020qsx ; Emparan:2020znc . The main
novelty of our approach with respect to previous work is that both the
boundary and the bulk spacetimes are constructed dynamically, one time step at
a time. Further differences include the fact that we do not introduce an
ultraviolet cut-off in the gauge theory or branes in the bulk Csaki:1999jh ;
Kehagias:1999vr ; Cline:1999ts ; Csaki:1999mp ; Gubser:1999vj ; Dvali:2000rv ;
Karch:2000ct ; Kiritsis:2005bm ; Dong:2011uf ; Emparan:2020znc  but work
directly with dynamical gravity at the boundary; we do not assume a perfect-
fluid form for the stress tensor Fischetti:2014hxa  or consider a derivative
expansion Buchel:2016cbj  but allow for arbitrarily-far-from-equilibrium
dynamics; and we do not make use of predetermined bulk solutions
Apostolopoulos:2008ru ; Erdmenger:2011sy ; Banerjee:2012dw  or restrict
ourselves to constant-curvature boundary metrics Ghosh:2020qsx .

For simplicity, in this paper we will focus on homogenous and isotropic gauge-
theory states, namely on Friedmann-Lemaître-Robertson-Walker geometries.
However, we expect that our approach can be extended to more general cases.
Our full code is publicly available at http://wilkevanderschee.nl/public-
codes.

##  2 Model, scheme and evolution

###  2.1 Model

We use the same model as in Ref. Casalderrey-Solana:2020vls , to which we
refer the reader for additional details. The four-dimensional gauge theory is
a large-$N$, strongly coupled, non-conformal theory with a mass scale $M$. We
will measure all the dimensionful gauge-theory quantities in units of $M$. The
five-dimensional bulk theory consists of gravity coupled to a scalar field
$\phi$, with action

$S=\frac{2}{8\pi G_{5}}\int_{\mathcal{M}}{\rm
d}^{5}x\sqrt{-\mathfrak{g}}\left(\frac{1}{4}\mathcal{R}-\frac{1}{2}(\partial\phi)^{2}-V(\phi)\right)+\frac{1}{8\pi
G_{5}}\int_{\partial\mathcal{M}}{\rm d}^{4}x\sqrt{-\gamma}K+S_{ct}\,.$ (3)

Here $G_{5}$ is the five-dimensional Newton’s constant, $\mathcal{R}$ is the
Ricci scalar associated to the five-dimensional bulk metric $\mathfrak{g}$ on
$\mathcal{M}$, $\gamma_{\mu\nu}$ is the metric induced on a four-dimensional
slice near the boundary $\partial\mathcal{M}$, and

$K=\gamma^{\mu\nu}K_{\mu\nu}=\gamma^{\mu\nu}\nabla_{\mu}n_{\nu}$ (4)

is the trace of the extrinsic curvature $K_{\mu\nu}$ associated to this slice.
The second term on the right-hand side of (3) is the familiar Gibbons–Hawking
term. The third term in (3) will be described shortly. The potential $V(\phi)$
encodes the properties of the dual gauge theory. As in Casalderrey-
Solana:2020vls , we choose

$V(\phi)=-\frac{4}{3}W\left(\phi\right)^{2}+\frac{1}{2}W^{\prime}\left(\phi\right)^{2}\,,$
(5)

where the superpotential is given by

$L\,W\left(\phi\right)=-\frac{3}{2}-\frac{\phi^{2}}{2}+\frac{\phi^{4}}{4\phi^{2}_{M}}\,.$
(6)

$L$ is a length scale. The dimensionless constant $\phi_{M}$ is a free
parameter that controls the degree of non-conformality of the model, for
example the maximum value of the bulk viscosity. For concreteness, in this
paper we will choose

$\phi_{M}=2\,.$ (7)

Both $V(\phi)$ and $W(\phi)$ have a maximum at $\phi=0$ and a minimum at
$\phi=\phi_{M}$. Each of these extrema yields an AdS solution of the equations
of motion with constant $\phi$ and radius $L^{2}=-3/V(\phi)$. In the gauge
theory each of these solutions is dual to a fixed point of the renormalisation
group with a number of degrees of freedom $N^{2}$ proportional to
$L^{3}/G_{5}$. In top-down models this relation is known precisely. For
example, in the case in which the gauge theory is $\mathcal{N}=4$ SYM with $N$
colours we would have

$\frac{L^{3}}{8\pi G_{5}}=\frac{N^{2}}{4\pi^{2}}\,.$ (8)

In our bottom-up model we will take this as a definition of the number of
degrees of freedom in the gauge theory, $N$, at each fixed point.

###  2.2 Scheme

Near the AdS boundary the metric can be written in the so-called
Fefferman–Graham (FG) gauge

$ds^{2}=L^{2}\,\frac{{\rm d}\rho^{2}}{4\rho^{2}}+\gamma_{\mu\nu}(\rho,x){\rm
d}x^{\mu}{\rm d}x^{\nu}\,.$ (9)

The boundary is located at $\rho=0$ and is parametrised by the coordinates
$x^{\mu}$ with $\mu=0,\ldots,3$. Near the boundary the metric and the scalar
behave as

$\gamma_{\mu\nu}(z,x)\sim\frac{g_{\mu\nu}(x)}{\rho}\,,\qquad\phi\sim
M\rho^{1/2}\,,$ (10)

where $g_{\mu\nu}(x)$ is the boundary metric and $M$ is the gauge theory
intrinsic scale. Substituting this in the first term of the action (3) we see
that it suffers from large-volume divergences. These divergences can be
regularised and renormalised by a procedure called holographic renormalisation
(see e.g. deHaro:2000vlm ; Bianchi:2001de ; Bianchi:2001kw ), which makes the
action finite and the variational principle well-defined. This procedure is
implemented by including in (3) the counterterm action

$\displaystyle S_{\mathrm{ct}}=\frac{L}{8\pi
G_{5}}\int_{\partial\mathcal{M}}{\rm d}^{4}x\sqrt{-\gamma}$
$\displaystyle\Bigg{[}\left(-\frac{1}{8}R-\frac{3}{2}-\frac{1}{2}\phi^{2}\right)+\frac{1}{2}\left(\log\rho\right)\mathcal{A}+$
$\displaystyle+L^{2}\Big{(}\,\alpha\mathcal{A}+\beta\phi^{4}+\varepsilon\,\phi^{2}R+\xi_{1}R^{2}+\xi_{2}\,\nabla^{2}R+\xi_{3}\nabla_{\mu}\nabla_{\nu}R^{\mu\nu}\Big{)}\Bigg{]},\qquad$

where $\alpha,\beta,\varepsilon,\xi_{i}$ are real constants and the factors of
$L$ are necessary for dimensional reasons. This action is integrated on a
timelike, constant-$\rho$ hypersurface near the boundary with induced metric
$\gamma_{\mu\nu}$. In this and in subsequent equations all metric-dependent
terms such as the Ricci scalar $R$, the covariant derivative $\nabla$, etc.
are those associated to $\gamma$. The second term of (3) is also understood to
be evaluated on this slice, the first term of (3) is understood to be
evaluated by integrating down to this slice, and the limit $\rho\to 0$ is
understood to be taken at the end of the calculation.

In (2.2), $\mathcal{A}(\gamma_{\mu\nu},\phi)$ is the so-called conformal
anomaly, which in our case is given by

$\mathcal{A}=\mathcal{A}_{g}+\mathcal{A}_{\phi}\,,$ (12)

where

$\mathcal{A}_{g}=\frac{1}{16}(R^{\mu\nu}R_{\mu\nu}-\frac{1}{3}R^{2})$ (13)

is the holographic gravitational conformal anomaly and

$\mathcal{A}_{\phi}=-\frac{\phi^{2}}{12}R$ (14)

is the conformal anomaly due to matter. In these and in subsequent equations
all the terms are functionals of the metric $\gamma_{\mu\nu}$ and of the
scalar field $\phi$ induced on the $\rho$-hypersurface. However, making use of
the near-boundary behaviour (10) we see that the product with the determinant
of the induced metric yields a finite contribution in the limit in which the
cut-off is removed, since

$\lim_{\rho\to
0}\,\sqrt{-\gamma}\,\mathcal{A}\left(\gamma_{\mu\nu},\phi\right)\,=\,\lim_{\rho\to
0}\,\Big{[}\rho^{-4}\,\sqrt{-g}\Big{]}\Big{[}\rho^{4}\mathcal{A}\left(g_{\mu\nu},M\right)\Big{]}\,=\,\sqrt{-g}\,\mathcal{A}\left(g_{\mu\nu},M\right)\,.$
(15)

For this reason we will often think of the anomaly, as well as of other
curvature invariants, as functionals of $M$ and the boundary metric
$g_{\mu\nu}$.

The fact that $\sqrt{-\gamma}\mathcal{A}$ yields a finite result has two
consequences. First, it means that the logarithmic term in (2.2) cancels a
purely logarithmic divergence from the bulk action. The requirement that this
cancellation takes place fixes uniquely the form of the anomaly, including the
values of all the numerical coefficients in (13) and (14). The presence of
this logarithmic term on the gravity side breaks diffeomorphism invariance and
is dual to the presence of the conformal anomaly in the dual gauge theory.

The second consequence is that the anomaly itself, without the $\log$, can be
added to the counterterm action with an arbitrary coefficient, which we named
$\alpha$ in (2.2). It is important to note that not just the anomaly but any
local, finite term that is invariant under the symmetries of the theory can be
added to the counterterm action with an arbitrary coefficient. The freedom to
add these terms with arbitrary coefficients is part of the general freedom in
the choice of renormalisation scheme. These terms can be constructed out of
non-negative powers of the scalar field and of curvature invariants of the
induced metric $\gamma_{\mu\nu}$ in such a way that their overall mass
dimension is four.111Derivatives of the scalar field should also be included
in situations with non-constant $M$. The second line of (2.2) is the most
general linear combination of terms of this type, except for the Kretschmann
scalar $R_{\mu\nu\tau\psi}R^{\mu\nu\tau\psi}$. We have not included the latter
because, in four dimensions, the integral

$\frac{1}{8\pi^{2}}\int\sqrt{-\gamma}\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\tau\psi}R^{\mu\nu\tau\psi}\right)=\chi\,,$
(16)

with $\chi$ the Euler character, is a topological invariant. A pedagogical
discussion of the independent curvature invariants in arbitrary dimension can
be found in http://kias.dyndns.org/crg/invariants.html.

The coefficients $\alpha$ and $\beta$ play special roles. In the first case,
this is because $\alpha$ can be shifted by a scale transformation, which is
implemented via the following rescaling of the coordinates

$x^{\mu}\to\lambda x^{\mu}\,,\qquad\rho\to\lambda^{2}\rho\,,$ (17)

where $\lambda$ is a positive real number. It is easy to see that the effect
of this transformation is to shift the counterterm action by a term of the
form $(\log\lambda)\mathcal{A}$, which in turn can be absorbed through the
redefinition $\alpha\to\alpha+\log\lambda$. The freedom to rescale $\rho$, or
equivalently to shift $\alpha$, is thus the freedom to choose a
renormalisation scale. We may therefore think of $\alpha$ as related to the
renormalization group scale or subtraction point $\mu$ through

$\alpha\sim\log\mu\,.$ (18)

In the second case the reason is that the counterterm associated to $\beta$ is
the only one that does not vanish for a flat boundary metric. In particular,
the value

$\beta=\frac{1}{4\phi_{M}^{2}}=\frac{1}{16}$ (19)

is special because in this case the $\beta\phi^{4}$ term combines with the
second and the third summands in the first term of (2.2) to give precisely the
superpotential (6). This means that, if the theory (3) is the bosonic
truncation of a supersymmetric theory with superpotential $W$, then this
choice of $\beta$ leads to a supersymmetric renormalisation scheme and, as a
consequence, the full boundary stress tensor vanishes identically if the
boundary metric is flat. We will come back to these points below.

The finite counterterms give contributions to the gauge theory stress tensor,
which therefore we can write as

$T_{\mu\nu}=T_{\mu\nu}^{(0)}+\alpha\left(T_{\mu\nu}^{(g)}+T_{\mu\nu}^{(\phi)}\right)+\beta\,T_{\mu\nu}^{(\beta)}+\varepsilon\,T_{\mu\nu}^{(\varepsilon)}+\xi_{i}\,T_{\mu\nu}^{(i)}\,,$
(20)

where $T_{\mu\nu}^{(0)}$ denotes the stress tensor in the scheme
$\alpha=\beta=\varepsilon=\xi_{i}=0$. This means that, in the absence of
dynamical gravity at the boundary, the boundary stress tensor is ambiguous to
the extent that the coefficients of the finite counterterms are arbitrary.
However, in the presence of dynamical boundary gravity, these coefficients
simply renormalize the gravitational couplings and the ambiguity is replaced
by the physical specification of the renormalized couplings Birrell:1982ix .
To see this, imagine first setting all coefficients to zero except for $\beta$
and consider the contribution to the full stress tensor of
$T_{\mu\nu}^{(\beta)}$, which takes the form

$T_{\mu\nu}^{(\beta)}=\frac{L^{3}}{8\pi
G_{5}}\,M^{4}\,g_{\mu\nu}=\frac{N^{2}}{4\pi^{2}}\,M^{4}\,g_{\mu\nu}\,,$ (21)

where we have made use of (8). Moving this term to the left-hand side of (1)
we can write Einstein’s equations in the form

$\frac{1}{8\pi
G}\left(R_{\mu\nu}-\frac{1}{2}\,Rg_{\mu\nu}\right)+\frac{\Lambda_{\rm{ren}}}{8\pi
G}\,g_{\mu\nu}=T_{\mu\nu}^{(0)}\,,$ (22)

with

$\frac{\Lambda_{\rm{ren}}}{8\pi G}=\frac{\Lambda}{8\pi G}-\frac{\beta
M^{4}}{4\pi^{2}}\,.$ (23)

Note that in this equation the $N^{2}$-factor coming from (21) has cancelled
out with the $N^{2}$-factor coming from (2) in such a way that (23) is
$N$-independent. We see that the effect of the $\beta$-counterterm is simply
to renormalize the bare cosmological constant222More precisely, the
combination $\Lambda/8\pi G$. $\Lambda$ in Einstein’s equations. In other
words, $\Lambda$ and $\beta$ are not separately meaningful, only the
combination $\Lambda_{\rm{ren}}$ is. It is therefore convenient to choose
$\beta$ as in (19), since in this case flat space is a solution of (1) with
$\Lambda=0$.

Consider now adding the contribution of the $\alpha$-term. Also, assume for
the moment that the boundary metric is of FRWL type, which will be our focus
below. On this subclass of boundary states the stress tensor associated to
$\mathcal{A}_{g}$ vanishes, namely $T_{\mu\nu}^{(g)}=0$, and the stress tensor
of $\mathcal{A}_{\phi}$ combines with that of the $\varepsilon$-counterterm to
give

$\frac{L^{3}}{8\pi
G_{5}}\Big{[}\alpha\left(T_{\mu\nu}^{g}+T_{\mu\nu}^{\phi}\right)+\varepsilon\,T_{\mu\nu}^{(1)}\Big{]}=-\frac{N^{2}}{4\pi^{2}}\frac{M^{2}}{6}\left(\alpha+2\varepsilon\right)\left(R_{\mu\nu}-\frac{1}{2}\,Rg_{\mu\nu}\right)\,,$
(24)

where we have made use of (8). Moving this term to the left-hand side of (1)
we get

$\frac{1}{8\pi
G_{\rm{ren}}}\left(R_{\mu\nu}-\frac{1}{2}\,Rg_{\mu\nu}\right)+\frac{\Lambda_{\rm{ren}}}{8\pi
G_{\rm{ren}}}\,g_{\mu\nu}=T_{\mu\nu}^{(0)}\,,$ (25)

with

$\frac{1}{8\pi G_{\rm{ren}}}=\frac{1}{8\pi
G}+\frac{N^{2}}{4\pi^{2}}\frac{M^{2}}{6}\left(\alpha+2\varepsilon\right)\,.\vspace{2mm}$
(26)

As in (23), the $N^{2}$-factors cancel out in (26). We see that the effect of
the $\alpha$\- and $\varepsilon$-terms is simply to renormalize the bare,
four-dimensional Newton’s constant $G$. In other words, $G$, $\alpha$ and
$\varepsilon$ are not separately meaningful, only the combination
$G_{\rm{ren}}$ is. Since the coefficient $\alpha$ is associated to
renormalization group transformations through (18), Eq. (26) can be seen as
the renormalization group equation for the running of Newton’s constant. For
convenience we will work in the scheme $\alpha=\varepsilon=0$.

Consider now what happens if the boundary metric is not FRWL or if we consider
the rest of the finite counterterms with coefficients $\zeta_{i}$. These terms
are of order higher than two in derivatives. If we were considering the most
general semiclassical gravitational theory, as we would do in the full
effective theory, then these contributions would simply renormalize the bare
values of the coefficients of higher-curvature terms that were omitted in (1).
As above, the only meaningful quantities would be the renormalized couplings
measured at the physical energy scale of interest $\mu$. The dynamical regime
that we wish to study is that of classical gravity coupled to quantum matter.
This means that $\mu\ll M_{p}$, with $M_{p}$ the Planck mass, since otherwise
we need to include quantum gravity effects. This is the regime of interest,
for example, if we want to understand how the QCD transition took place as the
Universe expanded and cooled. In this regime we expect that the renormalized
higher-derivative couplings will be suppressed by factors of $\mu/M_{p}$.
Therefore it is a consistent approximation to work with a truncated effective
theory in which we set these couplings to zero. Since only renormalized
couplings matter we may therefore declare that, in this approximation, all the
$\xi_{i}$ vanish and the left-hand side of (1) contains only two-derivative
terms.

###  2.3 Evolution

Having fixed the renormalization scheme, we can now discuss the dynamics. For
simplicity, we focus on homogeneous and isotropic states in the four-
dimensional theory, namely on Friedmann-Lemaître-Robertson-Walker cosmologies.
As a consequence, the boundary metric is completely determined by a scale
factor $a(t)$, and the only non-zero components of the stress tensor are the
energy density $\mathcal{E}(t)$ and the pressure $\mathcal{P}(t)$. Under these
conditions, (1) reduces to the Friedmann equation

$\left(\frac{\dot{a}}{a}\right)^{2}\equiv
H^{2}=\frac{1}{3}\,\Lambda+\frac{8\pi G}{3}\,\mathcal{E}$ (27)

and the continuity equation

$\dot{\mathcal{E}}=-3H\left(\mathcal{E}+\mathcal{P}\right)\,,$ (28)

with $H=\dot{a}/a$ the Hubble rate.

Below we will extract the stress tensor from the near-boundary fall-off of the
five-dimensional fields. For illustration, consider the bulk scalar field. In
an appropriate null holographic coordinate $r$ with the boundary at
$r\to\infty$ we have Casalderrey-Solana:2020vls

$\phi=\frac{M}{r}+\frac{\phi_{2}(t)}{r^{3}}+\frac{1}{r}\sum_{n\geq
3}\frac{\phi_{n}(t)}{r^{n}}+\frac{1}{r}\sum_{n\geq 2}\frac{\psi_{n}(t)\log
r}{r^{n}}+\cdots$ (29)

The logarithmic terms are specific to odd-dimensional bulk spacetimes
AST_1985__S131__95_0 . The near-boundary analysis only leaves undetermined
$\phi_{2}(t)$. The remaining coefficients $\phi_{n\geq 3}$ and $\psi_{n\geq
2}$ are given in terms of $\phi_{2}(t)$, $a(t)$ and their derivatives by
expressions of the form

$\displaystyle\phi_{n}\left(M,a,\dot{a},\ldots,a^{(n)},\phi_{2},\dot{\phi}_{2},\ldots,\phi_{2}^{(n-2)}\right),$
(30a)
$\displaystyle\psi_{n}\left(M,a,\dot{a},\ldots,a^{(n)}\right).\,\,\,\,\,\,\,\,$
(30b)

There is a similar expression for the fall-off of the five-dimensional bulk
metric with one undetermined coefficient $a_{4}(t)$. From the bulk viewpoint,
the function $\phi(r,t_{0})$ and the coefficient $a_{4}(t_{0})$ at an initial
time $t_{0}$ are free data. Moreover, if this data and the scale factor
$a(t_{0})$ are known, then integration of the constraints coming from the
Einstein-scalar equations in the bulk determines the entire five-dimensional
metric on the initial time slice at $t=t_{0}$.

Figure 2: Evolution of the Hubble rate (top), of the energy density and
pressure (middle), and of the effective temperature (bottom), for $G=1/2500$
and three different values of $\Lambda$.

Figure 3: Comparison between the holographic result for the pressure/energy
ratio (blue) and the ideal (red) and viscous (green) hydrodynamic
approximations, for $G=1/750$, $\Lambda=0$ and three different initial
conditions.

Eqs. (30), which arise from the bulk equations of motion, constitute a set of
constraints that relate the bulk initial condition $\phi(r,t_{0})$ and the
derivatives of the boundary scale factor. These corner conditions imply that
the initial data on the bulk slice and the boundary conditions on that slice
cannot be specified independently. For a non-dynamical boundary metric, as is
in e.g. Chesler:2008hg , $a(t)$ can be prescribed arbitrarily and these bulk
constraints can be used to determine the $\psi_{n}(t)$ coefficients. In
contrast, in the case of dynamical boundary gravity, it is highly non-trivial
that these bulk constraints can be made compatible with those coming from the
boundary Einstein equations (1). The latter arise as follows. The stress
tensor depends on the undetermined coefficients and on the scale factor via
expressions of the form Casalderrey-Solana:2020vls

$\mathcal{E}\left(a_{4},\phi_{2},a,\dot{a},\ddot{a}\right)\,,\qquad\mathcal{P}\left(a_{4},\phi_{2},a,\dot{a},\ddot{a}\right)\,.$
(31)

These, together with (27), (28) and (30a), can be shown to determine all the
derivatives of the scale factor at $t_{0}$ in terms of $a(t_{0})$,
$a_{4}(t_{0})$ and $\phi_{n}(t_{0})$. Through (30b), this fixes all the
logarithmic terms in $\phi(r,t_{0})$. This is particularly important for our
scheme because numerically we work with “subtracted” variables that differ
from the original ones by a number of logarithmic (and some non-logarithmic)
terms. Specifically, our evolution scheme is as follows. At $t_{0}$ we specify
$a(t_{0})$, $a_{4}(t_{0})$, $\phi_{2}(t_{0})$ and the subtracted version of
$\phi(r,t_{0})$. We then use the procedure above to find $a^{(n)}(t_{0})$ up
to $n=4$. These determine all the necessary logarithmic terms. Next we
integrate the Einstein-scalar constraints and find all the bulk data on the
initial time slice. By construction this is consistent with the corner
conditions up to the desired order. Finally, we use the bulk and the boundary
evolution equations to obtain $a$, $a_{4}$, $\phi_{2}$ and the subtracted
version of $\phi$ at $t_{0}+\Delta t$.

##  3 Results

We perform evolutions for three different values of the cosmological constant
$\Lambda=\\{-0.5,0,2\\}$. As initial data at $t_{0}=0$ we use $a(0)=1$ and a
radial profile $\phi(r,t_{0})$ that corresponds to a thermal equilibrium state
in flat space. In all cases we choose $a_{4}(0)=-2000$, except in Fig. 3, for
which $a_{4}(0)=-100$.

Fig. 2(top) shows the evolution of the Hubble rate. Negative $\Lambda$ leads
to a “Big Crunch” where the Hubble rate evolves towards minus infinity and the
spacetime collapses. For $\Lambda=0$ the Hubble rate decays to zero and the
spacetime approaches Minkowski space. Positive $\Lambda$ leads to an
exponentially expanding dS Universe.

Fig. 2(middle) shows $\mathcal{E}$ and $\mathcal{P}$. For $\Lambda<0$ the
energy density reaches a minimum, after which it diverges as the Big Crunch is
approached. For $\Lambda=0$, $\mathcal{E}$ and $\mathcal{P}$ decrease in a
power-law fashion that is well described by hydrodynamics (see below). For
$\Lambda>0$ the Universe approaches dS with a small Casimir contribution from
the non-conformal matter, $\mathcal{E}_{\rm dS}=-\mathcal{P}_{\rm dS}\approx
0.2667$. 333This value is consistent with Casalderrey-Solana:2020vls  after
taking into account a typo in Sec 4.3 of Casalderrey-Solana:2020vls , where we
wrote that we chose $\alpha=0$ while the actual value was $\alpha=3/4$.

In Fig. 2(bottom) we show the temperature of the gauge theory state,
$T=\kappa/2\pi$, computed from the surface gravity, $\kappa$, of the event
(EH) and of the apparent (AH) horizons of the bulk geometry. For $\Lambda<0$
the AH reaches the boundary of AAdS at a finite boundary proper time. The
boundary itself collapses at this point, and $T_{AH}$ diverges. We do not show
$T_{EH}$ because the definition of the EH is unclear in this case. For
$\Lambda>0$ the temperatures at late times approach $T_{EH}=-T_{AH}=H/2\pi$,
in agreement with Buchel:2017pto ; Casalderrey-Solana:2020vls . For
$\Lambda=0$ the horizon falls deep into the bulk and at late times $H\propto
t^{-1}$ and $T\propto t^{-1/2}$, as expected. In addition, Eqn. (27) implies
$\mathcal{E}/H^{4}\sim t^{2}/G\gg 1$, meaning that the dynamics is dominated
by the energy density. As a consequence, the late-time boundary state
approaches a thermal state in Minkowski space and the bulk EH and AH become
indistinguishable.

Holography can evolve strongly-coupled, far-from-equilibrium, quantum matter
which, after some time, is expected to enter a hydrodynamic regime (except in
dS Casalderrey-Solana:2020vls , see below). For $\Lambda=0$ this is
illustrated in Fig. 3, which shows the evolution of the pressure/energy ratio
for three different initial conditions, IC1, IC2 and IC3. For IC2 and IC3 we
added respectively $+2$ and $-1.5$ to the subtracted $\phi(r,0)$ of IC1. This
leads to evolutions that are just about numerically stable and hence as far
from equilibrium as our code allows. The blue curves are the holographic
results. The difference with the viscous hydrodynamic approximation
Casalderrey-Solana:2020vls  (green curves) at early times shows that the
initial dynamics is far from equilibrium. After $\Delta t\approx 2$ the
evolution becomes well described by viscous hydrodynamics, consistently with a
hydrodynamization time of $\mathcal{O}(1/T)$ Heller:2011ju ; Heller:2012km .
The comparison to ideal hydrodynamics in the right panel of Fig. 3 shows that
viscous corrections can be sizable even at late times.

The initial far-from-equilibrium period leaves an imprint on the scale factor.
This is illustrated in Fig. 4, which shows the Hubble rate for the three
evolutions of Fig. 3 as a function of the redshift
$z(t)=a(t_{\rm{obs}})/a(t)-1$. The time $t_{\rm obs}$ is defined for each
curve by the physical condition that $\mathcal{E}$ reaches some late-time
value, in this case $\mathcal{E}(t_{\rm obs})=0.02$. At small redshift the
evolutions are equivalent as a consequence of the applicability of
hydrodynamics at late times shown in Fig. 3. In contrast, at large redshift
the far-from-equilibrium dynamics at early times leads to significantly
different Hubble rates.

Figure 4: Evolution of the Hubble rate as function of redshift $z$ for the
three different initial conditions presented in Fig. 3.

In Fig. 5 we show the analogous results for $\Lambda=-0.5$.

Figure 5: Comparison between the holographic result for the pressure/energy
ratio (black) and the ideal (blue) and viscous (red) approximations, for
$G=1/2500$ and $\Lambda=-0.5$.

The dashed, grey line marks the time where $\mathcal{E}$ reaches a minimum and
$H=0$. The entire evolution is well described by viscous hydrodynamics. As
above, viscous corrections are non-negligible at late times.

Fig. 6 illustrates the asymtotically dS case.

Figure 6: Same as in Fig. 5 but for $\Lambda=2$.

At late times the backreaction is dominated by the cosmological constant,
which here includes a Casimir contribution that we subtract in the plot. Once
the expansion has diluted the energy density so that
$\mathcal{E}-\mathcal{E}_{\rm dS}\lesssim H^{4}$, the system is driven out of
equilibrium and the hydrodynamic approximation ceases to be valid, as expected
from the non-backreacted analysis Casalderrey-Solana:2020vls .

##  4 Discussion

We have provided the first example of holographic time evolution with
dynamical boundary gravity in which both the bulk and the boundary geometries
are constructed dynamically, one time step at a time.

In order to illustrate our approach in the simplest possible setting, we have
focused on homogeneous and isotropic states. However, we expect that our
scheme can be generalised to situations with no symmetry assumptions. Our work
thus suggests new possible applications of holography that we will develop
elsewhere. Here we just close with brief comments on two of them.

Inflation could be studied by promoting the boundary value of the bulk scalar
field to a dynamical boundary scalar field which would play the role of the
inflaton. This would allow us to use holography to study e.g. the pre- and re-
heating processes at the end of inflation Kofman:1994rk ; Kofman:1997yn .

In the absence of symmetry assumptions, cosmological backgrounds are expected
to suffer from instabilities Mukhanov:1996ak . This has been studied
holographically in the linear approximation Chesler:2020exl . Our approach
would allow us to determine the endpoint of these instabilities deep into the
nonlinear regime.

###### Acknowledgements.

It is a pleasure to thank R. Emparan, J. Garriga, G. Horowitz, E. Kiritsis and
J. Mas for discussions. JCS and DM are supported by grants SGR-2017-754,
PID2019-105614GB-C21, PID2019-105614GB-C22 and the “Unit of Excellence MdM
2020-2023” award to the Institute of Cosmos Sciences (CEX2019-000918-M).

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#  On the super edge-magicness of graphs with a specific degree sequence

Rikio Ichishima  Department of Sport and Physical Education, Faculty of
Physical Education, Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo
206-8515, Japan<EMAIL_ADDRESS>and  Francesc A. Muntaner-Batle
Graph Theory and Applications Research Group, School of Electrical Engineering
and Computer Science, Faculty of Engineering and Built Environment, The
University of Newcastle, NSW 2308 Australia<EMAIL_ADDRESS>
(Date: Aug 13, 2023)

###### Abstract.

A graph $G$ is said to be super edge-magic if there exists a bijective
function $f:V\left(G\right)\cup
E\left(G\right)\rightarrow\left\\{1,2,\ldots,\left|V\left(G\right)\right|+\left|E\left(G\right)\right|\right\\}$
such that
$f\left(V\left(G\right)\right)=\left\\{1,2,\ldots,\left|V\left(G\right)\right|\right\\}$
and $f\left(u\right)+f\left(v\right)+f\left(uv\right)$ is a constant for each
$uv\in E\left(G\right)$. In this paper, we study the super edge-magicness of
graphs of order $n$ with degree sequence $s:4,2,2,\ldots,2$. We also
investigate the super edge-magic properties of certain families of graphs.
This leads us to propose some open problems.

###### Key words and phrases:

(super) edge-magic graph, (super) edge-magic labeling, vertex degree, degree
sequence, graph labeling

###### 1991 Mathematics Subject Classification:

05C07, 05C78

##  1\. Introduction

Only graphs without loops or multiple edges will be considered in this paper.
Undefined graph theoretical notation and terminology can be found in [4] or
[17]. The _vertex set_ of a graph $G$ is denoted by $V\left(G\right)$, while
the _edge set_ of $G$ is denoted by $E\left(G\right)$.

Motivated by the notion of magic valuation of graphs introduced by Kotzig and
Rosa [15], Enomoto, Lladó, Nakamigawa, and Ringel [5] introduced the concept
of super edge-magic graphs. A graph $G$ is said to be _super edge-magic_ if
there exists a bijective function $f:V\left(G\right)\cup
E\left(G\right)\rightarrow\left\\{1,2,\ldots,\left|V\left(G\right)\right|+\left|E\left(G\right)\right|\right\\}$
such that
$f\left(V\left(G\right)\right)=\left\\{1,2,\ldots,\left|V\left(G\right)\right|\right\\}$
and $f\left(u\right)+f\left(v\right)+f\left(uv\right)$ is a constant (called
the _valence_ $\text{val}\left(f\right)$ of $f$) for each $uv\in
E\left(G\right)$. Such a function is called a _super edge-magic labeling_.

The following lemma found in [6] is a very useful characterization of super
edge-magic graphs.

######  Lemma 1.1.

A graph $G$ is super edge-magic if and only if there exists a bijective
function
$f:V\left(G\right)\rightarrow\left\\{1,2,\ldots,\left|V\left(G\right)\right|\right\\}$
such that the set

$S=\left\\{f\left(u\right)+f\left(v\right):uv\in E\left(G\right)\right\\}$

consists of $\left|E\left(G\right)\right|$ consecutive integers. In such a
case, $f$ extends to a super edge-magic labeling of $G$ with valence
$k=\left|V\left(G\right)\right|+\left|E\left(G\right)\right|+s$, where
$s=\min\left(S\right)$ and

$S=\left\\{k-\left(\left|V\left(G\right)\right|+1\right),k-\left(\left|V\left(G\right)\right|+2\right),\ldots,k-\left(\left|V\left(G\right)\right|+\left|E\left(G\right)\right|\right)\right\\}\text{.}$

It is worth to mention that Acharya and Hegde [1] introduced the concept of
strongly indexable graph that turns out to be equivalent to the concept of
super edge-magic graph (see [8]). The study of super edge-magic labelings of
graphs has proven to be crucial in the last two decades, since many relations
with other types of labelings have been found (see [6]), and relations with
other concepts such as Skolem and Langford sequences (see [9]), and dual
shuffle primes and Jacobsthal sequences (see [11] and [14]).

For a thorough study of graph labeling problems, see the extensive survey by
Gallian [7]. For more information on super edge-magic graphs and related
topics, see the books by Bača and Miller[2], Chartrand, Egan, and Zhang [3],
López and Muntaner-Batle[10], and Marr and Wallis [16].

One of the main goals of this paper is to study the super edge-magic
properties of graphs of order $n$ with degree sequence $s:4,2,2,\ldots,2$. In
particular, we would like to stress the family of graphs that we define next
that will be a point of study in the subsequent pages of this paper. For
integers $m\geq 3$ and $n\geq 3$, let $C(m,n)$ denote the graph consisting of
two cycles $C_{m}$ and $C_{n}$ of orders $m$ and $n$ in such a way that the
two cycles share exactly one vertex in common. Observe that this family of
graphs is the simplest case of the family of graphs for which all blocks are
cycles. In fact, another goal for this paper is to spread around the interest
about the super edge-magic properties of graphs of this type.

Next, we proceed with the following lemma found in [6] that will be useful to
obtain certain results in this paper. For this purpose, the degree of a vertex
$v$ is denoted by $\deg_{G}(v)$ or simply by $\deg(v)$ if the graph $G$ under
discussion is clear.

######  Lemma 1.2.

Let $G$ be a graph such that $\deg(v)$ is even for all $v\in V\left(G\right)$
and $\left|E\left(G\right)\right|\equiv 2\pmod{4}$. Then $G$ is not super
edge-magic.

##  2\. Main results

We are now ready to state and prove our first result.

######  Theorem 2.1.

Let $G$ be a graph of even order $n\geq 6$ with degree sequence
$s:4,2,2,\ldots,2$. Then $G$ is not super edge-magic.

###### Proof.

Consider such a graph $G$ with a super edge-magic labeling $f$. Then

$\left|E\left(G\right)\right|=\sum_{v\in
V\left(G\right)}\deg(v)=\frac{4+2\left(n-1\right)}{2}=n+1\text{.}$

Thus, the valence $\text{val}\left(f\right)$ of $f$ can be computed as
follows:

$\begin{split}\text{val}\left(f\right)&=\frac{2\sum_{i=1}^{n}i+\sum_{i=n+1}^{2n+1}i+2\alpha}{n+1}\\\
&=\frac{\sum_{i=1}^{n}i+\sum_{i=1}^{2n+1}i+2\alpha}{n+1}=\frac{5n^{2}+7n+\left(2+4\alpha\right)}{2\left(n+1\right)}\text{,}\end{split}$

where $\alpha\in\left[1,n\right]$. We next examine for which valences of
$\alpha\in\left[1,n\right]$, $4\alpha$ is a multiple of $2\left(n+1\right)$
or, equivalently, $2\alpha$ is a multiple of $n+1$. Observe that if
$\alpha\in\left[1,n/2\right]$, then we have $2\alpha\leq n<n+1$. On the other
hand, if $\alpha\in\left[n/2+1,n\right]$, then we have

$n+1<n+2\leq 2\alpha\leq 2n<2n+1\text{.}$

Consequently, in either case, $2\alpha$ cannot be a multiple of $n+1$. Now,

$5n^{2}+7n+\left(2+4\alpha\right)=2\left(n+1\right)\left(\frac{5n}{2}+1\right)+4\alpha$

so that

$\text{val}\left(f\right)=\frac{5n^{2}+7n+\left(2+4\alpha\right)}{2\left(n+1\right)}=\left(\frac{5n}{2}+1\right)+\frac{2\alpha}{n+1}\text{.}$

Note that $5n/2+1$ is a positive integer, since $n$ is an even integer $n\geq
6$. However, since $2\alpha$ is not a multiple of $n+1$, it follows from the
last equation that $\text{val}\left(f\right)$ is not a positive integer, which
is impossible. ∎

Observe that the preceding result excludes many graphs from being super edge-
magic. In particular, we immediately obtain the following corollary.

######  Corollary 2.1.

For every two integers $m\geq 3$ and $n\geq 3$, the graph $C\left(m,n\right)$
is not super edge-magic when $m+n$ is odd.

At this point, a question raises naturally as the next problem indicates.

######  Problem 1.

What can be said about the super edge-magicness of graphs of odd order $n\geq
5$ with degree sequence $s:4,2,2,\ldots,2$?

A partial solution to Problem 1 comes directly from Lemma 1.2. It is clear
that a graph of odd order $n$ with degree sequence $s:4,2,2,\ldots,2$ has even
size. If the size is not only even, but also not divisible by $4$, then it
meets the hypothesis of Lemma 1.2, and hence it is not super edge-magic. Then
Problem 1 becomes as follows.

######  Problem 2.

Let $G$ be a graph of odd order $n\geq 7$ with
$\left|E\left(G\right)\right|\equiv 0\pmod{4}$ and degree sequence
$s:4,2,2,\ldots,2$. What can be said about the super edge-magicness of $G$?

We know so far that the family of graphs $C\left(m,n\right)$ cannot be super
edge-magic except possibly in the case when $m+n\equiv 0\pmod{4}$. This leads
to propose the following problem.

######  Problem 3.

What can be said about the super edge-magicness of $C\left(m.n\right)$ when
$m+n\equiv 0\pmod{4}$?

An exhaustive computer search shows that $C\left(3,5\right)$ is not super
edge-magic but is this true in general? It is especially interesting to know
whether $C\left(3,4k-3\right)$ is super edge-magic for integers $k\geq 3$.

##  3\. Conclusions

We have established the super edge-magicness of the family of graphs
$C\left(m,n\right)$ when $m+n$ is odd (see Corollary 2.1), and different
problems have emerged out of this study. In particular, we want to highlight
the following problem.

######  Problem 4.

What can be said about the super edge-magicness of graphs that consist of
blocks, which are all isomorphic to a cycle?

We formulate the preceding problem in the most general sense possible, that
is, the following concepts were introduced in [12]. Let $G$ be a graph of
order $p$ and size $q$ and $S_{G}$ denote the set

$S_{G}=\Biggl{\\{}\frac{\sum_{u\in
V\left(G\right)}\text{deg}(u)g(u)+\sum_{i=p+1}^{p+q}i}{q}\text{:}\\\ \text{
the function }g:V\left(G\right)\rightarrow\left[1,p\right]\text{ is
bijective}\Biggr{\\}}\text{.}$

If $\lceil\min S_{G}\rceil\leq\lfloor\max S_{G}\rfloor$, then the _super edge-
magic interval_ of $G$ is the set

$I_{G}=[\lceil\min S_{G}\rceil,\lfloor\max
S_{G}\rfloor]\cap\mathbb{N}\text{,}$

where $\mathbb{N}$ denotes the set of natural numbers. The _super edge-magic
set_ of $G$ is

$\sigma_{G}=\left\\{k\in I_{G}\right.\text{: there exists a super edge-magic
labeling of }G\text{ with valence }k\left.\right\\}.$

López et al. called a graph $G$ _perfect super edge-magic_ if
$I_{G}=\sigma_{G}$.

Hence, we would like to encourage researchers to study which graphs, if any,
in this family are perfect super edge-magic. It is worth to mention to finish
this paper that similar ideas to the ones of perfect super edge-magic graphs
had already appeared in some other works. The interested reader may consult
the results in [12] and [13].

### Acknowledgment

The authors are gratefully indebted to Yukio Takahashi for his continuous
encouragement and technical assistance during this work.

## References

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