# Mapping 1+1-dimensional black hole thermodynamics to finite volume effects

Jean Alexandre<EMAIL_ADDRESS>Department of Physics, King’s College
London, Strand, London, WC2R 2LS, United Kingdom  Drew Backhouse
<EMAIL_ADDRESS>Department of Physics, King’s College London, Strand,
London, WC2R 2LS, United Kingdom  Eleni-Alexandra Kontou
<EMAIL_ADDRESS>Department of Mathematics, King’s College London,
Strand, London WC2R 2LS, United Kingdom  Diego Pardo Santos
<EMAIL_ADDRESS>Department of Mathematics, King’s College London,
Strand, London WC2R 2LS, United Kingdom  Silvia Pla
<EMAIL_ADDRESS>Department of Physics, King’s College London,
Strand, London, WC2R 2LS, United Kingdom

###### Abstract

Both black hole thermodynamics and finite volume effects in quantum field
theory violate the null energy condition. Motivated by this, we compare
thermodynamic features between two 1+1-dimensional systems: (i) a scalar field
confined to a periodic spatial interval of length $a$ and tunneling between
two degenerate vacua; (ii) a dilatonic black hole at temperature $T$ in the
presence of matter fields. If we identify $a\propto T^{-1}$, we find similar
thermodynamic behaviours, which suggests some deeper connection arising from
the presence of non-trivial boundary conditions in both systems.

KCL-PH-TH/2024-29

###### Contents

  1. I Introduction
  2. II Finite size effects in 1+1 dimensional flat spacetime
    1. II.1 Convexity from tunnelling
    2. II.2 Tunneling and the Casimir effect
      1. II.2.1 Semi-classical approximation and true vacuum
      2. II.2.2 Static saddle points
      3. II.2.3 Time-dependent saddle points
      4. II.2.4 Partition function
    3. II.3 Null Energy Condition
    4. II.4 Entropy
  3. III Black holes in 1+1 dilaton gravity
    1. III.1 Introduction to dilaton gravity
    2. III.2 Adding quantum matter: static black holes
    3. III.3 Without backreaction, the CGHS model
      1. III.3.1 Null energy condition
      2. III.3.2 Entropy and thermodynamics
    4. III.4 With backreaction, the BPP model
      1. III.4.1 Null energy condition
      2. III.4.2 Entropy and thermodynamics
  4. IV Comparison
  5. V Conclusions
  6. A Casimir effect for static saddle points
    1. A.1 Zero temperature
    2. A.2 Finite-temperature corrections

##  I Introduction

One of the seminal results in semi-classical gravity is Hawking radiation, and
subsequent black hole evaporation Hawking:1974rv . Part of its importance is
that it shows a clear quantum effect in a regime where the curvature is small
(the black hole horizon) and thus it can be treated classically. An important
consequence of Hawking radiation is the violation of the null energy condition
(NEC) around the black hole horizon.

The NEC is part of the classical energy conditions which are what we call
pointwise: they restrict some contraction of the stress tensor at every
spacetime point (see Curiel:2014zba  and Kontou:2020bta  for reviews). The NEC
is the weakest of them and it is written as

$T_{\mu\nu}\ell^{\mu}\ell^{\nu}\geq 0\,,$ (1)

where $T_{\mu\nu}$ is the stress-energy tensor and $\ell^{\mu}$ a null vector
field. The NEC is obeyed by most classical fields111It is however violated by
scalars with non-minimal coupling to gravity Barcelo:2000zf ; Fliss:2023rzi .
and it is often considered an important property of physical matter. Its
geometric form, obtained by the use of the Einstein equation, is called the
null convergence condition and it implies that a non-rotating null geodesic
congruence locally converges. It was famously used in Penrose’s singularity
theorem Penrose:1964wq , Hawking’s black hole area theorem Hawking:1971vc  and
other classical relativity results. If the stress-energy tensor has the form
of a perfect fluid, $T_{\mu\nu}=(\rho+p)v^{\mu}v^{\nu}+pg^{\mu\nu}$, where
$\rho$ is the energy density, $p$ the pressure and $v^{\mu}$ is the fluid’s
unit four-velocity vector field, the NEC becomes

$\rho+p\geq 0\,.$ (2)

The NEC, as is the case for all pointwise energy conditions, is violated in
the context of semi-classical gravity; with the most prominent case being the
Hawking radiation. More generally, quantum field theories (QFTs) obeying some
reasonable axioms always have states that admit negative energy as shown by
Epstein, Glaser and Jaffe in the 1960’s Epstein:1965zza .

Interestingly, the NEC is violated in a different setting, involving finite
volume effects in scalar QFT. A finite volume in QFT implies two fundamental
features: quantisation of momentum and tunnelling between multiple vacua. The
first feature is at the origin of the Casimir effect (see Bordag  for a
review), which is known to induce NEC violation. As shown more recently and
reviewed in the next section, the second feature also leads to NEC violation,
in relation to convexity of the effective potential Alexandre:2022qxc ;
Alexandre:2023iig ; Alexandre:2023pkk ; Alexandre:2023bih ; Ai:2024taz . For
both the Casimir effect and tunnelling, NEC violation arises from a ground
state energy which is not extensive, i.e. not simply proportional to the size
of the system.

We investigate the possibility of a correspondence between these two sources
of NEC violation. In particular: (i) a scalar field confined to a periodic
spatial interval of length $a$ and tunnelling between two degenerate vacua in
the limit of zero temperature; (ii) a dilatonic black hole at temperature $T$
in the presence of matter fields in an infinite spatial volume.

For simplicity, the particular systems we are considering are both
1+1-dimensional222Other studies of finite volume QFT effects in 1+1
dimensional spacetime can be found in Hu:2021fjq .. The motivation is to find
common features between two non-trivial thermodynamical systems due to quantum
effects. Our main results are summarised in Table 1, in the regime $ma\propto
M/T\lesssim 1$, where $m,a$ are respectively the mass and length scales in the
tunnelling description, and $M,T$ are respectively the mass and temperature of
the dilatonic black hole (which are independent parameters). The two systems
are thermodynamically similar under the matching condition of $a\propto 1/T$,
suggesting a mapping between finite size and finite temperature. It is
important to note that this analogy cannot be attributed to dimensional
considerations though, since several length/mass scales are present in both
models.

Thermodynamical System |  Null Energy Condition violation |  Capacitance
---|---|---
Tunneling in 1+1 flat spacetime | $-\dfrac{\pi}{3a^{2}}$ | $\dfrac{1}{2}-\dfrac{2m}{3}a$
1+1 Dilatonic Black Holes | $-\dfrac{N\pi^{2}}{12}T^{2}$ | $-\dfrac{N}{12}-\dfrac{M}{\pi T}$
Table 1: Summary of results comparing the two thermodynamical systems: (i)
Tunneling in 1+1 flat spacetime (cf. Eqs. (31) and (93)); (ii) 1+1 Dilatonic
Black Holes (cf. Eqs. (68) and (91)). $N$ is the number of massless scalar
fields.

In Sec. II we calculate the free energy for the ground state resulting from
the scalar field tunnelling between degenerate minima. We explain why in one
space dimension, the Casimir and tunnelling effects are of the same order of
magnitude. In the limit of vanishing temperature, NEC violation can be
decomposed as the sum of two contributions: one from the Casimir effect and
one from tunnelling. We show that the latter is actually more important than
the former if $ma\gtrsim 1$, which is a new feature with relevance potentially
going beyond the present study.

Sec. III presents the derivation of the thermodynamical properties of the
dilatonic black hole in the presence of non-self-interacting matter fields. A
detailed explanation is given for the role of the environment regarding the
entropy of the system once backreaction of the matter fields on the background
metric is taken into account.

In Sec. IV we compare the two studies and we find that the relation $a\propto
T^{-1}$ provides a mapping between the two systems.

Tunnelling at finite temperature is described with a Euclidean metric whereas
the metric sign convention for the black hole description is $(-,+)$. Natural
units of $\hbar=c=1$ are used throughout.

##  II Finite size effects in 1+1 dimensional flat spacetime

We consider a massive self-interacting real scalar field theory defined on a
one-dimensional periodic interval $x\in[0,a]$ at a temperature $T\equiv
1/\beta$ with a corresponding Euclidean action

$I=\frac{1}{2}\int_{0}^{\beta}\text{d}\tau\int_{0}^{a}\text{d}x\left((\dot{\phi})^{2}+(\phi^{\prime})^{2}+\frac{m^{2}}{4}(\phi^{2}-1)^{2}\right)~{},$
(3)

where $m$ is the mass scale of the theory. Starting from this we study the
thermodynamics of the true vacuum of the effective theory.

###  II.1 Convexity from tunnelling

It is known that the one-Particle-Irreducible (1PI) effective potential is
necessarily convex if one takes several vacua into account Symanzik:1969ek ;
Coleman:1974jh ; Iliopoulos:1974ur ; Haymaker:1983xk ; Fujimoto:1982tc ;
Bender:1983nc ; Hindmarsh:1985nc ; Alexandre:2012ht ; Plascencia:2015pga ;
Millington:2019nkw . Focusing on two degenerate vacua at $\pm v$, the dynamics
of this feature relies on tunnelling between these vacua Alexandre:2022qxc ,
which restores symmetry and induces a true ground state corresponding to a
vanishing field expectation value $\left<\phi\right>=0$ (see Fig. 1).
Equivalently, by symmetry of the bare potential, the true vacuum corresponds
to a vanishing source $j=0$, and

$\left<\phi\right>\equiv-\frac{1}{Z[j]}\left.\frac{\delta Z[j]}{\delta
j}\right|_{j=0}=0~{},$ (4)

where $Z[j]$ is the partition function. The picture described here can be
interpreted as backreaction: the double-well bare potential allows quantum
fluctuations to tunnel between the minima, which in turns modifies the vacuum
structure by imposing convexity. The resulting symmetric vacuum corresponds
then to a non-perturbative process, which can be described by the semi-
classical approximation for $Z[j]$, as explained below.

Symmetry restoration is possible in a finite volume only, though, since an
infinite volume implies Spontaneous Symmetry Breaking instead. But a finite
volume requires a discrete set of momenta for quantum fluctuations and, as we
show in this article, in the limit of vanishing temperature, finite-size
effects can be decomposed as the sum of two contributions: (i) discretisation
of momentum for quantum fluctuations, that we will refer to as the Casimir
effect; (ii) symmetry restoration due to tunnelling, that we will refer to as
the tunnelling effect.

Allowing tunnelling between two degenerate vacua, the 1PI effective potential
induced by a dilute gas of instantons was calculated in Alexandre:2023iig ,
based on an expansion in $\left<\phi\right>$ to the quadratic order. This
result explicitly shows a convex effective potential, with a positive mass
term and a true vacuum at $\left<\phi\right>=0$. Focusing on this true vacuum,
the complete one-loop quantisation of the dilute gas with discrete momentum is
calculated in Ai:2024taz  for a 3-torus, providing the full picture of the
interplay between Casimir and tunnelling effects.

These calculations were done in 3+1 dimensions though, and we consider here
the 1+1 dimensional case, where both effects are comparable. Indeed, for a
finite length $a$ and a mass scale $m$, the instanton action is of the order
$ma$, leading for $ma\gg 1$ to a suppression of the tunnelling effect of the
order $\exp(-ma)$, similarly to what happens in the Casimir effect.

As we show in this section, one feature of convexity obtained from quantum
fluctuations is NEC violation in the true ground state. If we allowed the
system to evolve freely, this violation would imply an increase in the length
$a$, similarly to spacetime expansion due to tunnelling-induced NEC violation,
as described in Alexandre:2023pkk ; Alexandre:2023bih  (see for example
Rubakov:2014jja ; Easson:2024fzn  for reviews of NEC violation in the context
of Cosmology). In the present work, we do not take into account spacetime
dynamics though, and we stick to static QFT. This implies that some external
system fixes the length $a$, which requires some energy. As a consequence,
although the confined scalar field violates the NEC, the Averaged NEC is not
violated, which can be seen by integrating the NEC along a null geodesic going
through the confining walls Sopova:2002cs ; Graham:2002yr ; Graham:2005cq .

The main part of this section focuses on the zero-temperature case, since
thermal effects tend to restore the NEC. Nevertheless we start the
calculations at finite temperature $\beta^{-1}$, and consider then the limit
$\beta\to\infty$.

Figure 1: The classical potential (solid blue) and effective potential (dashed
orange) in a finite spatial volume. The effective potential is necessarily
convex due to tunneling between the two degenerate vacua, restoring symmetry.

###  II.2 Tunneling and the Casimir effect

####  II.2.1 Semi-classical approximation and true vacuum

Defining the dimensionless variables

$t\equiv m\tau~{}~{}~{}~{}\mbox{and}~{}~{}~{}r\equiv mx~{},$ (5)

the action of Eq. (3) becomes

$I=\frac{1}{2}\int_{0}^{m\beta}\text{d}t\int_{0}^{ma}\text{d}r\left((\dot{\phi})^{2}+(\phi^{\prime})^{2}+\frac{1}{4}(\phi^{2}-1)^{2}\right)~{},$
(6)

where dots and primes now represent derivatives in $t$ and $r$ respectively.
One can see that this action depends on the two dimensionless parameters $ma$
and $m\beta$, and is invariant under the simultaneous rescaling

$a\to\lambda a~{}~{},~{}~{}\beta\to\lambda\beta~{}~{},~{}~{}m\to
m/\lambda~{},$ (7)

and the quantum theory should also respect this symmetry, as we confirm in
what follows. The equation of motion (EoM) with solutions $\phi_{i}$ is

$\ddot{\phi}_{i}+\frac{1}{2}\left(\phi_{i}-\phi_{i}^{3}\right)=0~{},$ (8)

where in this work we consider only Euclidean-time-dependent and homogeneous
configurations $\phi_{i}$, since vacuum bubbles forming from degenerate vacua
would have an infinite radius Coleman:1977py ; Callan:1977pt . There are
several solutions to the classical EoM (8), each to be studied in following
subsections.

For a vanishing source $j=0$ and in the semi-classical approximation, the
partition function $Z[0]\equiv Z$ can be approximated as a sum of path
integrals over regions in field space around the action minimising saddle
points $\varphi_{i}$ via the field decomposition $\varphi=\varphi_{i}+\psi$,
integrating over fluctuations $\psi$. This assumes that the fluctuations do
not overlap, and the one-loop approximation for the fluctuation factors leads
to

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\int\mathcal{D}[\phi]~{}\exp\Big(-I[\phi]\Big{missing})$
$\displaystyle\simeq$
$\displaystyle\sum_{i}\int\mathcal{D}[\psi]~{}\exp\Big(-I[\varphi_{i}+\psi]\Big{missing})$
$\displaystyle=$
$\displaystyle\sum_{i}\Big{(}\mbox{det}(\delta^{2}I[\phi_{i}])\Big{)}^{-1/2}\exp\Big(-I[\phi_{i}]\Big{missing})$
$\displaystyle\equiv$
$\displaystyle\sum_{i}\exp\Big(-W[\phi_{i}]\Big{missing})~{},$

where the individual connected graph generating functionals are

$W[\phi_{i}]\equiv
I[\phi_{i}]+\frac{1}{2}\Tr\left\\{\ln\left(\delta^{2}I[\phi_{i}]\right)\right\\}~{}.$
(10)

####  II.2.2 Static saddle points

There are two static saddle points in the present model, $\phi_{s}=\pm 1$. The
corresponding individual connected graph generating functionals (10) can be
evaluated using known methods developed for the study of the thermal Casimir
effect on a 1D periodic interval Bordag . The steps are outlined in Appendix A
and lead to

$W_{\text{stat}}(a,\beta)\equiv
W[\phi_{s}]=a\beta\Lambda^{2}-\frac{m^{2}a\beta}{\pi}\int_{1}^{\infty}\text{d}u\frac{\sqrt{u^{2}-1}}{e^{mau}-1}+\sum_{n\in\mathbb{Z}}\ln\left(1-\mathrm{e}^{-\beta\omega_{n}}\right)~{},$
(11)

where $\Lambda^{2}$ is an ultraviolet cutoff, corresponding to the vacuum
energy of unbounded space, and the quantised frequencies/wave vectors are

$\omega_{n}=\sqrt{m^{2}+k_{n}^{2}}\quad,\quad k_{n}=\frac{2\pi n}{a}~{}.$ (12)

We note here that quantum corrections indeed depend on $a,\beta,m$ through the
products $ma$ and $m\beta$ only.

####  II.2.3 Time-dependent saddle points

The fundamental time-dependent saddle point is the (anti-)instanton relating
the two vacua of the bare potential

$\phi_{\text{inst}}(t)=(\pm)\tanh\left(\frac{t-\text{$t_{1}$}}{2}\right)~{},$
(13)

where $t_{1}$ is the time of the jump. At a finite temperature, field
configurations are periodic in Euclidean time and hence instantons and anti-
instantons can only exist in pairs. Such field configurations are well
approximated by a product of individual (anti-)instanton configurations

$\phi_{\text{n-pair}}(\tau)\simeq\prod_{j=1}^{2n}(-1)^{j}\tanh\left(\frac{t-t_{j}}{2}\right)~{},$
(14)

provided that the jumps at $t_{i}$ and $t_{j}$ are sufficiently distant
$(|t_{i}-t_{j}|\gg 1)$. The factor $-1$ ensures that an instanton is always
followed by an anti-instanton and the product is taken to $2n$ to enforce
periodicity in Euclidean time. In the limit of small temperature, $m\beta\gg
1$, a large amount of instanton/anti-instanton pairs is allowed and we assume
in what follows the instanton dilute gas approximation Kleinert:2004ev , where
the width of each jump is negligible compared to $m\beta$. Also,
(anti-)instantons are far enough from each other for them to keep their shape,
which for $n$ pairs leads to the total action

$I_{\text{n-pairs}}\simeq 2nI_{\text{inst}}~{},$ (15)

where the action for one (anti-)instanton is

$I_{\text{inst}}\equiv I[\phi_{\text{inst}}]=\frac{2ma}{3}~{}.$ (16)

The fluctuation factor for $n$ instanton/anti-instanton pairs can then be
approximated by the product of fluctuation factors for each static saddle
point evaluated over half the total Euclidean time interval $\beta/2$, times
the fluctuation factors for each instanton jump. The corresponding connected
graph generating functional is then

$W_{\text{n-pairs}}\equiv W[\phi_{\text{n-pair}}]\simeq
2W_{\text{stat}}(a,\beta/2)+2nW_{\text{jump}}\,,$ (17)

where we know from tunnelling in Quantum Mechanics Kleinert:2004ev  that

$W_{\text{jump}}\equiv
I_{\text{inst}}-\frac{1}{2}\ln(\frac{6I_{\text{inst}}}{\pi})~{}.$ (18)

We note that the expression for $W_{\text{jump}}$ takes into account time-
dependent quantum fluctuations over the instantons, and it neglects the space-
dependence of these fluctuations. However, it was shown in Ai:2024taz  that
the main contribution of the instanton jump comes from the zero-modes,
validating the approximation made here.

####  II.2.4 Partition function

Assuming the semi-classical approximation and a dilute gas of instantons/anti-
instantons, the partition function can be expressed as a sum over all the
possible $n$-pair configurations

$Z\simeq
2\exp\big(-W_{\text{stat}}(a,\beta)\big{missing})+2\sum_{n=1}^{\infty}\left(\prod_{i=1}^{2n}\int_{t_{i-1}}^{m\beta}\mathrm{d}t_{i}\right)\exp(-2W_{\text{stat}}(a,\beta/2)-2nW_{\text{jump}})~{}.$
(19)

The first term in the right-hand side corresponds to the two static saddle
points. In the second term the product of integrals accounts for the
invariance of the total action $I_{\text{n-pairs}}$ under the translations of
each successive instanton jump over the remaining dimensionless Euclidean time
interval $t\in[t_{i},m\beta]$, defining $t_{0}\equiv 0$ since the first
instanton can exist over the whole interval. This invariance under translation
of the jumps corresponds to the zero modes of the fluctuation factors for each
$n-$pair configuration. The factor 2 takes into account the instanton
configurations starting and ending at either $+1$ or $-1$. Using the known
result Coleman:1977py

$\prod_{i=1}^{2n}\int_{t_{i-1}}^{m\beta}\mathrm{d}t_{i}=\frac{(m\beta)^{2n}}{(2n)!}~{},$
(20)

the partition function (19) can be expressed as

$Z\simeq
2\exp\big(-W_{\text{stat}}(a,\beta)\big{missing})+2\exp\big(-2W_{\text{stat}}(a,\beta/2)\big{missing})\sum_{n=1}^{\infty}\frac{\bar{N}^{2n}}{(2n)!}~{},$
(21)

where

$\bar{N}\equiv
m\beta\sqrt{\frac{6I_{\text{inst}}}{\pi}}e^{-I_{\text{inst}}}=2m\beta\sqrt{\frac{ma}{\pi}}~{}\exp\left(-\frac{2ma}{3}\right)~{},$
(22)

with $\bar{N}/2$ corresponding to the average number of instanton/anti-
instanton pairs over the whole Euclidean time $\beta$ Alexandre:2022qxc . In
the small temperature limit $(m\beta\gg 1)$ the last term in $W_{\text{stat}}$
(11) can be neglected, such that it can be taken as linear in $\beta$

$W_{\text{stat}}(a,\beta)\simeq 2W_{\text{stat}}(a,\beta/2)~{},$ (23)

and the partition function (21) becomes 333The approximation (23) is applied
to the static saddle point contribution, and not to the instanton
contribution, since it is sub-dominant at low temperatures.

$Z\simeq
2\exp\big(-2W_{\text{stat}}(a,\beta/2)\big{missing})\sum_{n=0}^{\infty}\frac{\bar{N}^{2n}}{(2n)!}=2\exp\big(-2W_{\text{stat}}(a,\beta/2)\big{missing})\cosh(\bar{N})~{}.$
(24)

Finally, in the limit $m\beta\gg 1$ for finite $ma$ (such that $\bar{N}\gg 1$)
the total free energy is

$F_{\text{true}}\equiv-T\ln(Z)\simeq
2TW_{\text{stat}}(a,\beta/2)-T\bar{N}~{},$ (25)

and corresponds to the sum of the usual free-field Casimir contribution
$2TW_{\text{stat}}(a,\beta/2)$ and the tunneling contribution $-T\bar{N}$.

###  II.3 Null Energy Condition

We show here that the true ground state of the system we consider violates the
NEC, as a consequence of the true vacuum energy not being extensive: the free
energy (25) is not simply proportional to $a$.

We assume here that the dilute instanton gas described by the partition
function (24) may be treated as a perfect fluid, such that the resulting null
energy condition reduces to the simpler form (2). The thermodynamic energy
density and pressure are then defined as

$\displaystyle\rho$ $\displaystyle\equiv$
$\displaystyle\frac{1}{a}\left(F_{\text{true}}+\beta\frac{\partial
F_{\text{true}}}{\partial\beta}\right),$ (26) $\displaystyle p$
$\displaystyle\equiv$ $\displaystyle-\frac{\partial F_{\text{true}}}{\partial
a}\,.$ (27)

Here we show that the NEC is violated by the finite volume effects we
consider.

The sum $\rho+p$ may be evaluated from the free energy (25) and satisfies

$\displaystyle\frac{\rho+p}{m^{2}}$ $\displaystyle=$
$\displaystyle-\frac{ma}{\pi}\int_{1}^{\infty}du\frac{u\mathrm{e}^{mau}\sqrt{u^{2}-1}}{\left(\mathrm{e}^{mau}-1\right)^{2}}-\frac{4ma+3}{3\sqrt{\pi
ma}}\exp\left(-\frac{2ma}{3}\right)$
$\displaystyle+\sum_{n\in\mathbb{Z}}\frac{m^{2}a^{2}+8n^{2}\pi^{2}}{m^{2}a^{3}\omega_{n}}\left(\mathrm{e}^{\beta\omega_{n}/2}-1\right)^{-1}~{},$

where we can identify the following terms:

  1. (i)

the first term (integral over $t$) corresponds to the known Casimir effect,
obtained for a free field, with a negative contribution;

  2. (ii)

the second term corresponds to tunnelling arising from degenerate vacua, with
a negative contribution;

  3. (iii)

the third term (sum over Matsubara modes) corresponds to finite temperature
effects providing a positive contribution, and becomes the usual contribution
from black body radiation $(\propto T^{2})$ in the massless and infinite
length limit.

The expression (II.3) is plotted in Fig. 2 as a function of inverse
dimensionless spatial length $1/ma$ with finite temperature corrections.

Figure 2: Numerical plot of $(\rho+p)/m^{2}$ (II.3) for inverse dimensionless
length $1/ma$ with finite temperature corrections. At zero temperature, NEC
violation increases as the spatial interval reduces, corresponding to an
increased tunneling rate, and approaches zero in the limit of infinite spatial
length where tunneling is completely suppressed. For a given length scale,
finite temperature effects increase $\rho+p$ and can lead to NEC satisfaction
at large length scales.

Thermal effects decrease exponentially for small temperatures: for $m\beta\gg
1$ we have

$\sum_{n\in\mathbb{Z}}\frac{m^{2}a^{2}+8n^{2}\pi^{2}}{m^{2}a^{3}\omega_{n}}\left(\mathrm{e}^{\beta\omega_{n}/2}-1\right)^{-1}\simeq\frac{\mathrm{e}^{-m\beta/2}}{am}~{},$
(29)

and in what follows we take the limit $\beta\to\infty$, in order to focus on
NEC violating finite-length effects. It is then interesting to look at two
asymptotic regimes for the length $a$:

  * –

$ma\gg 1$ In this case we have

$\frac{\rho+p}{m^{2}}\approx-\frac{\mathrm{e}^{-am}}{\sqrt{2\pi
am}}-\frac{4}{3}\sqrt{\frac{am}{\pi}}~{}\mathrm{e}^{-2ma/3}~{},$ (30)

and we can see that tunnelling effects are more important than the free-field
Casimir effect;

  * –

$ma\lesssim 1$ In this case we have

$\frac{\rho+p}{m^{2}}\approx-\frac{\pi}{3(am)^{2}}~{},$ (31)

where tunnelling is negligible and the result is identical to the one obtained
for a massless free field Bordag .

###  II.4 Entropy

At zero temperature the classical thermal entropy vanishes, which can be seen
with

${\cal S}_{\text{classical}}=-\lim_{T\to 0}\frac{\partial
F_{\text{true}}}{\partial T}=0~{}.$ (32)

There is a quantum contribution left though, which can be interpreted as the
entropy for the dilute gas of instantons/anti-instantons which relate the two
vacua. Taking into account the instanton fluctuation factors described above,
the probability of having $n$ instanton/anti-instanton pairs may be read off
from the partition function $Z$

$p_{n}=\frac{2}{Z}\frac{(\bar{N})^{2n}}{(2n)!}\mathrm{e}^{-2W_{\text{stat}}(\beta/2)}=\frac{1}{\cosh(\bar{N})}\frac{(\bar{N})^{2n}}{(2n)!}~{},$
(33)

where $Z$, $\bar{N}$ and $W_{\text{stat}}$ are given by Eqs. (24), (22) and
(11) respectively. The entropy of the dilute gas should be extensive and thus
proportional to its number of degrees of freedom (although it is not
proportional to the length $a$). The entropy for the systems of instantons and
anti-instantons ${\cal S}_{\text{tun}}$ is then twice the entropy for the
system of pairs, which is given by the usual sum over probabilities

${\cal S}_{\text{tun}}=-2\sum_{n=0}^{\infty}p_{n}\ln(p_{n})~{}.$ (34)

In the limit $\bar{N}\gg 1$, we find numerically that the entropy (34) is
asymptotically equivalent to

$\displaystyle{\cal S}_{\text{tun}}$ $\displaystyle\approx$
$\displaystyle\ln(\bar{N})$ $\displaystyle=$
$\displaystyle\ln(m\beta)-I_{\text{inst}}+\frac{1}{2}\ln\left(\frac{6}{\pi}I_{\text{inst}}\right)~{},$

where $I_{\text{inst}}$ is the instanton action as given in (16), and the
result matches the usual microcanonical entropy for $\bar{N}$ microstates. As
expected, one can also check that $\cal{S}_{\text{tun}}$ vanishes in the limit
$a\to\infty$ (where $\bar{N}\to 0$, even for zero temperature), since
tunnelling is then completely suppressed. This behaviour is in correspondence
with an “effective third law of thermodynamics”, where $1/a$ plays the role of
a temperature. We will come back to this analogy later in this article.

For a finite length $a$, the entropy is non-zero with a logarithmic divergence
in the zero temperature limit. Such logarithmic divergences in the zero
temperature entropy of a quantum system are not new, such as the massless
Casimir effect Cho:2022ngf  and it was argued in Brevik:2002gh  that such
divergences should be removed.

The isothermal compressibility, defined as

$K\equiv-\frac{1}{a}\frac{\partial a}{\partial
p}\equiv\frac{1}{a}\left(\frac{\partial^{2}F_{\text{true}}}{\partial
a^{2}}\right)^{-1}~{},$ (36)

is negative for all $a$, which is usually interpreted as a sign of
instability. One may think that this instability is similar to the one
obtained for a Van der Waals fluid experiencing an isothermal compression. If
one assumes homogeneity of the Van der Waals fluid in a volume $V$, the bulk
modulus $-V\partial p/\partial V$ is negative in a given range of volumes,
which is not physical. What happens is that the fluid separates into two
phases, liquid and vapour, leading to the Maxwell construction, which
corresponds to a constant-pressure plateau. The position of this plateau is
determined by identifying the chemical potentials in each phase. Also, a
constant pressure leads to a vanishing compressibility, and the true free
energy is convex, as expected for the Legendre transform of the internal
energy. This constant-pressure plateau allows random regions of one vacuum or
the other, in a proportion given by the volume, which varies between values
corresponding to the first drop of liquid and the last bubble of vapour.

In our case, the system remains homogeneous though: the effective potential
does not feature any plateau, but it has a unique minimum at
$\left<\phi\right>=0$. An intuitive description is provided by weakly-
interacting spins on a lattice, each with a random direction and a vanishing
average value444This is different from the high-temperature limit, where
thermal fluctuations dominate over spin interactions and lead to a random spin
distribution. The vanishing average spin discussed here happens at zero
temperature, and is due to tunnelling instead. A flat effective potential
would be obtained in the limit $ma\to\infty$, where the tunnelling rate
exponentially vanishes though, in which case one would have to wait an
infinite amount of time for the true vacuum to settle. As mentioned in Sec.
II.1, in our situation the instability indicated by the negative
compressibility would correspond to a spacetime expansion, if no environment
was present to fix the length $a$.

##  III Black holes in 1+1 dilaton gravity

The study of Hawking radiation including its backreaction on the spacetime
geometry is an extremely difficult problem in $3+1$ dimensions. Motivated by
dimensional reduction, it is possible to simplify the problem by studying the
$1+1$ dimensional case. In particular, we focus on the classical Callan,
Giddings, Harvey and Strominger (CGHS) two-dimensional dilatonic black hole
model 1992_Callan ; 1994_Fiola . For the semi-classical description of the
theory including backreaction, we consider the standard Polyakov term which
represents the leading order quantum fluctuations in a $1/N$ expansion where
$N$ is the number of matter fields. To find analytical solutions to the
semiclassical theory it is necessary to introduce suitable counterterms in the
action. To consider these counterterms, we introduce the one-parameter family
of models that ranges between the Russo, Susskind and Thorlacius (RST) model
1992_Russo  and the Bose, Parker, Peleg (BPP) model 1995_Bose , following the
parameterization of the action presented in Cruz:1995zt . We focus our study
on the BPP model since it results in simpler expressions for the metric and
the dilaton.

After describing the solutions of the semi-classical theory, this section
delves into the implications for the stress-energy tensor and the entropy of
two-dimensional black holes.

###  III.1 Introduction to dilaton gravity

In two dimensions, the Einstein-Hilbert action is just the Euler
characteristic of the manifold (accordingly, $G_{\mu\nu}$ vanishes
identically), and $1+1$ dimensional gravity is trivial. Since we want to
capture aspects of the $3+1$ dimensional theory within a $1+1$ dimensional
description, we use the dilaton field 1984_Jackiw ; 2002_Grumiller  which
emerges from the compactification of higher dimensions. Here, we derive the
dilaton from the dimensional reduction of spherically symmetric gravity in
$3+1$ dimensions.

We consider the 3+1 Einstein-Hilbert action

$I^{(4)}_{EH}=\frac{1}{16\pi
G^{(4)}}\int\text{d}^{4}x\sqrt{-g^{(4)}}R^{(4)}\,,$ (37)

where (4) indicates the spacetime dimension and $G^{(4)}$ is Newton’s
constant. We consider the spherically symmetric ansatz

$\text{d}s_{(4)}^{2}=g_{ab}\text{d}x^{a}\text{d}x^{b}+\frac{e^{-2\phi(x^{a})}}{\lambda^{2}}(\text{d}\theta^{2}+\sin^{2}\theta\text{d}\varphi^{2})\,,$
(38)

where the radius $r$ of the 2-sphere has been parametrized via a dilaton field
$\phi(x^{a})$, $r=\lambda^{-1}e^{-\phi}$. The parameter $\lambda$ is
dimensionful and is introduced to get a dimensionless dilaton. Using this
ansatz, we can write $R^{(4)}$ in terms of $R\equiv R^{(2)}$ and the four
dimensional volume element in in terms of the two-dimensional volume element
times the angular terms fabbri-navarro ; 2022_Djordjevic

$\displaystyle R^{(4)}$ $\displaystyle=$ $\displaystyle
R+2(\nabla\phi)^{2}+2\lambda^{2}e^{2\phi}-2e^{2\phi}\Box e^{-2\phi}\,,$
$\displaystyle\text{d}^{4}x\sqrt{-g^{(4)}}$ $\displaystyle=$
$\displaystyle\text{d}^{2}x\,\text{d}\theta\,\text{d}\varphi\sqrt{-g}\frac{e^{-2\phi}}{\lambda^{2}}\sin^{2}\theta\,.$
(39)

Integrating out the angular part, the resulting dilaton action is fabbri-
navarro ; 2022_Djordjevic

$I_{D}=\frac{1}{4\pi\lambda^{2}G^{(4)}}\int\text{d}^{2}x\sqrt{-g}\left(e^{-2\phi}\left(R+2(\nabla\phi)^{2}\right)+2\lambda^{2}\right)~{},$
(40)

where we see that $\lambda$ plays the role of a cosmological constant in the
reduced theory. We define a two dimensional Newton’s constant
$G^{(2)}=\lambda^{2}G^{(4)}$ and work in units where
$G^{(4)}=\frac{1}{2\lambda^{2}}$.

To simplify the theory in such a way that it is possible to find an exact
analytical solution, we work with the action555$I_{\phi}$ can be exactly
derived by integrating out the angular part of near-extreme, magnetically
charged black holes in four-dimensional dilaton gravity 1992_Giddings ;
1992_Callan .

$I_{\phi}=\frac{1}{2{\pi}}\int\text{d}^{2}x\sqrt{-g}e^{-2\phi}\left(R+4(\nabla\phi)^{2}+4\lambda^{2}\right)\,,$
(41)

where we have modified the potential term of the dilation and the coefficient
of the kinetic term as compared with (40). Despite the changes, this dilaton
theory still has black holes and Hawking radiation 1992_Strominger ;
1992_Callan ; 1994_Fiola . We work in conformal gauge,

$\text{d}s^{2}=-e^{2\eta\,}\text{d}x^{+}\text{d}x^{-}\,,$ (42)

with null coordinates $x^{\pm}$. The EoM resulting from the variation of the
action with respect to $\eta$ and $\phi$ can be conveniently written in terms
of $2(\eta-\phi)$ and $e^{-2\phi}$, namely

$\displaystyle\partial_{+}\partial_{-}e^{-2\phi}+\lambda^{2}e^{2(\eta-\phi)}$
$\displaystyle=$ $\displaystyle 0\,,$ (43) $\displaystyle
2e^{-2\phi}\partial_{+}\partial_{-}(\eta-\phi)+\partial_{+}\partial_{-}e^{-2\phi}+\lambda^{2}e^{2(\eta-\phi)}$
$\displaystyle=$ $\displaystyle 0\,.$ (44)

Additionally, we derive the following constraints from the variation of the
action with respect to the $(\pm,\pm)$ components of the metric $g_{\mu\nu}$

$\partial_{\pm}^{2}e^{-2\phi}-2\partial_{\pm}(\eta-\phi)\partial_{\pm}e^{-2\phi}=0\,.$
(45)

Combining equations (44) and (43) we get the free field equation

$2\partial_{+}\partial_{-}(\eta-\phi)=0\,,$ (46)

which has solutions of the form $2(\eta-\phi)=h(x^{+})+s(x^{-})$. In the
Kruskal gauge, the remaining freedom is fixed by making $h(x^{+})=s(x^{-})=0$,
i.e. $\eta=\phi$. This model has black hole solutions which, in the Kruskal
gauge, are (see 1992_Callan ; fabbri-navarro  for a detailed discussion)

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{dx^{+}dx^{-}}{(M/\lambda-\lambda^{2}x^{+}x^{-})}\,,$ (47)
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\phi=-\frac{1}{2}\ln(\frac{M}{\lambda}-\lambda^{2}x^{+}x^{-})\,,$
(48)

where $M$ is an integration constant that corresponds to the ADM mass of the
black hole 1992_Callan . This metric has a curvature singularity at
$\lambda^{2}x^{+}x^{-}=M/\lambda$ and horizons at $\lambda^{2}x^{+}x^{-}=0$,
see Fig. 3. The surface gravity can be easily computed, and reads
$\kappa=\lambda$. Therefore, the black hole temperature is

$T=\frac{\lambda}{2\pi}\,.$ (49)

In contrast to the black hole temperature in four dimensions, the two-
dimensional black hole temperature is independent of the mass $M$. We use
these results to evaluate the NEC and the entropy.

IIIIIIIV$i^{+}$$i^{-}$$i^{0}$$\cal{J}^{+}$$\cal{J}^{-}$$i^{+}$$i^{-}$$i^{0}$$x^{-}=0$$\cal{J}^{+}$$\cal{J}^{-}$$x^{+}=0$
Figure 3: Penrose diagram for a static two-dimensional dilatonic black hole.

###  III.2 Adding quantum matter: static black holes

Now we add $N$ massless scalar fields $f_{i}$ and we have the total action

$I_{0}=I_{\phi}+I_{f}=\frac{1}{2\pi}\int\text{d}^{2}x\sqrt{-g}\left[e^{-2\phi}\left(R+4(\nabla\phi)^{2}+4\lambda^{2}\right)-\frac{1}{2}\sum_{i=1}^{N}(\nabla
f_{i})^{2}\right]\,.$ (50)

This action corresponds to the CGHS model. We continue the analysis in
conformal gauge (42). In these coordinates, the classical stress-energy tensor
for the fields $f_{i}$ is

$T_{\pm\pm}=\frac{1}{2}\sum_{i=0}^{N}(\partial_{\pm}f_{i})^{2}\,.$ (51)

The next step is to quantise the theory. We want to focus on static solutions,
$\langle f_{i}\rangle=0$, and look at the one-loop quantum corrections to the
stress-energy tensor in different vacuum states. The quantum corrections of
the different fields that we have seen in the CGHS model contribute to the
semiclassical theory. In order to make the analysis feasible, we assume that
the number of matter fields $N$ is very large and calculate the effective
action at leading order in an expansion in $1/N$. In this limit, the quantum
fluctuations of the dilaton and the metric can be ignored and we only have to
consider the one-loop correction of the matter fields to the stress-energy
tensor 1992_Harvey ; 1994_Fiola ; Mikovic:1997mz . This one-loop correction to
$T_{\mu\nu}$ due to the $N$ massless scalar fields can be evaluated using the
trace anomaly, which relates the expectation value of the stress-energy tensor
and the Ricci scalar 1977_Christensen

$\langle{T}\rangle=\frac{N}{24}R\,.$ (52)

In conformal gauge, the trace anomaly leads to

$\langle{T}_{+-}\rangle=-\frac{N}{12}\partial_{+}\partial_{-}\eta\,.$ (53)

In addition, we can use $\langle{T}_{+-}\rangle$ in (53) together with the
conservation of the stress-energy tensor to determine
$\langle{T}_{\pm\pm}\rangle$

$\langle{T}_{\pm\pm}\rangle=-\frac{N}{12}\,\left(\partial_{\pm}\eta\partial_{\pm}\eta-\partial^{2}_{\pm}\eta+t_{\pm}\right)\,,$
(54)

where $t_{\pm}$ is fixed by boundary conditions (vacuum choice). We will
analyse two vacuum choices: the Hartle-Hawking vacuum, which describes thermal
equilibrium at infinity and is given by $t_{\pm}=0$, and the Boulware vacuum,
which describes empty space at infinity and is given by the boundary
conditions $t_{\pm}=-\frac{1}{4(x^{\pm})^{2}}$.

Alternatively, the expectation value of the stress-energy tensor can be
obtained by functional differentiation of an effective action, the Polyakov
action

$\langle T_{\mu\nu}\rangle=-\frac{2\pi}{\sqrt{-g}}\frac{\delta I_{P}}{\delta
g^{\mu\nu}}\,,$ (55)

where

$I_{P}=-\frac{N}{96\pi}\int\textrm{d}^{2}x\sqrt{-g(x)}\int\textrm{d}^{2}y\sqrt{-g(y)}\,R(x)G(x,y)R(y)\,.$
(56)

$G(x,y)$ is the Green’s function for the differential operator $\Box_{g}$.
$I_{P}$ incorporates the backreaction of the quantum fluctuations of the
matter fields on the metric. By writing $I_{P}$ in the conformal gauge, we can
derive immediately Eqs. (53) and (54) from (55). We will use these expressions
later on in the evaluation of the NEC.

It is convenient to convert the non-local Polyakov action into a local one by
introducing an auxiliary scalar field $\varphi$ constrained to obey the
equation $\Box_{g}\varphi=R$ (see for example Eq. (5.56) in fabbri-navarro ).
By doing this, the local action is

$I_{P}=-\frac{N}{96\pi}\int\textrm{d}^{2}x\sqrt{-g(x)}\left(\varphi\Box_{g}\varphi+2\varphi
R\right)\,.$ (57)

In conformal gauge, the equation of motion for $\varphi$ has the following
solution

$\varphi(x^{\pm})=-2\eta(x^{\pm})+2\left(\varphi_{+}(x^{+})+\varphi_{-}(x^{-})\right)\,,$
(58)

where $\varphi_{+}(x^{+})$ and $\varphi_{-}(x^{-})$ are solutions of

$-(\partial_{\pm}\varphi_{\pm})^{2}+\partial_{\pm}^{2}\varphi_{\pm}=t_{\pm}(x^{\pm})\,.$
(59)

We are also interested in the entropy of the system. As we showed in (47),
this theory has black hole solutions. Therefore, the total entropy of the
system consists of two terms: the geometric black hole entropy, which is the
$1+1$ dimensional equivalent of the Bekenstein-Hawking entropy, and the von
Neuman entropy, associated to the quantum fields outside the horizon, usually
called fine-grained entropy 1994_Fiola ; 1994_Myers ; 2023_Hirano . For
general diffeomorphism invariant theories, it is possible to evaluate both of
these entropies using the method developed in Ref. 1994_Iyer  and
particularised to the context of two-dimensional gravity in Refs. 1994_Myers ;
2022_Yu ; 2023_Hirano . In this way, both entropies can be evaluated in a
geometrical way by calculating the derivatives of the Lagrangian associated
with the different contributions to the action with respect to the curvature

$\displaystyle\mathcal{S}_{\phi}$ $\displaystyle=$
$\displaystyle\left.\frac{4\pi}{\sqrt{-g}}\frac{\partial\mathcal{L}_{\phi}}{\partial
R}\,\right|_{H}=2e^{-2\phi}\,\Big{|}_{H}$ (60) $\displaystyle\mathcal{S}_{P}$
$\displaystyle=$
$\displaystyle\frac{4\pi}{\sqrt{-g}}\frac{\partial\mathcal{L}_{P}}{\partial
R}\Bigg{|}_{H}=-\frac{N}{12}\,\varphi\,\Big{|}_{H}\,,$ (61)

where $|_{H}$ means that these quantities should be evaluated at the horizon.
It can be shown that $\mathcal{S}_{\phi}$ and $\mathcal{S}_{P}$ are equivalent
to the black hole and the fine-grained entropy respectively 1994_Fiola ;
2021_Pedraza ; 2023_Hirano . Equivalently, the entropy can be evaluated using
the Euclidean path integral approach, as done in Ref. Hayward:1994dw  for the
RST model.

In what follows, we proceed to evaluate the NEC and the entropy. We start with
the simpler case without backreaction, and then we study how backreaction
modifies the results.

###  III.3 Without backreaction, the CGHS model

For the case without backreaction, the background spacetime is described by
(50), i.e. the CGHS model. Since we are interested in static black holes, we
focus on the case where $\langle f\rangle=0$. As we have seen, under these
conditions, the solution is an eternal black hole of mass $M$.

####  III.3.1 Null energy condition

The null energy condition

$\langle T_{\mu\nu}\rangle\ell^{\mu}\ell^{\nu}\geq 0\,,$ (62)

gives two equations that, in the conformal gauge, are proportional to the
diagonal components of the stress-energy tensor $\langle T_{\pm\pm}\rangle$ .
In what follows, we will focus on the exterior region, so we assume $x^{+}>0$
and $x^{-}<0$. For the Hartle-Hawking vacuum ($t_{\pm}=0$), using the
dilatonic metric (48) in Eq. (54), we find666The condition $\langle
T_{\mu\nu}\rangle\ell^{\mu}\ell^{\nu}\geq 0$ is defined up to a positive
overall factor. In this case, we find it useful to compute $(\lambda
x^{\pm})^{2}\langle T_{\pm\pm}\rangle$ instead of $\langle T_{\pm\pm}\rangle$,
since it is the quantity related to an asymptotic observer at infinity. This
quantity comes directly from the transformation law for tensors
$T_{vv}=(dx^{+}/dv)^{2}\,T_{++}$ (and similarly for the other components).

$NEC^{\pm}_{H}=(\lambda x^{\pm})^{2}\langle
H|T_{\pm\pm}|H\rangle=\frac{N\,\lambda^{2}}{48}\,(\lambda x^{\pm})^{2}(\lambda
x^{\mp})^{2}\,e^{4\eta}>0\,,$ (63)

while for the Boulware vacuum we obtain

$NEC^{\pm}_{B}=(\lambda x^{\pm})^{2}\langle
B|T_{\pm\pm}|B\rangle=-\frac{N\,\lambda^{2}}{48}\left(1-(\lambda
x^{\pm})^{2}(\lambda x^{\mp})^{2}\,e^{4\eta}\right)<0\,.$ (64)

We can easily check that the difference between the Hartle–Hawking and
Boulware stress energy tensors is just a thermal distribution of massless
particles at the Hawking temperature777See Ref. Visser:1996ix  for a similar
analysis in a Schwarzschild background.

$NEC_{H}^{\pm}-NEC_{B}^{\pm}=N\frac{\lambda^{2}}{48}=N\frac{\pi^{2}}{12}\,T^{2}\,,$
(65)

where $T=\lambda/2\pi$ is the black hole temperature as given in Eq. (49). The
extra $N$ factor appears because we are considering $N$ conformal fields. For
convenience, we can write the stress-energy tensor in the Hartle-Hawking
vacuum as

$NEC_{H}^{\pm}=\frac{N\lambda^{2}}{48}-\frac{N\lambda^{2}}{48}\left(1-(\lambda
x^{\pm})^{2}(\lambda x^{\mp})^{2}\,e^{4\eta}\right)>0\,.$ (66)

In this expression, we see that there is a negative contribution coming from
pure vacuum effects plus a positive contribution coming from thermal effects.
As a final note, we point out that the quantity $\left(1-(\lambda
x^{\pm})^{2}(\lambda x^{\mp})^{2}\,e^{4\eta}\right)$ vanishes as
$x^{\pm}\to\pm\infty$ and tends to 1 as $x^{\pm}\to 0$. It means that
$(\lambda x^{+})^{2}\langle H|T_{++}|H\rangle\to N\lambda^{2}/48$
asymptotically (constant thermal flux), and goes to zero at the horizon
(thermal bath in thermal equilibrium with the black hole, so the fluxes
cancel). We can remove the contribution from the thermal bath to compute the
NEC vacuum contribution of the black hole

$NEC_{\text{BH}}=(\lambda x^{\pm})^{2}\langle
H|T_{\pm\pm}|H\rangle-\left(\frac{N\,\lambda^{2}}{48}\right)=-\frac{N\,\lambda^{2}}{48}\left(1-(\lambda
x^{\pm})^{2}(\lambda x^{\mp})^{2}\,e^{4\eta}\right)<0\,,$ (67)

which results in a negative contribution to the NEC due to vacuum effects. On
the horizon

$NEC_{\text{BH}}=-\left(\frac{N\,{\pi^{2}}T^{2}}{12}\right)\,.$ (68)

In Fig. 4 we summarise the results for the NEC.

Figure 4:  We plot the (normalized) energy condition
$\frac{48}{N\lambda^{2}}$(NEC) as a function of $\lambda^{2}x^{+}\,x^{-}$ for
the BPP and classical models. The green curve corresponds to the thermal bath
(65), the orange curve corresponds to the contribution from vacuum effects
(64), and the total NEC (66) is represented in blue. The black hole mass is
$M/\lambda=2$.

####  III.3.2 Entropy and thermodynamics

Inserting the dilaton solution (48) into (60), the black hole entropy is

$\mathcal{S}_{BH}=\mathcal{S}_{\phi}=\frac{2M}{\lambda}=\frac{M}{\pi T}\,.$
(69)

From the dimensional reduction (38) relating the radius and the dilaton, we
can interpret this term as $1/4$ of the horizon area of the classical
$3+1$-dimensional black hole. Remember that in the dimensional reduction (38),
$r=\lambda^{-1}e^{-\phi}$ and
$\mathcal{S}_{\phi}=2e^{-2\phi_{H}}=\frac{2\lambda^{2}}{\pi}\,\frac{4\pi
r_{H}^{2}}{4}$ [see Eq. (60)].

To evaluate the entropy associated with the $N$ massless fields surrounding
the black hole we use (61). Although we are not considering backreaction in
this subsection, we can still evaluate $\mathcal{S}_{P}$ using the classical
solution (48) to obtain the fine-grained entropy of the matter fields. In the
Hartle-Hawking vacuum ($t_{\pm}(x^{\pm})=0$), the auxiliary field $\varphi$ is
[see Eqs. (58) and (59)]

$\varphi=-2\eta-\ln\left(-\lambda^{2}x^{+}x^{-}\right)+const\,.$ (70)

which means that there is a logarithmic divergence when evaluated at the
horizon. However, its contribution to the entropy can be easily understood in
terms of the entropy of a thermal bath. Indeed, if we rewrite the log term in
light-cone coordinates

$x^{+}=\lambda^{-1}e^{\lambda\sigma^{+}}\,,\quad\text{and}\quad
x^{-}=-\lambda^{-1}e^{-\lambda\sigma^{-}}\,,$ (71)

we obtain

$-\frac{N}{12}\,\ln(-\lambda^{2}x^{+}x^{-})\Big{|}_{H}=\frac{N}{12}\lambda(\sigma^{-}-\sigma^{+})\Big{|}_{H}=\frac{N}{12}\lambda(2L)=\frac{N\pi}{6}T(2L)=\mathcal{S}_{thermal}$
(72)

which is exactly the entropy of a thermal bath in an one dimensional box of
length $2L=(\sigma^{-}-\sigma^{+})|_{H}$, as seen by an asymptotic observer.
Since the length is infinite, ($\sigma^{-}-\sigma^{+}=-2x\to\infty$ at the
horizon), the entropy diverges. This result allows us to rewrite the entropy
of the Polyakov term as

$\mathcal{S}_{P}=\mathcal{S}_{quantum}+\mathcal{S}_{thermal}\,,$ (73)

where

$\mathcal{S}_{quantum}=\frac{N}{6}\eta|_{H}=-\frac{N}{12}\ln\left(\frac{M}{\lambda}\right)\,.$
(74)

The Polyakov entropy matches the fine-grained entropy given in Eq. (93) of
1994_Fiola  for an eternal black hole. This is easily done by identifying our
$\mathcal{S}_{thermal}$ with the
$\frac{N}{12}\ln\left(-x^{+}_{\text{max}}x^{-}_{\text{max}}/\delta^{2}\right)$
term. In Ref. 1994_Fiola , the $x_{\max}^{\pm}$ are infrared cut-offs for the
right and left moving modes and $\delta$ is a short distance cut-off
introduced to regularize the logarithmic ultraviolet divergence arising from
the entanglement of the short-wavelength field fluctuations at the edge of the
horizon 2021_Pedraza .

In what follows, we will subtract the thermal contribution to the entropy,
since we want to focus on vacuum effects. Now, the total entropy of the system
is (excluding $\mathcal{S}_{thermal}$)

$\mathcal{S}_{tot}=\mathcal{S}_{BH}+\mathcal{S}_{quantum}=\frac{2M}{\lambda}-\frac{N}{12}\ln\left(\frac{M}{\lambda}\right)\,,$
(75)

where we have used that, in the Kruskal gauge, $\phi=\eta$ is given by (48).

###  III.4 With backreaction, the BPP model

As we discussed, the backreaction on the metric of the quantum fluctuations of
the matter field can be considered by adding $I_{P}$ to the action $I_{\phi}$.
However, the EoM from $I_{\phi}+I_{P}$ can not be solved analytically. To
address this problem, we can add the following extra term to the action
Cruz:1995zt

$\displaystyle I_{extra}$ $\displaystyle=$
$\displaystyle\frac{N}{24\pi}\int\textrm{d}^{2}x\sqrt{-g}\left[(1-2b)(\nabla\phi)^{2}+(b-1)\phi
R\right]$ (76) $\displaystyle=$
$\displaystyle\frac{N}{24\pi}\int\textrm{d}^{2}x\left(-2(1-2b)\partial_{+}\phi\partial_{-}\phi+4(b-1)\phi\partial_{+}\partial_{-}\eta\right)\,.$

This results in a family of models characterized by the parameter $b$.
$I_{extra}$ being local modifies the local dynamics but not the global
properties. For $b=1/2$ we recover the RST model 1992_Russo  and for $b=0$ we
recover the BPP model 1995_Bose . By introducing the Liouville fields
Cruz:1995zt

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sqrt{\frac{N}{12}}\,b\phi+\sqrt{\frac{12}{N}}e^{-2\phi},$ (77)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\sqrt{\frac{N}{12}}\,\eta+\sqrt{\frac{N}{12}}(b-1)\phi+\sqrt{\frac{12}{N}}e^{-2\phi}\,,$
(78)

we can rewrite our family of models as a Liouville theory that can be solved
analytically. The EoM for the action $I_{\phi}+I_{P}+I_{extra}$ in terms of
the Liouville variables are

$\displaystyle\partial_{+}\partial_{-}\chi$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{\sqrt{\kappa}}e^{\frac{2}{\sqrt{\kappa}}(\chi-\Omega)},$
(79) $\displaystyle\partial_{+}\partial_{-}\Omega$ $\displaystyle=$
$\displaystyle-\frac{\lambda^{2}}{\sqrt{\kappa}}e^{\frac{2}{\sqrt{\kappa}}(\chi-\Omega)},$
(80)

which implies

$\partial_{+}\partial_{-}(\chi-\Omega)=0\,.$ (81)

The constraint equations become

$-\partial_{\pm}\chi\partial_{\pm}\chi+\sqrt{\frac{N}{12}}\partial_{\pm}^{2}\chi+\partial_{\pm}\Omega\partial_{\pm}\Omega-\frac{N}{12}t_{\pm}=0\,.$
(82)

In the Kruskal gauge $\Omega=\chi$ and for the Hartle-Hawking vacuum
($t_{\pm}=0$) we have the solution

$\sqrt{\frac{N}{12}}\,\Omega=\frac{M}{\lambda}-\lambda^{2}x^{+}x^{-}\,.$ (83)

From now on, we focus on the model $b=0$, i.e. the BPP model, since it results
in a simpler solution. Taking $b=0$ and (83) into the Liouville variables (77)
and (78), it is immediate to find

$\phi=\eta=-\frac{1}{2}\left(\frac{M}{\lambda}-\lambda^{2}x^{+}x^{-}\right)\,.$
(84)

These solutions are the same as the classical solutions (48), which indicates
that the metric and the dilaton do not have quantum corrections in the BPP
model.

####  III.4.1 Null energy condition

The results for the NEC in section III.3.1 don’t change when we consider the
BPP model in the Hartle-Hawking vacuum since the metric is not affected by the
quantum corrections. Fig. 4 summarizes the results for the NEC in the BPP
model and the classical results.

####  III.4.2 Entropy and thermodynamics

The evaluation of the entropy follows the discussion in section III.3.2. Since
we included the $I_{bpp}=I_{extra}(b=0)$ term in the action, we have an
additional contribution to the entropy

$\mathcal{S}_{bpp}=\frac{4\pi}{\sqrt{-g}}\frac{\partial\mathcal{L}_{bpp}}{\partial
R}\Bigg{|}_{H}=-\frac{N}{6}\eta\Bigg{|}_{H}=\frac{N}{12}\ln\left(\frac{M}{\lambda}\right)\,.$
(85)

The total entropy of the semi-classical system is (as before, we omit the
$\mathcal{S}_{thermal}$ contribution of $\mathcal{S}_{P}$)

$\mathcal{S}_{tot}=\mathcal{S}_{\phi}+\mathcal{S}_{bpp}+\mathcal{S}_{quantum}=2e^{-2\phi}\Big{|}_{H}=\frac{2M}{\lambda}\,,$
(86)

where we have used that, in the Kruskal gauge $\phi=\eta$, and the dilaton is
given by (48). Note that $\mathcal{S}_{bpp}$ cancels with
$\mathcal{S}_{quantum}$. We find that $\mathcal{S}_{tot}$ is equal to the
$\mathcal{S}_{BH}$ for the case without backreaction (69). It means that the
total entropy of the semi-classical system is exactly the entropy of a
dilatonic black hole of mass $M$ at a temperature $\lambda/2\pi$. This is not
surprising since the solution to the semi-classical equations in the Hartle-
Hawking vacuum state is precisely a black hole of mass $M$ at a temperature
$\lambda/2\pi$. However, we can still split the total entropy into two parts:
the entropy of the black hole, and the entropy of the matter sector

$\displaystyle\mathcal{S}_{BH}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{\phi}+\mathcal{S}_{bpp}=\frac{2M}{\lambda}+\frac{N}{12}\ln\left(\frac{M}{\lambda}\right)\,,\qquad$
(87) $\displaystyle\mathcal{S}_{quantum}$ $\displaystyle=$
$\displaystyle-\frac{N}{12}\ln\frac{M}{\lambda}\,.$ (88)

The matter entropy $\mathcal{S}_{quantum}$ is negative just because we removed
the log divergent contribution coming from the thermal bath. Let us note that
$\mathcal{S}_{BH}$ has a quantum correction as compared to the black hole
entropy without backreaction (69) coming from $\mathcal{S}_{bpp}$.

##  IV Comparison

In Secs. II and III we discussed the thermodynamics of finite volume effects
in QFT and dilatonic black holes respectively. These effects seem at first
glance unrelated. However, in both systems we have NEC violation and finite
quantum entropy. But the analogy runs even deeper; we observe that the effect
of the temperature $T$ in black holes is similar to the effect of the inverse
length scale $1/a$ in tunneling.

We begin the comparison with the NEC. For tunneling, the NEC was computed in
Sec. II.3. We will compare in the regime where $ma\lesssim 1$, where the NEC
becomes

$NEC_{\text{tun}}\approx-\frac{\pi}{3a^{2}}\,.$ (89)

It is interesting to note that in this regime the Casimir effect is dominant
over the tunneling effects.

Given the correspondence $a\sim 1/T$, the equivalent regime for the black hole
is the one where $T/M\gtrsim 1$. We observe that this is not necessarily a
high temperature regime, as the mass of the black hole is an independent
parameter. The result on the black hole horizon for any ratio $T/M$ in the BPP
model is computed in Sec. III

$NEC_{\text{BH}}=-\frac{N\,\pi^{2}T^{2}}{12}\,.$ (90)

First, we should comment on the left hand side of the expressions. For the
black hole, it corresponds to the renormalized stress-energy tensor contracted
with null vectors where the expression has been evaluated at the horizon. For
the tunnelling case, the system is homogeneous, so the stress energy tensor is
that of a perfect fluid, which implies that the NEC is proportional to the sum
of pressure and energy density.

Then we notice the clear equivalence of the dominant terms in the regime of
interest: the NEC violation is proportional to $1/a^{2}$ and $T^{2}$ for the
tunneling and the black hole respectively. Physically, this is partly a
manifestation of the effect that the NEC violation is due to finite volume
effects in one case and the non-zero temperature of the black hole in the
other.

The second comparison involves the entropy of the two systems. First, we
observe that in both systems there is a thermal term in the entropy that is
infinite. For the tunneling case, this is $\ln{m\beta}$ [see Eq. (II.4)] which
diverges in the limit of zero temperature. For the black hole case, it is $\pi
NT(2L)/6$ [see Eq. (72)] which diverges for infinite volume.

To proceed with the comparison without the infinite quantities, we first
compute the heat capacitance of the black hole entropy (87)

$C_{\text{BH}}\equiv T\frac{\partial{\mathcal{S}_{\text{BH}}}}{\partial
T}=-\frac{N}{12}-\frac{M}{\pi T}<0\,.$ (91)

We note that a negative $C_{BH}$ is an indication of instability.888An
alternative definition for the heat capacitance is $C_{BH}=\frac{\partial
M}{\partial T}$, which is infinite since the 1+1 BH temperature $\lambda$ is
independent of $M$. This is the result of the black hole moving towards
equilibrium with the thermal bath at temperature $T$. On the other hand, if we
evaluate the heat capacitance for the complete system including the thermal
bath, i.e. Eqs. (86) and (72), we obtain

$C_{tot}=T\frac{\partial(\mathcal{S}_{tot}+\mathcal{S}_{thermal})}{\partial
T}=\frac{\pi NTL}{3}-\frac{M}{\pi T}\,,$ (92)

which is positive due to the dominant contribution of the thermal bath.

Using the correspondence $a\sim 1/T$ we define a new capacitance for the
tunnelling case in terms of the entropy (II.4)

$C_{\text{tun}}\equiv a\frac{\partial\mathcal{S}_{\text{tun}}}{\partial
a}=\frac{1}{2}-\frac{2m}{3}a~{}\,.$ (93)

This quantity is also negative for $ma>3/4$. As explained in Sec. II, this
does not lead to a thermodynamical instability, and the system remains
homogeneous.

As in the case of the NEC, there is a clear equivalence between the systems
with a constant term and a negative term $\sim a$ for tunneling and $\sim 1/T$
for the black hole. In both cases, this is not an effect that can be explained
by simple dimensional analysis. The quantities $ma$ and $T/M$ are
dimensionless and thus any combination could appear in these expressions. A
comparison between the Casimir effect and black hole thermodynamics but using
isothermal compressibility instead of capacitance was done in Ref.
Widom:1998pn ; Sassaroli:1998ug .

##  V Conclusions

Motivated by the description of Hawking radiation in terms of NEC violation,
we mapped a few aspects of black hole thermodynamics with finite volume
effects in QFT arising from tunnelling in a confined space. Both descriptions
are done in $1+1$ dimensions, and feature similar behaviours in terms of
energy and entropy, when one identifies the inverse of the BH temperature with
the finite size of the confining space for a scalar field. The origin of the
mapping is the presence of boundaries in both systems, either in the form of a
horizon for the BH, or in the form of periodic boundary conditions in QFT.

We focused the comparison on the regime $ma\propto M/T\lesssim 1$, and not on
the regime $ma\propto M/T\gg 1$. In the latter case, the non-trivial effects
vanish exponentially with $a$ for tunnelling, whereas for the black hole they
vanish as a power law with $T$. In this example, the mapping we discussed does
not hold and a more thorough discussion is necessary to include all the
regimes in this study, which is left for a future work.

An essential point in our study is the presence of an environment which is
necessary to justify the static regimes we study. For tunnelling, this
environment fixes the spatial period, whereas for the black hole it plays the
role of a heat source and fixes the temperature. Removing this environment
could also be an avenue to explore, in which case the equilibrium assumption
is no longer valid; since the black hole evaporates and the confining space
for the scalar field is modified by energetics of NEC violation.

Another interesting connection point that was not discussed in this work, is
that Hawking radiation can be studied as tunneling of particles through the
black hole horizon Parikh2000 . Then both systems can be viewed as tunneling,
one on a flat background with a finite volume and one on a curved background
with an infinite volume. This connection could be explored further in future
work, especially in the $3+1$ dimensional case.

Given the above analogies, we hope to find a more formal description for this
mapping in light of the AdS/CFT correspondence, although both systems here
have the same dimensionality. Hopefully such a mapping could be extended to
3+1 dimensions, and might be relevant to either astrophysics or analogue
condensed matter systems.

###### Acknowledgements.

The authors would like to thank Dionysios Anninos, Jose Navarro-Salas, and
Andrew Svesko for useful discussions. The Science and Technology Facilities
Council supports the work of JA and DB (grant No. STFC-ST/X000753/1), the
Engineering and Physical Sciences Research Council supports the work of JA
(grant No. EP/V002821/1) and the Leverhulme Trust supports the work of JA and
SP (grant No. RPG-2021-299). DPS is supported by the EPSRC studentship grant
EP/W524475/1. For the purpose of Open Access, the authors have applied a CC BY
public copyright licence to any Author Accepted Manuscript version arising
from this submission.

##  Appendix A Casimir effect for static saddle points

This appendix is based on the book Bordag , and we consider periodic boundary
conditions in space. We first show the derivation for vanishing temperature
(limit $\beta\to\infty$), for which the ground state energy contains an
ultraviolet divergence. We then show how to include finite-temperature
effects, which do not introduce new divergences.

Evaluating the individual connected graph generating functional $W[\phi_{i}]$
(10) at the static saddle points $\phi_{s}=\pm 1$ for vanishing source $j=0$
yields

$W[\phi_{s}]\equiv
W_{\text{stat}}(a,\beta)=\frac{1}{2}\sum_{n\in\mathbb{Z}}\sum_{l\in\mathbb{Z}}\ln\left(\frac{\nu_{l}^{2}+\omega_{n}^{2}}{\nu_{l}^{2}}\right)~{},$
(94)

where $\omega_{n}=\sqrt{m^{2}+k_{n}^{2}}$, and

$\nu_{l}=\frac{2\pi l}{\beta}~{}~{},~{}~{}k_{n}=\frac{2\pi n}{a}~{}.$ (95)

The origin of energies is chosen in such a way as to recover the usual sum of
ground state energies of harmonic oscillators, at zero temperature.

###  A.1 Zero temperature

In the limit of zero temperature the summation over Matsubara modes becomes an
integral

$\lim_{\beta\to\infty}\big{\\{}W[\phi_{s}]\big{\\}}=\frac{\beta}{2}\sum_{n\in\mathbb{Z}}\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}\ln\left(\frac{\nu^{2}+\omega_{n}^{2}}{\nu^{2}}\right)=\frac{\beta}{2}\sum_{n\in\mathbb{Z}}\omega_{n}~{}.$
(96)

Using the Abel-Plana formula

$\sum_{n\in\mathbb{N}}F(n)\equiv-\frac{1}{2}F(0)+\int_{0}^{\infty}F(t)dt+i\int_{0}^{\infty}\frac{dt}{e^{2\pi
t}-1}\Big{[}F(it)-F(-it)\Big{]}~{},$ (97)

the sum over the frequencies can be expressed as

$\sum_{n\in\mathbb{Z}}\omega(n)=2a\Lambda^{2}+2i\int_{0}^{\infty}\frac{dt}{e^{2\pi
t}-1}\Big{[}\omega(it)-\omega(-it)\Big{]}~{},$ (98)

where the ultraviolet divergence is

$\Lambda^{2}\equiv\frac{m^{2}}{2\pi}\int_{0}^{\infty}\sqrt{t^{2}+1}~{}dt~{}.$
(99)

By considering the principle branch $z\in]-\infty,0]$ of $\ln(z)$, we have

$\omega(it)-\omega(-it)=\frac{4\pi i}{a}\sqrt{t^{2}-\mu^{2}}\theta(t-\mu)~{},$
(100)

where $\mu\equiv ma/2\pi$. The individual connected graph generating
functional for static saddle points therefore reads, in the limit of zero
temperature,

$\displaystyle\lim_{\beta\to\infty}\big{\\{}W[\phi_{s}]\big{\\}}$
$\displaystyle=$ $\displaystyle
a\beta\Lambda^{2}-\beta\frac{4\pi}{a}\int_{\mu}^{\infty}\frac{dt}{e^{2\pi
t}-1}\sqrt{t^{2}-\mu^{2}}$ (101) $\displaystyle=$ $\displaystyle\beta
a\Lambda^{2}-\beta\frac{m^{2}a}{\pi}\int_{1}^{\infty}\frac{du}{e^{mau}-1}\sqrt{u^{2}-1}~{}.$
(102)

###  A.2 Finite-temperature corrections

In order to calculate thermal corrections to the individual connected graph
generating functional for static saddle points, we employ the zeta-function
regularisation and write the expression (11) as

$W_{\text{stat}}(a,\beta,s)=-\frac{1}{2}\frac{\partial}{\partial
s}\left(\sum_{l\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}(\beta
a)^{-s}\left(\nu_{l}^{2}+\omega_{n}^{2}\right)^{-s}\right)~{},$ (103)

where the ultraviolet divergence is turned into a divergence in the limit
$s\to 0$. The above can be written in terms of a parametric integral

$W_{\text{stat}}(a,\beta,s)=-\frac{1}{2}\frac{\partial}{\partial
s}\left(\int_{0}^{\infty}\frac{dt}{t}\frac{t^{s}}{\Gamma(s)}\sum_{l\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}\mathrm{e}^{-t\beta
a\left(\nu_{l}^{2}+\omega_{n}^{2}\right)}\right)~{}.$ (104)

From the Poisson summation formula, one can derive the following identity

$\sum_{l\in\mathbb{Z}}\mathrm{e}^{-zl^{2}}=\sqrt{\frac{\pi}{z}}\sum_{l\in\mathbb{Z}}\mathrm{e}^{-\pi^{2}l^{2}/z}~{},$
(105)

which, when applied to the Matsubara sum in Eq. (104) with $z=\beta at(2\pi
T)^{2}$, leads to

$W_{\text{stat}}(a,\beta,s)=-\frac{\beta}{2}\frac{\partial}{\partial
s}\left(\sum_{l\in\mathbb{Z}}\int_{0}^{\infty}\frac{dt}{t}\frac{t^{s}}{\Gamma(s)\sqrt{4\pi\beta
at}}\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\frac{l^{2}\beta^{2}}{4\beta at}-\beta
at\omega_{n}^{2}}\right)~{}.$ (106)

The ultraviolet divergence is contained within the temperature-independent
integral for $l=0$. We thus make the following decomposition of Eq. (106)

$W_{\text{stat}}(a,\beta,s)=\lim_{\beta\to\infty}\big{\\{}W_{\text{stat}}(a,\beta,s)\big{\\}}+W_{\text{stat}}^{T}(a,\beta)~{},$
(107)

where the temperature independent part is calculated in the previous section,
and the temperature-dependent part is given by

$W_{\text{stat}}^{T}(a,\beta)\equiv-\frac{\beta}{\sqrt{4\pi\beta
a}}\sum_{l\in\mathbb{N}}\int_{0}^{\infty}\frac{dt}{t^{3/2}}\sum_{n\in\mathbb{Z}}\mathrm{e}^{-\frac{l^{2}\beta^{2}}{4\beta
at}-\beta at\omega_{n}^{2}}~{}.$ (108)

Note that the regulator has been removed from Eq. (108) using

$\underset{s\to 0}{\lim}\left[\frac{\partial}{\partial
s}\frac{f(s)}{\Gamma(s)}\right]=f(0)~{}.$ (109)

The integral and summation over $l$ in Eq. (108) can be then evaluated,
leading to

$W_{\text{stat}}^{T}(a,\beta)=\sum_{n\in\mathbb{Z}}\ln\left(1-\mathrm{e}^{-\beta\omega_{n}}\right)~{}.$
(110)

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