# Electrically charged spherical matter shells in higher dimensions: Entropy,
thermodynamic stability, and the black hole limit

Tiago V. Fernandes<EMAIL_ADDRESS>Centro de
Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior
Técnico - IST, Universidade de Lisboa \- UL,
Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal  José P. S. Lemos
<EMAIL_ADDRESS>Centro de Astrofísica e Gravitação - CENTRA, Departamento
de Física, Instituto Superior Técnico - IST, Universidade de Lisboa \- UL,
Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

###### Abstract

We study the thermodynamic properties of a static electrically charged
spherical thin shell in $d$ dimensions by imposing the first law of
thermodynamics on the shell. The shell is at radius $R$, inside it the
spacetime is Minkowski, and outside it the spacetime is Reissner-Nordström. We
obtain that the shell thermodynamics is fully described by giving two
additional reduced equations of state, one for the temperature and another for
the thermodynamic electrostatic potential. We choose the equation of state for
the temperature as essentially a power law in the gravitational radius $r_{+}$
with exponent $a$, such that the $a=1$ case gives the temperature of a shell
with black hole thermodynamic properties, and for the electrostatic potential
we choose an equation of state characteristic of a Reissner-Nordström
spacetime. The entropy of the shell is then found to be proportional to
$A_{+}^{a}$, where $A_{+}$ is the gravitational area corresponding to $r_{+}$,
with the exponent $a$ obeying $a>0$ to have a physically reasonable entropy.
We are then able to perform the black hole limit $R=r_{+}$, find the Smarr
formula for $d$-dimensional electrically charged black holes, and generically
recover the thermodynamics of a $d$-dimensional Reissner-Nordström black hole.
We further study the intrinsic thermodynamic stability of the shell with the
chosen equations of state. We obtain that for $0<a\leq\frac{d-3}{d-2}$ all the
configurations of the shell are thermodynamically stable, for
$\frac{d-3}{d-2}<a<1$ stability depends on the mass and electric charge, for
$a=1$ the configurations are unstable, unless the shell is at its own
gravitational radius, i.e., at the black hole limit, in which case it is
marginally stable, and that for $1<a<\infty$ all configurations are unstable.
Rewriting the stability conditions with variables that can be measured in the
laboratory, it is found that the sufficient condition for the stability of
these shells is when the isothermal electric susceptibility $\chi_{p,T}$ is
positive, marginal stability happening when this quantity is infinite, and
instability, and thus depart from equilibrium, arising for configurations with
a negative electric susceptibility.

self-gravitating thin shells, entropy, thermodynamics, black holes

††preprint: APS/123-QED

##  I Introduction

Self-gravitating thin shell physics in general relativity has proved of great
value in the understanding of the many aspects involving the interaction
between gravitational and matter fields. We mention a few of these aspects.
First, the main features of gravitational collapse and black hole formation
has been described in detail from the dynamics of thin shells in Schwarzschild
and Reissner-Nordström spacetimes, as well as in some of their generalizations
to higher dimensions israelshellcollapse ; kuchar ; kijowski ; diasgaolemos ;
gaolemos ; eiroasimeone1 . Second, the issue of the compactness of stars has
been studied through the help of neutral and electrically charged thin shells
andreasson . Third, the understanding of stars with outer normals to their
surface pointing to decreasing radius can be clearly formulated through the
maximum analytical extensions of shell solutions and their appearance in the
other side of the Carter-Penrose diagram lemosluz1 . Fourth, wormhole
spacetimes can be constructed through the support of thin shells diaslemos ;
bejaranoeiroa , the complementary of wormholes and bubble universes is clearly
displayed with the use of thin shells lemosluz2 . Fifth, regular black holes
can be found employing thin shells lemoszanchin ; uchikata ; brandenberger .

Self-gravitating thin shells allow a precise mathematical treatment not only
in a dynamic context, but also in a thermodynamic framework embedded within
general relativity, and as such they are of great interest in the
understanding of the thermodynamics of matter in a gravitational field, as
well as in the understanding of the thermodynamics of the gravitational field
itself. Remarkably, general relativity by itself fixes the equation of state
for pressure of the matter in a static spherical shell. On the other hand,
general relativity does not determine an equation of state for the
temperature. To find one, the first law of thermodynamics, valid for the
matter on the shell, has to be postulated with all thermodynamic quantities
that enter into it having a precise and correct meaning. Then, the
integrability conditions applied to the first law of thermodynamics restrict
the form of the temperature equation of state, leaving nonetheless some
freedom for its choice, that can be performed through some deduced reasonable
guess or via some more fundamental description of matter, and which, in turn,
narrows the possible types for the entropy function. For a shell with an
exterior given by a Schwarzschild spacetime this was performed in
Martinez:1996 , also treated in bergliaffa1 , and generalized to
$d$-dimensions in Andre:2019 . The inclusion of electric charge means that the
exterior spacetime is Reissner-Nordström, and although one needs to take care
of a further equation of state for the thermodynamic electric potential, it
also allows an exact treatment as done in Lemos:2015a , also studied in
bergliaffa2 , and with the extremal state being analyzed in Lemos:2015c ;
lemoshellextremal12 . One important fact about thermodynamics of shells is
that the shell can be put to its own gravitational radius. One can then argue
that at this stage its properties should be black hole like, and indeed they
are.

The interest in thermodynamics of spacetimes, in particular in thermodynamics
of shell spacetimes, comes from black holes. Black holes radiate at the
Hawking temperature, have the Bekenstein-Hawking entropy, and thus, in
appropriate settings, can be described as thermodynamic objects. In a path
integral statistical mechanics formulation for spherical black holes, the
suitable ensemble to use is the canonical ensemble, where one fixes a
temperature at a cavity of a given radius, and from which the full
thermodynamic properties of the system can then be deduced. For a
Schwarzschild vacuum spacetime one finds a small black hole solution which is
thermodynamically unstable and a large black hole solution, of radius near the
cavity radius, which is stable, see York:1986 ; can  and its $d$-dimensional
generalization andrelemos1 ; andrelemos2 . In this path integral formulation,
one can include a matter shell surrounding the black hole ym , put electric
charge into the black hole spacetime Braden:1990 , and work with AdS
spacetimes pecalemos .

Alternatively, one can find black hole thermodynamics through matter
thermodynamics via the quasiblack hole approach Lemos:2007 ; Lemos:2010 ;
Lemos:2011 ; lemoszaslavskii . In general, to have a thermodynamic
formulation, knowledge of the matter equations of state is required. However,
using the quasiblack hole approach one is able to skip the setting of specific
equations of state. In this approach, one keeps the gravitational radius
fixed, and changes the proper mass and the radius of the configuration,
maintaining it near the black hole threshold. One can then integrate the first
law of thermodynamics over this set of configurations, finding that the result
is indeed model independent, and retrieving fully the black hole properties.

Thus, thin shells, black holes, and quasiblack holes are of importance in the
understanding of thermodynamics of spacetimes. It is certainly of significance
to proceed with these themes. In particular, it is of interest to study
further thermodynamic shell properties. Here we analyze the entropy and the
thermodynamic stability of a static electrically charged spherical thin shell
in $d$ dimensions, as well as its black hole limit, extending the analysis and
the results given in Andre:2019 ; Lemos:2015a . The study of systems in $d$
dimensions is of relevance in obtaining knowledge of features that are
permanent, independent of dimension, and is of use as several theories of
gravitation imply the existence of extra dimensions. To make progress in our
analysis, we impose an equation of state such that the entropy of the shell is
a power in the gravitational area, i.e., the area corresponding to the
gravitational radius, and find that stability requires positive heat capacity,
positive isothermal compressibility, and positive isothermal electric
susceptibility. Putting the shell at its own gravitational radius we find the
corresponding black hole thermodynamic properties including the Smarr formula
and its thermodynamic stability. The essential thermodynamic formalism used is
in the book by Callen callen .

This paper is organized as follows. In Sec. II, we apply the thin shell
formalism to obtain the internal energy and the pressure of the shell in terms
of the spacetime variables. In Sec. III, we apply the first law of
thermodynamics to the shell and compute its entropy after providing two
equations of state, and afterward we analyze the black hole limit. In Sec. IV,
we study the intrinsic stability of the shell with such equations of state and
entropy, analyzing when applicable the black hole limit. In Sec. V, we
establish the connection of intrinsic stability to physical quantities such as
the heat capacity, the isothermal compressibility and the isothermal electric
susceptibility. In Sec. VI we conclude. There are several appendices that
complete the paper, including one where all the necessary plots to understand
the thermodynamic stability of the shell are displayed.

##  II Electrically charged spherical shell spacetime

###  II.1 Thin shell spacetime formalism

General relativity coupled to Maxwell electromagnetism has the equations

$\displaystyle G_{ab}=8\pi GT_{ab}\,,$ (1) $\displaystyle\nabla_{b}\hskip
1.13791pt{F_{a}}^{b}=J_{a}\,,$ (2)

where, $G_{ab}$ is the Einstein tensor given in terms of the metric $g_{ab}$
and its first two derivatives, $G$ is the gravitational constant in $d$
dimensions, the speed of light $c$ is set to one, $c=1$, $T_{ab}$ is the
stress-energy tensor, $\nabla_{b}$ is the covariant derivative of the metric
$g_{ab}$, $F_{ab}$ is the Maxwell tensor, $J_{a}$ is the electric current, and
indices $a,b$, are $d$-dimensional indices, running from 0 to $d-1$. The
Maxwell tensor $F_{ab}$ also obeys the internal equations
$\nabla_{[c}F_{ab]}=0$, where brackets in indices means total
antisymetrization, and allow us to write $F_{ab}$ in terms of an
electromagnetic potential vector $A_{a}$ as ${F}_{ab}=\frac{\partial
A_{b}}{\partial x^{a}}-\frac{\partial A_{a}}{\partial x^{b}}$. For an
electrovacuum spacetime the stress-energy tensor $T_{ab}$ is given by

$\displaystyle
T_{ab}=\varepsilon\left({F_{a}}^{c}F_{bc}-\frac{1}{4}g_{ab}F^{cd}F_{cd}\right)\,,$
(3)

where the parameter $\varepsilon$ is defined as
$\varepsilon=\epsilon\frac{(d-3)}{\Omega}$, with $\epsilon$ being the
electromagnetic coupling constant, and
$\Omega=\frac{\pi^{\frac{d-2}{2}}}{\Gamma(\frac{d}{2})}$ is the area of a
$d-2$ unit sphere, which in four dimensions yields the usual $4\pi$. In a thin
shell spacetime, one has an interior region, $\mathcal{V}_{i}$ say, that obeys
Eqs. (1) and (2), an exterior region, $\mathcal{V}_{e}$, that also obeys Eqs.
(1) and (2), and a boundary surface, i.e., a thin shell, in-between these two
regions that has properties found by appropriate junction conditions to match
the two different spacetime regions.

The interior $\mathcal{V}_{i}$ has coordinates $x_{i}^{a}$ assigned to it and
a metric ${{g}_{i}}_{ab}$. We denote ${{n}_{i}}_{a}$ as the covector
orthogonal to a hypersurface. An important quantity is given by the way in
which ${{n}_{i}}_{a}$ changes along the hypersurface, i.e.,
${{\nabla_{i}}}_{a}{{n_{i}}}_{b}$, where ${{\nabla_{i}}}_{a}$ is the covariant
derivative in the interior region. If the coordinates of the hypersurface are
denoted by $y^{\alpha}$, where Greek indices $\alpha,\beta$, are
$d-1$-dimensional indices and run from 0 to $d-2$, then the tangent vectors at
the hypersurface are $(e_{i})^{a}_{\alpha}=\frac{\partial x_{i}^{a}}{\partial
y^{\alpha}}$. The interior $\mathcal{V}_{i}$ is assumed to have an
electromagnetic vector potential ${{A}_{i}}_{a}$ and a corresponding field
strength
${{F}_{i}}_{ab}=\frac{\partial{A_{i}}_{{b}}}{\partial{x_{i}}^{{a}}}-\frac{\partial{A_{i}}_{{a}}}{\partial{x_{i}}^{{b}}}$.

The exterior $\mathcal{V}_{e}$ has coordinates $x_{e}^{a}$ assigned to it and
a metric ${{g}_{e}}_{ab}$. We denote ${{n}_{e}}_{a}$ as the covector
orthogonal to a hypersurface. An important quantity is related to how
${{n}_{e}}_{a}$ changes along the hypersurface,
${{\nabla_{e}}}_{a}{{n_{e}}}_{b}$, where ${{\nabla_{e}}}_{a}$ is the covariant
derivative in the exterior region. If the coordinates of the hypersurface are
denoted by $y^{\alpha}$ then the tangent vectors at the hypersurface are
$(e_{e})^{a}_{\alpha}=\frac{\partial x_{e}^{a}}{\partial y^{\alpha}}$. The
exterior $\mathcal{V}_{e}$ is assumed to have an electromagnetic vector
potential ${{A}_{e}}_{a}$ and a corresponding field strength
${{F}_{e}}_{ab}=\frac{\partial{A_{e}}_{b}}{\partial{x_{e}}^{a}}-\frac{\partial{A_{e}}_{a}}{\partial{x_{e}}^{b}}$.

The boundary hypersurface, with coordinates $y^{\alpha}$ assigned to it, and
which can be a thin shell, is assumed to be timelike and common to
$\mathcal{V}_{i}$ and $\mathcal{V}_{e}$. The pull-back of a covariant
tensorial quantity in each region allows the definition of the covariant
tensorial quantity at this boundary hypersurface. Then, the junction
conditions give that tensorial quantity uniquely at the common boundary
hypersurface. For the interior, the metric ${g}_{i}$ has the pull-back
$(\phi^{*}{g_{i}})_{\alpha\beta}={{g_{i}}}_{ab}(e_{i})^{a}_{\alpha}(e_{i})^{b}_{\beta}\equiv{{h_{i}}}_{\alpha\beta}$,
the quantity $\nabla_{i}n_{i}$ has a pull-back yielding the hypersurface
extrinsic curvature ${K_{i}}_{\alpha\beta}$, namely,
$(\phi^{*}\nabla_{i}n_{i})_{\alpha\beta}={{\nabla_{i}}}_{a}{{n_{i}}}_{b}e^{a}_{\alpha}e^{b}_{\beta}={{\nabla_{i}}}_{\alpha}{{n_{i}}}_{\beta}\equiv{K_{i}}_{\alpha\beta}$,
${A_{i}}$ has the pull-back
$(\phi^{*}{A_{i}})_{\alpha}={{A_{i}}}_{a}(e_{i})^{a}_{\alpha}\equiv{{A_{i}}}_{\alpha}$,
${\bm{F}}_{i}$ has the pull-back
$(\phi^{*}{\bm{F}_{i}})_{\alpha\beta}={{F_{i}}}_{ab}(e_{i})^{a}_{\alpha}(e_{i})^{b}_{\beta}\equiv{{F_{i}}}_{\alpha\beta}$,
and ${F}_{i}$ defined such that ${{F}_{i}}_{a}\equiv{{F_{i}}}_{ab}{n_{i}}^{b}$
has the pull-back
$(\phi^{*}{F_{i}})_{\alpha}={{F_{i}}}_{ab}{n_{i}}^{b}(e_{i})^{a}_{\alpha}\equiv{{F_{i}}}_{\alpha}$.
For the exterior, the metric ${g}_{e}$ has the pull-back
$(\phi^{*}{g_{e}})_{\alpha\beta}={{g_{e}}}_{ab}(e_{e})^{a}_{\alpha}(e_{e})^{b}_{\beta}\equiv{{h_{e}}}_{\alpha\beta}$,
the quantity $\nabla_{e}n_{e}$ has a pull-back yielding the extrinsic
curvature ${K_{e}}_{\alpha\beta}$, namely,
$(\phi^{*}\nabla_{e}n_{e})_{\alpha\beta}={{\nabla_{e}}}_{a}{{n_{e}}}_{b}e^{a}_{\alpha}e^{b}_{\beta}={{\nabla_{e}}}_{\alpha}{{n_{e}}}_{\beta}\equiv{K_{e}}_{\alpha\beta}$,
${A_{e}}$ has the pull-back
$(\phi^{*}{A_{e}})_{\alpha}={{A_{e}}}_{a}(e_{e})^{a}_{\alpha}\equiv{{A_{e}}}_{\alpha}$,
${\bm{F}}_{e}$ has the pull-back
$(\phi^{*}{\bm{F}_{e}})_{\alpha\beta}={{F_{e}}}_{ab}(e_{e})^{a}_{\alpha}(e_{e})^{b}_{\beta}\equiv{{F_{e}}}_{\alpha\beta}$,
and ${F}_{e}$ defined such that ${{F}_{e}}_{a}\equiv{{F_{e}}}_{ab}{n_{e}}^{b}$
has the pull-back
$(\phi^{*}{F_{e}})_{\alpha}={{F_{e}}}_{ab}{n_{e}}^{b}(e_{e})^{a}_{\alpha}\equiv{{F_{e}}}_{\alpha}$.
The first junction condition is the continuity of the metric

$\displaystyle[h_{\alpha\beta}]=0\,,$ (4)

where $[h_{\alpha\beta}]$ means
$[h_{\alpha\beta}]={h_{e}}_{\alpha\beta}-{h_{i}}_{\alpha\beta}$, and the same
for any other quantity. Equation (4) means that one can define a metric
$h_{\alpha\beta}$ at the boundary surface with coordinates $y^{\alpha}$ and of
course such that it obeys
$h_{\alpha\beta}={h_{i}}_{\alpha\beta}={h_{e}}_{\alpha\beta}$. The second
junction condition is

$\displaystyle-\Big{(}[K_{\alpha\beta}]-[K]h_{\alpha\beta}\Big{)}={8\pi
G}S_{\alpha\beta}\,,$ (5)

where $K$ is the trace of the extrinsic curvature $K_{\alpha\beta}$, and
$S_{\alpha\beta}$ is the stress-energy tensor for matter in the shell. We
consider that the thin shell is made of a perfect fluid having the stress-
energy tensor

$\displaystyle
S_{\alpha\beta}=(\sigma+p)u_{\alpha}u_{\beta}+ph_{\alpha\beta}\,,$ (6)

where $\sigma$ is the energy density, $p$ is the pressure and $u^{\alpha}$ is
the velocity of the fluid on the boundary.

There are also junction conditions for the pull backs of the covector
potential and of the electromagnetic field strength. They are given by

$\displaystyle[A_{\alpha}]=0\,,$ (7)
$\displaystyle[F_{\alpha\beta}]=0\,\,,\,\,\,[F_{\alpha}]=j_{\alpha}\,,$ (8)

where $F_{\alpha}=F_{ab}n^{b}e^{a}_{\alpha}$ is defined on each side of the
boundary hypersurface, i.e, from the interior and exterior region sides, and
the electric current $j_{\alpha}$ is given by

$\displaystyle j_{\alpha}={\zeta}\sigma_{e}u_{\alpha},$ (9)

with $\zeta$ being defined as $\zeta=\frac{\Omega}{\epsilon_{q}}$,
$\epsilon_{q}$ being the electric permittivity, and with $\sigma_{e}$ being
the electric charge density.

Now we apply this formalism to a particular $d$-dimentional spacetime, namely,
a Minkowski interior, a thin shell, and a $d$-dimensional Reissner-Nordström,
also called Reissner-Nordström-Tangherlini, exterior. We use the convention
that the coupling constant $\epsilon$ appearing implicitly in Eq. (3) and the
electric permittivity $\epsilon_{q}$ appearing implicitly in Eq. (9) are set
to one, $\epsilon=1$ and $\epsilon_{q}=1$, and the $d$-dimensional constant of
gravitation $G$ is not set to one, it is left generic.

###  II.2 The spacetime solution

####  II.2.1 The interior

The interior region $\mathcal{V}_{i}$ is a vacuum $d$-dimensional spherically
symmetric Minkowski region with spherical coordinates $x_{i}^{a}$ assigned to
it such that $x_{i}^{a}=(t_{i},r_{i},\theta_{i}^{A})$ with
$A\in\\{1,...,d-2\\}$, and with line element

$\displaystyle ds_{i}^{2}=-dt_{i}^{2}+dr^{2}+r^{2}d\Omega^{2}\,,\quad 0\leq
r_{i}\leq R_{i}\,,$ (10)

where we have put $r\equiv r_{i}$, $d\Omega^{2}$ is the line element of a
$(d-2)$ unit sphere, and $R_{i}$ is the radius of the shell as measured from
the interior.

The Maxwell potential covector $A_{a}$ is given in the interior region as

$\displaystyle(A_{i})_{t_{i}}=A_{i}\,,$ (11)

where $A_{i}$ is a constant and the other components are zero.

####  II.2.2 The exterior

The exterior region $\mathcal{V}_{e}$ is a vacuum $d$-dimensional spherically
symmetric Reissner-Nordström-Tangherlini region with spherical coordinates
$x_{e}^{a}$ assigned to it such that $x_{e}^{a}=(t_{e},r_{e},\theta_{e}^{A})$,
with $A\in\\{1,...,d-2\\}$, and with line element

$\displaystyle
ds_{e}^{2}=-f(r)\,dt_{e}^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2}\,,\quad
R_{e}\leq r_{e}\leq\infty\,,$ (12)

where we have put $r\equiv r_{e}$, a redefinition that can be done, $R_{e}$ is
the radius of the shell as measured from the exterior, and

$\displaystyle f(r)=1-\frac{2\mu m}{r^{d-3}}+\frac{qQ^{2}}{r^{2(d-3)}}\,,$
(13)

where $m$ is the spacetime, also called ADM, mass, and $Q$ is the total
electric charge, and with $\mu$ and $q$ being given by

$\displaystyle\mu=\frac{8\pi G}{(d-2)\Omega}\,,\quad\quad q=\frac{8\pi
G}{(d-2)\Omega}\,,$ (14)

with $G$ being the $d$-dimensional gravitational constant, and again
$\Omega=\pi^{\frac{d-2}{2}}\Gamma(\frac{d}{2})^{-1}$ is the area of a $(d-2)$
unit sphere. Thus, $\mu=q$. Note that without putting $\epsilon$ and
$\epsilon_{q}$ to one, $q$ in Eq. (14) is $q=\frac{8\pi
G{\epsilon}}{(d-2)\Omega{\epsilon_{q}^{2}}}$.

The Reissner-Nordström metric has its gravitational radius and Cauchy radius
at its coordinate singularities given by

$\displaystyle r_{\pm}^{d-3}=\mu m\pm\sqrt{\mu^{2}m^{2}-qQ^{2}}\,,$ (15)

where $r_{+}$ corresponds to the gravitational radius and $r_{-}$ to the
Cauchy radius. Note that the gravitational radius and the Cauchy radius in
general are not horizon radii, they are only horizon radii for a full
electrovacuum solution of the Einstein-Maxwell system of equations, in which
case they are the event horizon radius and the Cauchy horizon radius. From Eq.
(15), one sees that the extremal case, defined as $r_{+}=r_{-}$ yields a mass
to charge relation given by $\sqrt{\mu}m=Q$, which from Eq. (14) yields in
four dimensions $\sqrt{G}m=Q$. One can put $G=1$ to give in four dimensions
$m=Q$, but we keep the $d$-dimensional $G$ in our calculations to not fall
into awkward units along the calculations. The area $A_{+}$ corresponding to
the gravitational radius $r_{+}$ is an important quantity, defined as the
gravitational area, and given by

$\displaystyle A_{+}=\Omega r_{+}^{d-2}\,.$ (16)

One can invert Eq. (15), giving

$\displaystyle m=\frac{1}{2\mu}(r_{+}^{d-3}+r_{-}^{d-3})\,,\quad\quad
Q=\frac{(r_{+}r_{-})^{\frac{d-3}{2}}}{\sqrt{q}}\,.$ (17)

Note that $q$ in Eq. (17) can be swapped for $\mu$ due to Eq. (14), but we
stick to $q$ whenever the coefficient is associated to the electric charge
$Q$. Now, with the two characteristic radii $r_{\pm}$ defined in Eq. (15) we
can rewrite Eq. (13) as

$\displaystyle
f(r)=\left(1-\left(\frac{r_{+}}{r}\right)^{d-3}\right)\left(1-\left(\frac{r_{-}}{r}\right)^{d-3}\right)\,.$
(18)

The Maxwell potential covector $A_{a}$ is given in the exterior region as

$\displaystyle(A_{e})_{t_{e}}=-\frac{Q}{(d-3)r_{e}^{d-3}}\,,$ (19)

where we have set without loss of generality a constant of integration $A_{e}$
to zero, $A_{e}=0$, and the other components are zero. Note than that the
outer electric field is $(F_{e})^{t_{e}r}=\frac{Q}{r_{e}^{d-2}}$. If we do not
set $\epsilon_{q}=1$ then Eq. (19) is
$(A_{e})_{t_{e}}=-\frac{Q}{{(d-3)\epsilon_{q}}r_{e}^{d-3}}+A_{e}$ and the
outer electric field is $(F_{e})^{t_{e}r}=\frac{Q}{\epsilon_{q}r_{e}^{d-2}}$.

####  II.2.3 The thin shell

The boundary hypersurface $\Sigma$ is spherically symmetric and has in
principle a thin shell in it and it is useful to give to it an intrinsic
metric $h_{\alpha\beta}$ such that its line element
$ds^{2}=h_{\alpha\beta}dx^{\alpha}dx^{\beta}$ can be written as

$\displaystyle ds_{\Sigma}^{2}=-d\tau^{2}+R(\tau)^{2}d\Omega^{2}\,,$ (20)

where the coordinate system $y^{\alpha}=(\tau,\theta^{A})$ has been chosen,
with $A\in\\{1,...,d-2\\}$, the coordinate $\tau$ is the proper time of the
shell, and $R(\tau)$ is the radius of the shell. The Maxwell potential
covector $A_{\alpha}$ is given at the thin shell as

$\displaystyle(A_{\Sigma})_{\tau}=A_{\Sigma}\,,$ (21)

respectively, where $A_{\Sigma}$ is a constant and the other components are
zero. Recall that we use the latin indices to designate quantities in the
regions $\mathcal{V}_{e}$ and $\mathcal{V}_{i}$ whereas greek indices
designate quantities at the hypersurface. The pull-back of the metric in the
region $\mathcal{V}_{i}$ on the hypersurface $\Sigma$,
$(\phi^{*}\bm{g}_{i})_{\alpha\beta}$, assumes the line element form

$ds_{i,\Sigma}^{2}=\left(-\dot{t}_{i}^{2}+\dot{R}_{i}^{2}\right)d\tau^{2}+R_{i}(\tau)^{2}d\Omega^{2}\,,$
(22)

where the boundary hypersurface $\Sigma$ has an history defined in the
interior by $R_{i}(t_{i})$ and $\dot{\,\,}=\derivative{\,\,}{\tau}$. The pull-
back of the metric in the region $\mathcal{V}_{e}$ on the hypersurface
$\Sigma$, $(\phi^{*}\bm{g}_{e})_{\alpha\beta}$, is given by the line element

$\displaystyle ds_{e,\Sigma}^{2}=$
$\displaystyle\Big{[}-f(R_{e}(\tau))\dot{t}_{e}^{2}+f(R_{e}(\tau))^{-1}\dot{R}_{e}^{2}\Big{]}d\tau^{2}$
$\displaystyle+R_{e}(\tau)^{2}d\Omega^{2}\,,$ (23)

where the boundary hypersurface $\Sigma$ has an history defined in the
exterior by $R_{e}(t_{e})$. Now, we apply the first junction condition, Eq.
(4). i.e.,
$[h_{\alpha\beta}]=(\phi^{*}\bm{g}_{e})_{\alpha\beta}-(\phi^{*}\bm{g}_{i})_{\alpha\beta}=0$
to Eqs. (10), (12), and (20), to give two equations,

$R_{e}(\tau)=R_{i}(\tau)=R(\tau)\,,$ (24)
$-\dot{t}_{i}^{2}+\dot{R}^{2}=-f(R)\dot{t}_{e}^{2}+f(R)^{-1}\dot{R}^{2}=-1\,.$
(25)

Clearly, Eq. (24) entitles us to use the same radial coordinate $r$ for the
interior, the boundary surface, and the exterior, as we did a priori, and
permits to define a unique area $A$ to the shell,

$\displaystyle A=\Omega R^{d-2}\,.$ (26)

The extrinsic curvature of the hypersurface in both regions can be computed
from $K_{\alpha\beta}=(\phi^{*}\nabla
n)_{\alpha\beta}=\nabla_{a}n_{b}e^{a}_{\alpha}e^{b}_{\beta}$, where $n$ is the
unit outward normal covector to the hypersurface. For the interior, $n_{i}$ is
given by
$n_{i}=\left(1-\left(\derivative{R}{t_{i}}\right)^{2}\right)^{-\frac{1}{2}}\left(-\derivative{R}{t_{i}}dt_{i}+dr_{i}\right)$.
In the hypersurface, it is useful to write these components in terms of
$\tau$, so using Eq. (25), we have
$\evaluated{\left(1-\left(\derivative{R}{t_{i}}\right)^{2}\right)^{-\frac{1}{2}}}_{\Sigma}=\sqrt{1+\dot{R}^{2}}$
and
$\evaluated{\derivative{R}{t_{i}}}_{\Sigma}=\frac{\dot{R}}{\sqrt{1+\dot{R}^{2}}}$,
so that $n_{i}=\left(-\dot{R},\sqrt{1+\dot{R}^{2}},0,0\right)$. For the
exterior, $n_{e}$ is given by $n_{e}=\
\sqrt{f(r_{e})}\left(f(r_{e})^{2}-\left(\derivative{R}{t_{e}}\right)^{2}\right)^{-\frac{1}{2}}\left(-\derivative{R}{t_{e}}dt_{e}+dr_{e}\right)$.
In the hypersurface, it is useful to write these components in terms of
$\tau$, so using Eq. (25) we have
$\sqrt{f(r_{e})}\evaluated{\left(f(r_{e})^{2}-\left(\derivative{R}{t_{e}}\right)^{2}\right)^{-\frac{1}{2}}}_{\Sigma}=\frac{\sqrt{f(R)+\dot{R}^{2}}}{f(R)}$
and
$\evaluated{\derivative{R}{t_{e}}}_{\Sigma}=\frac{f(R)\dot{R}}{\sqrt{f(R)+\dot{R}^{2}}}$,
so that
$n_{e}=\left(-\dot{R},\frac{\sqrt{f(R)+\dot{R}^{2}}}{f(R)},0,0\right)$, where
from Eq. (13) one has $f(R)=1-\frac{2\mu
m}{R^{d-3}}+\frac{qQ^{2}}{R^{2(d-3)}}$. Then, for the interior the nonzero
components of the extrinsic curvature are

$\displaystyle(K_{i})^{\tau}_{\,\,\,\tau}=\frac{\ddot{R}}{\sqrt{1+\dot{R}^{2}}}\,,\quad(K_{i})^{\theta^{A}}_{\,\,\,\theta^{A}}=\frac{\sqrt{1+\dot{R}^{2}}}{R}\,,$
(27)

and for the exterior are

$\displaystyle(K_{e})^{\tau}_{\,\,\,\tau}=\frac{\ddot{R}+\frac{\partial_{R}f(R)}{2}}{\sqrt{f(R)+\dot{R}^{2}}}\,,\quad(K_{e})^{\theta^{A}}_{\,\,\,\theta^{A}}=\frac{\sqrt{f(R)+\dot{R}^{2}}}{R}\,.$
(28)

We assume that the shell is static and in equilibrium, thus $\dot{R}=0$,
$\ddot{R}=0$, and $u^{\alpha}=(1,0,0)$. From Eqs. (5) and (6), and Eqs.
(27)-(28), the energy density and the pressure of the shell are obtained as
$\sigma=\frac{d-2}{8\pi GR}\left(1-\sqrt{f(R)}\right)$ and $p=\frac{d-3}{8\pi
GR}\left(\sqrt{f(R)}-1\right)+\frac{f^{\prime}(R)}{16\pi G\sqrt{f(R)}}$,
respectively, where ${}^{\prime}=\frac{\partial\;}{\partial R}$. These two
expressions for $\sigma$ and $p$ can be put in the form

$\displaystyle\sigma=\frac{1-k}{\mu\Omega R}\,,$ (29) $\displaystyle
p=\frac{1}{2\mu\Omega
R^{2d-5}k}\frac{d-3}{d-2}\Bigg{[}(1-k)^{2}R^{2(d-3)}-qQ^{2}\Bigg{]}\,,$ (30)

where here $k$ is the redshift function evaluated at $R$, i.e.,

$\displaystyle k=\sqrt{f(R)}\,.$ (31)

Note also that $q$ in Eq. (30) could be swapped for $\mu$ due to Eq. (14), but
again, we keep $q$ whenever the coefficient is associated to the electric
charge $Q$. It is useful to define the rest mass of the shell by

$\displaystyle M=\Omega R^{d-2}\sigma\,.$ (32)

Then, knowing the energy density $\sigma$ from Eq. (29) and using Eq. (32) one
gets $M=\frac{R^{d-3}}{\mu}\left(1-k\right)$. This in turn can be manipulated
to give a relation for the spacetime mass $m$, namely,

$\displaystyle m=M-\frac{\mu M^{2}}{2R^{d-3}}+\frac{Q^{2}}{2R^{d-3}}\,,$ (33)

where Eq. (31) in the form $k(M,R,Q)=\sqrt{1-\frac{2\mu
m}{R^{d-3}}+\frac{qQ^{2}}{R^{2(d-3)}}}$ has been used, see Eq. (13). We see
that the total energy $m$ of the shell is given by the rest mass $M$ plus a
second and third terms that represent the gravitational potential energy and
the electric potential energy, respectively. For generic $\epsilon$ and
$\epsilon_{q}$, i.e., not set to one, Eq. (33) is $m=M-\frac{\mu
M^{2}}{2R^{d-3}}+\frac{{\epsilon}Q^{2}}{2{\epsilon_{q}^{2}}R^{d-3}}$.

Since the shell is static, the junction condition for the covector field
$A_{a}$ given in Eq. (7), i.e., $[A_{\tau}]=0$, together with Eqs. (11) and
(19) yield $-\frac{Q}{(d-3)R^{d-3}k}-A_{i}=0$, or

$\displaystyle A_{i}=-\frac{Q}{(d-3)R^{d-3}k}\,.$ (34)

For generic $\epsilon$ and $\epsilon_{q}$, Eq. (34) is
$A_{i}=-\frac{Q}{(d-3){\epsilon_{q}}R^{d-3}k}$ So, Eq. (34) sets the constant
value of the potential in the region $\mathcal{V}_{i}$ as a function of the
electric charge, the position of the shell and the electric potential at
infinity which we have put to zero, $A_{e}=0$, and moreover gives the value of
the electric potential at the hypersurface, see Eq. (21), $A_{\Sigma}=A_{i}$.
In Eq. (8), the relevant condition for the Faraday tensor is
$[F_{\alpha}]=j_{\alpha}$, which upon using Eq. (9) becomes
$-(F_{e})_{tr}k\dot{t}_{e}=\zeta\sigma_{e}$, with
$\zeta=\frac{\Omega}{\epsilon_{q}}$, see Eq. (9), and since we are putting
$\epsilon_{q}=1$, $\zeta=\Omega$. Then, from ${F}_{ab}=\frac{\partial
A_{b}}{\partial x^{a}}-\frac{\partial A_{a}}{\partial x^{b}}$ together with
Eqs. (11) and (19) implies

$\displaystyle Q=R^{d-2}\sigma_{e}\,.$ (35)

This is the relation between the total electric charge and its corresponding
charge density.

##  III Entropy of the charged thin shell

###  III.1 The first law of thermodynamics

The analysis of the thermodynamics of an electrically charged thin shell is
performed by imposing the first law of thermodynamics on the shell. Noting
that the internal energy of the shell is its rest mass, the first law of
thermodynamics is

$\displaystyle TdS=dM+pdA-\Phi dQ\,,$ (36)

where $T$ is the local temperature throughout the shell, $S$ is the entropy of
the shell, $M$ is the rest mass of the shell calculated in the last
subsection, $p$ is the pressure on the shell calculated in the last
subsection, $A$ is the area of the shell, $\Phi$ is the thermodynamic electric
potential at the shell, and $Q$ is the electric charge of the shell. Defining
the inverse temperature $\beta$ as

$\displaystyle\beta=\frac{1}{T}\,,$ (37)

the first law can be written as

$\displaystyle dS=\beta dM+\beta pdA-\beta\Phi dQ\,.$ (38)

To solve the first law, three equations of state have to be provided. A first
equation of state is for the inverse temperature in terms of $M$, $A$, and
$Q$,

$\displaystyle\beta=\beta(M,A,Q)\,.$ (39)

A second equation of state is for the pressure in terms of $M$, $A$, and $Q$,

$\displaystyle p=p(M,A,Q)\,.$ (40)

This equation of state has already been found. Indeed, Eq. (30) is a dynamic
as well as a thermodynamic equation, it is $p(M,A,Q)=\frac{1}{2\mu\Omega
R^{2d-5}k}\frac{d-3}{d-2}\Big{[}(1-k)^{2}R^{2(d-3)}-qQ^{2}\Big{]}$, as $k$ is
a function of $M$, $A$, and $Q$, and $R$ can be swapped for $A$. A third
equation of state is for the thermodynamic electric potential $\Phi$ in terms
of $M$, $A$, and $Q$,

$\displaystyle\Phi=\Phi(M,A,Q)\,.$ (41)

Note that $\beta$ and $\Phi$ are restricted by the integrability conditions
but otherwise free, and $p$ is fixed by the equations of motion, showing that
Einstein equations have already thermodynamics in-built into them. Given the
functions $\beta(M,A,Q)$, $p(M,A,Q)$, and $\Phi(M,A,Q)$ of Eqs. (39)-(41) we
are interested in calculating the entropy $S$ as a function of $M$, $A$, and
$Q$, i.e., $S(M,A,Q)$, through the first law of thermodynamics given in Eq.
(38).

###  III.2 Integrability conditions

The entropy $S$ is a function of the thermodynamic parameters $(M,A,Q)$ and
its differential is exact by definition. This places the condition that the
Hessian matrix of $S$ needs to be symmetric. Since from Eq. (38) the first
derivatives of $S(M,A,Q)$ are

$\displaystyle\Big{(}\partialderivative{S}{M}\Big{)}_{A,Q}=\beta\,,$
$\displaystyle\Big{(}\partialderivative{S}{A}\Big{)}_{M,Q}=\beta p\,,$
$\displaystyle\Big{(}\partialderivative{S}{Q}\Big{)}_{M,A}=-\beta\Phi\,,$ (42)

the condition on the Hessian of $S$ is explicitly

$\displaystyle\Big{(}\partialderivative{\beta}{A}\Big{)}_{M,Q}=\Big{(}\partialderivative{\beta
p}{M}\Big{)}_{A,Q}\,,$
$\displaystyle\Big{(}\partialderivative{\beta}{Q}\Big{)}_{M,A}=-\Big{(}\partialderivative{\beta\Phi}{M}\Big{)}_{A,Q}\,,$
$\displaystyle\Big{(}\partialderivative{\beta
p}{Q}\Big{)}_{M,A}=-\Big{(}\partialderivative{\beta\Phi}{A}\Big{)}_{M,Q}\,.$
(43)

These are the integrability conditions, necessary to have the entropy $S$ as
an exact differential.

###  III.3 Parameter transformation and the entropy

In order to compute $S$, we can make parameter transformations to simplify the
differential. The parameter space ($M,A,Q$) can be easily transformed into
($M,R,Q$) since $A$ depends solely on $R$ through Eq. (26), i.e., $A=\Omega
R^{d-2}$. We can also express $S$ in the parameters ($r_{+},r_{-},R$), which
will be more convenient. This transformation can be performed by using Eq.
(17). Then, we can use Eq. (32) together with (29) and with the redshift
function $k=\sqrt{f(R)}$ of Eq. (31) put in the form
$k(r_{+},r_{-},R)=\sqrt{\left(1-\left(\frac{r_{+}}{R}\right)^{d-3}\right)\left(1-\left(\frac{r_{-}}{R}\right)^{d-3}\right)}$,
see Eq. (18). The derivatives of the entropy $S(r_{+},r_{-},R)$ can be found
by the chain rule, so that
$\left(\partialderivative{S}{R}\right)_{r_{+},r_{-}}=\beta\left(\partialderivative{M}{R}\right)_{r_{+},r_{-}}+\beta
p\left(\partialderivative{A}{R}\right)_{r_{-},r_{+}}$ and
$\left(\partialderivative{S}{r_{\pm}}\right)_{r_{\mp},R}=\beta\left(\partialderivative{M}{r_{\pm}}\right)_{r_{\mp},R}-\beta\Phi\left(\partialderivative{Q}{r_{\pm}}\right)_{r_{\mp},R}$.
Moreover, from Eqs. (26), (30), and (32), we can find that
$\left(\partialderivative{M}{R}\right)_{r_{+},r_{-}}=-p\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}$.
So, clearly, the partial derivative in $R$ vanishes,
$\left(\partialderivative{S}{R}\right)_{r_{+},r_{-}}=0$. This means that the
entropy that in general is a function $S=S(r_{+},r_{-},R)$ in this case has no
dependence on $R$, only in $r_{+}$ and $r_{-}$, and so

$\displaystyle S=S(r_{+},r_{-})\,,$ (44)

i.e., the entropy is independent of $R$ in this parameter space.

###  III.4 The temperature and the electric potential

The integrability conditions or Euler relations impose restrictions on the
expressions of $\beta$ and $\Phi$, that can be worked out in the parameters
$(r_{+},r_{-},R)$.

Beginning with $\beta$, we can calculate by the chain rule its derivative with
respect to $R$ with $r_{+}$ and $r_{-}$ fixed, i.e.,
$\left(\partialderivative{\beta}{R}\right)_{r_{+},r_{-}}=\left(\partialderivative{\beta}{M}\right)_{A,Q}\left(\partialderivative{M}{R}\right)_{r_{+},r_{-}}+\left(\partialderivative{\beta}{A}\right)_{M,Q}\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}$.
Using the first equation in Eq. (43) and that
$\left(\partialderivative{M}{R}\right)_{r_{+},r_{-}}=-p\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}$,
which comes from Eqs. (26), (30), and (32), and that
$\left(\partialderivative{p}{M}\right)_{A,Q}=\frac{1}{(d-2)\Omega
kR^{(d-3)}}\left(\partialderivative{k}{R}\right)_{r_{+},r_{-}}$, which comes
from Eq. (30), we find
$\left(\partialderivative{\beta}{R}\right)_{r_{+},r_{-}}=\frac{\beta}{k}\left(\partialderivative{k}{R}\right)_{r_{+},r_{-}}$,
which upon integration gives

$\displaystyle\beta(r_{+},r_{-},R)=b(r_{+},r_{-})k\,,$ (45)

where $b(r_{+},r_{-})$ is a reduced equation of state, an intrinsic quantity
that will depend solely on the nature of the matter in the shell. From Eq.
(45) one sees that $b(r_{+},r_{-})=\beta(r_{+},r_{-},\infty)$. The redshift
function $k=\sqrt{f(R)}$ of Eq. (31) here is put in the form
$k(r_{+},r_{-},R)=\sqrt{\left(1-\left(\frac{r_{+}}{R}\right)^{d-3}\right)\left(1-\left(\frac{r_{-}}{R}\right)^{d-3}\right)}$,
see Eq. (18). The dependence of $\beta$ on $k$ just found is in agreement with
the Tolman’s formula for the temperature in a static gravitational field.

Now, for the case of $\Phi(r_{+},r_{-},R)$, the chain rule for the derivative
in $R$ together with
$\left(\partialderivative{M}{R}\right)_{r_{+},r_{-}}=-p\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}$
gives
$\left(\partialderivative{\Phi}{R}\right)_{r_{+},r_{-}}=\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}\left(\partialderivative{\Phi}{A}\right)_{Q,M}-p\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}\left(\partialderivative{\Phi}{M}\right)_{A,Q}$.
The three equations in Eq. (43) can be rearranged to substitute the right-hand
side of the latter equation into
$\left(\partialderivative{\Phi}{R}\right)_{r_{+},r_{-}}=-\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}\left(\partialderivative{p}{Q}\right)_{M,A}-\Phi\left(\partialderivative{A}{R}\right)_{r_{+},r_{-}}\left(\partialderivative{p}{M}\right)_{A,Q}$.
Then, we can use Eqs. (26) and the expressions of the derivatives of the
pressure, i.e.,
$\left(\partialderivative{p}{M}\right)_{A,Q}=\frac{1}{(d-2)\Omega
kR^{(d-3)}}\left(\partialderivative{k}{R}\right)_{r_{+},r_{-}}$ and
$\left(\partialderivative{p}{Q}\right)_{M,A}=-\frac{Q(d-3)}{(d-2)\Omega
kR^{2d-5}}$, to find explicitly the equation for $\Phi(r_{+},r_{-},R)$, which
becomes
$\left(\partialderivative{k\Phi}{R}\right)_{r_{+},r_{-}}=\frac{(d-3)Q}{R^{d-2}}$.
Upon integration one finds

$\displaystyle\Phi(r_{+},r_{-},R)=\frac{Q}{k}\Big{[}c(r_{+},r_{-})-\frac{1}{R^{d-3}}\Big{]}\,,$
(46)

where $c(r_{+},r_{-})$ is a reduced equation of state, an intrinsic quantity
that will depend solely on the nature of the matter in the shell. From Eq.
(46) one sees that $c(r_{+},r_{-})=\frac{\Phi(r_{+},r_{-},\infty)}{Q}$.

###  III.5 The differential of the entropy

The differential $dS$ in the parameters $(r_{+},r_{-},R)$ can be written
considering Eqs. (45) and (46). It follows that

$\displaystyle dS=$
$\displaystyle\frac{{(d-3)}b(r_{+},r_{-})}{2\mu}\Big{[}\Big{(}1-r_{-}^{d-3}c(r_{+},r_{-})\Big{)}r_{+}^{d-4}dr_{+}$
$\displaystyle+\Big{(}1-r_{+}^{d-3}c(r_{+},r_{-})\Big{)}r_{-}^{d-4}dr_{-}\Big{]}\,.$
(47)

To ensure the integrability of the differential, we apply once again the
symmetric characteristic of the Hessian matrix in these coordinates, which
gives the condition

$\displaystyle\partialderivative{b}{r_{-}}\,(1-c\,r_{-}^{d-3})r_{+}^{d-4}-\partialderivative{b}{r_{+}}\,(1-c\,r_{+}^{d-3})r_{-}^{d-4}$
$\displaystyle=\partialderivative{c}{r_{-}}br_{-}^{d-3}r_{+}^{d-4}-\partialderivative{c}{r_{+}}br_{+}^{d-3}r_{-}^{d-4}\,.$
(48)

Hence, the entropy $S(r_{+},r_{-})$ will depend on two functions
$b(r_{+},r_{-})$ and $c(r_{+},r_{-})$ that are related by a partial
differential equation, Eq. (48). These functions cannot be specified by the
first law of thermodynamics together with general relativity as they depend on
the class of matter that composes the shell. To make progress we need to
specifiy the two reduced equations of state for $b(r_{+},r_{-})$ and
$c(r_{+},r_{-})$.

It is also interesting to notice that the differential for the entropy can be
rewritten in a simpler form as
$dS=\frac{b}{2\mu}d(r_{+}^{d-3}+r_{-}^{d-3})-\frac{b}{2\mu}c\,d[(r_{+}r_{-})^{d-3}]$,
and thus from Eq. (17) one has $dS=bdm-b\phi dQ$, where we have defined the
electric potential $\phi=Q\,c$, and have used our convention $q=\mu$. So the
entropy and its derivatives are functions of the ADM mass $m$ and the modulus
of the electric charge of the shell, i.e., $S=S(m,Q)$, and, as well, the
equations of state will be functions of $m$ and $Q$, namely, $b=b(m,Q)$ and
$c=c(m,Q)$, where $Q$ here means the modulus of the electric charge. This
shows that the dependence on the rest mass and on the pressure and the area in
the first law of thermodynamics as given in Eq. (38), i.e., $\beta dM+\beta
pdA$, comes from the ADM mass $m$, since in this version these terms are
compressed to $b\,dm$, and is aligned with the fact that rest mass and
pressure are forms of energy in general relativity.

###  III.6 The reduced equations of state: Specific choice

To proceed, we have now to give the two reduced equations of state, one for
$b(r_{+},r_{-})$ and the other for $c(r_{+},r_{-})$.

We choose the reduced equation of state for the temperature of the shell, or
better, for its inverse temperature as

$\displaystyle
b(r_{+},r_{-})=\frac{a\gamma\Omega^{a-1}}{d-3}\frac{r_{+}^{a(d-2)}}{r_{+}^{d-3}-r_{-}^{d-3}}\,\,,$
(49)

where $a$ is a free exponent and $\gamma$ is a free parameter. The reduced
equation of state given in Eq. (49) imposes the restriction $r_{-}\leq r_{+}$,
i.e., $r_{+}$ and $r_{-}$ have real values. This means that the shell can be
undercharged or, in the limit, extremely charged, but not overcharged. Thus,
this reduced equation of state cannot be applied to overcharged shells.

Inserting the $b(r_{+},r_{-})$ given in Eq. (49) into the integrability
condition given in Eq. (48), one finds that one of the solutions for
$c(r_{+},r_{-})$ is

$\displaystyle c(r_{+},r_{-})=\frac{1}{r_{+}^{d-3}}\,,$ (50)

which yields the typical Reissner-Nordström equation of state for the electric
potential, and so we choose it as the second equation of state.

The equation of state for the temperature, Eq. (49), introduces two free
parameters, namely, $a$ and $\gamma$, which can be chosen at will as long as
the choice is physically reasonable, with the power law exponent $a$ being the
more relevant. Envisaging the equation of state for the temperature as arising
from quantum effects in the matter of the shell, it implies that the Planck
constant $\hbar$ appears intrinsically in the formula for $b$, as well as
Boltzmann constant $k_{B}$ since the whole setup involves thermodynamics. In
this case $b$ is a length scale, a thermal one. We set $\hbar=1$ and
$k_{B}=1$, so the parameter $\gamma$ has units of length to the power
$(d-2)(1-a)$, and in this case, incidentally, the $d$-dimensional constant of
gravitation $G$ has units of length to the power $d-2$. The equation of state
for the electric potential, Eq. (50), has no new free parameters. Both
equations of state have another parameter which is the dimension of spacetime
$d$. Being a special parameter, it can nonetheless be treated as a free one,
if one wishes. As long as the dimension $d$ is finite one can put the desired
dimension, be it $d=4$, $d=11$, or any other finite $d$, into the formulas to
obtain the corresponding expressions for the physical quantities for that
dimension. The infinite $d$ limit is a specific case that depends on how the
limit is taken and so requires special attention. Here, we are just interested
in finite $d$ however large it is.

###  III.7 The entropy formula

The differential equation for the entropy given in Eq. (47) can then be
integrated to yield $S=\frac{\gamma}{2\mu(d-2)\Omega}(\Omega r_{+}^{d-2})^{a}$
where we have chosen the constant of integration to be zero, and using Eq.
(16), it gives, $S=\frac{\gamma}{2\mu(d-2)\Omega}A_{+}^{a}$, or restoring the
constant of gravitation $G$ from Eq. (14), one has

$\displaystyle S=\ \frac{\gamma}{16\pi G}A_{+}^{a}\,,$ (51)

i.e., the entropy of the shell, a dimensionless quantity, is proportional to a
power of the gravitational area $A_{+}$. Due to the chosen equations of state,
namely Eqs. (49) and (50), the generic dependence of $S$ on $r_{+}$ and
$r_{-}$, $S=S(r_{+},r_{-})$, see Eq. (44), is now reduced to a dependence on
$r_{+}$ alone, $S=S(r_{+})$ or, adopting the gravitational area instead of the
gravitational radius as the variable for the entropy, one has $S=S(A_{+})$.
Furthermore, in Eq. (51), we should perhaps impose that $a>0$ so that the
entropy does not diverge in the no black hole limit $r_{+}\rightarrow 0$. Note
that here $A_{+}$ is not the event horizon area since there is no event
horizon, there is no black hole, there is only the spacetime gravitational
radius $r_{+}$.

We can now see the motivation for the choice of the reduced equations of state
given in Eqs. (49) and (50). It is twofold. First, power laws in
thermodynamics and statistical mechanics are ubiquitous, so it is natural to
take for the reduced equations of state $b(r_{+},r_{-})$ and $c(r_{+},r_{-})$
power laws in $r_{+}$ and $r_{-}$, which themselves are functions of $M$, $A$,
and $Q$. Second, there is the motivation that by choosing such equations of
state they give the possibility of taking the black hole limit $R=r_{+}$.
Thus, $b(r_{+},r_{-})$ given in Eq. (49) has that, for $a=1$, one gets a
functional dependence equal to the Hawking temperature of the black hole. The
equation for the reduced potential Eq. (50), is simply the same as the
corresponding black hole. These two choices yield in the $a=1$ case
$S=\frac{\gamma}{16\pi G}A_{+}$, see Eq. (51), i.e., an entropy for the shell
proportional to the gravitational radius, which has the same functional
dependence as the Bekenstein-Hawking black hole entropy. Note that other power
laws could be chosen. For instance, one could choose a power of Eq. (49)
itself and another different power of Eq. (50), and these equations would
still yield black hole features for the appropriate choice of the exponents.
Yet a different equation of state for the reduced inverse temperature $b$, is
to choose $b$ as a power law in the ADM mass, in which case it permits to
treat not only undercharged and extremal charged shells, but also overcharged
shells, see Appendix A for such a choice. Of course, other choices with
physical meaning can be thought of.

Another important point brought about by Eq. (51) is that as long as $r_{+}$
is fixed, the entropy is the same for any radius $R$ of the shell. To
understand the process involved we use Eq. (32), or better, the equation
before it, namely, $M=\frac{R^{d-3}}{\mu}\left(1-k\right)$, in full,
$M=\frac{R^{d-3}}{\mu}\left(1-\sqrt{\left(1-\left(\frac{r_{+}}{r}\right)^{d-3}\right)\left(1-\left(\frac{r_{-}}{r}\right)^{d-3}\right)}\right)$.
To simplify the discussion, put $d=4$ and $r_{-}=0$, i.e., $Q=0$. Then,
$M=R\left(1-\sqrt{1-\frac{r_{+}}{R}}\right)$. For $r_{+}$ fixed we see that
for $R=r_{+}$ one has $M=R=r_{+}$, and for $R\to\infty$ one has
$M=\frac{1}{2}\,r_{+}$, plus the derivative of $M$ in $R$ is strictly
negative. So, for fixed $r_{+}$, as $R$ increases the rest mass $M$ of the
shell decreases. In this process of changing the radius of the shell
maintaining $r_{+}$ fixed, one has, from Eq. (51), that the entropy does not
change. Since the size and the energy of the system change, one increases, the
other decreases, or vice versa, but the entropy does not change, one is in the
presence of an isentropic process.

###  III.8 Euler theorem

According to Eq. (29) together with (32), the rest mass $M$ can be written in
terms of $r_{+}$, $r_{-}$, and $R$, see also Eq. (33). Moreover, using Eqs.
(16) and (51) the gravitational radius, $r_{+}$, can be written in terms of
$S$ as $r_{+}=\frac{1}{\Omega^{\frac{1}{d-2}}}\left(\frac{16\pi
GS}{\gamma}\right)^{\frac{1}{a(d-2)}}$, then using Eq. (17) the Cauchy radius,
$r_{-}$, can be written in terms of $S$ and $Q$ as
$r_{-}=\frac{(qQ^{2})^{\frac{1}{d-3}}\Omega^{\frac{1}{d-2}}}{\left(\frac{16\pi
GS}{\gamma}\right)^{\frac{1}{a(d-2)}}}$, and finally using Eq. (26) $R$ can be
written in terms of $A$ as $R=\left(\frac{A}{\Omega}\right)^{\frac{1}{d-2}}$.
Substituting these latter three results into Eq. (32) together with (29),
i.e., $M=\frac{R^{d-3}}{\mu}\left(1-k\right)$, one has that the rest mass $M$
seen as a function of $S$, $A$, and $Q$, i.e. $M(S,A,Q)$, is given by
$M=\frac{1}{\mu}\left(\frac{A}{\Omega}\right)^{\frac{d-3}{d-2}}\left[1-\sqrt{\left(1-s_{1}\right)\left(1-s_{2}\right)}\right]$,
where we have defined $s_{1}=\left(\frac{16\pi GS}{\gamma
A^{a}}\right)^{\frac{d-3}{a(d-2)}}$ and
$s_{2}=\frac{qQ^{2}\Omega^{2\frac{d-3}{d-2}}}{\left(\frac{16\pi
GSA^{a}}{\gamma}\right)^{\frac{d-3}{a(d-2)}}}$. Now, from this expression for
$M(S,A,Q)$ one can see that $M\left(\lambda S^{\frac{1}{a}},\lambda A,\lambda
Q^{\frac{d-2}{d-3}}\right)=\lambda^{\frac{d-3}{d-2}}M\left(S^{\frac{1}{a}},A,Q^{\frac{d-2}{d-3}}\right)$,
for some arbitrary factor $\lambda$. Since the derivatives of $M$ are
described by the differential $dM=TdS-pdA+\Phi dQ$, i.e., the first law of
thermodynamics, one obtains by the Euler relation theorem for homogeneous
functions that

$\displaystyle\frac{d-3}{d-2}M=aTS-pA+\frac{d-3}{d-2}\Phi Q\,\,.$ (52)

This relation is an integrated version of the first law of thermodynamics for
the thin shell with the specific entropy given in Eq. (51).

###  III.9 Shell with black hole features, the black hole limit, and Smarr
formula

####  III.9.1 Shell with black hole features

To get a shell with black hole features we see that taking $a=1$ we obtain
from Eq. (49) that
$b(r_{+},r_{-})=\frac{\gamma}{d-3}\frac{r_{+}^{(d-2)}}{r_{+}^{d-3}-r_{-}^{d-3}}$,
so that $T_{0}$ defined as $T_{0}=\dfrac{1}{b}$ is given by
$T_{0}(r_{+},r_{-})=\frac{d-3}{\gamma}\frac{r_{+}^{d-3}-r_{-}^{d-3}}{r_{+}^{d-2}}$.
With the Planck length defined as $\ell_{p}=(\frac{\hbar
G}{c^{3}})^{\frac{1}{d-2}}$, and since here we have $c=1$ and $\hbar=1$, we
have $\ell_{p}=G^{\frac{1}{d-2}}$. Putting in addition $\gamma=4\pi$ one gets
$T_{0}(r_{+},r_{-})=\frac{d-3}{4\pi}\frac{r_{+}^{d-3}-r_{-}^{d-3}}{r_{+}^{d-2}}$,
which is the Hawking temperature $T_{+}$ for the matter on the shell. The
reduced electric potential is still given by Eq. (50),
$c(r_{+},r_{-})=\frac{1}{r_{+}^{d-3}}$. Thus, for the shell with black hole
features, one has that the reduced inverse temperature and electric potential
are given by

$\displaystyle\hskip-11.38092ptb_{+}(r_{+},r_{-})\hskip-1.42271pt=\hskip-1.42271pt\frac{4\pi}{d-3}\frac{r_{+}^{d-2}}{r_{+}^{d-3}-r_{-}^{d-3}},\;c_{+}(r_{+},r_{-})\hskip-1.42271pt=\hskip-1.42271pt\frac{1}{r_{+}^{d-3}},\hskip-8.5359pt$
(53)

where a subscript + are for quantities characteristic of black holes. Then,
the entropy of the shell given in Eq. (51) turns into

$\displaystyle S_{+}=\frac{1}{4}\frac{A_{+}}{A_{p}}\,,$ (54)

where $A_{p}$ is the Planck area defined as $A_{p}=l_{p}^{d-2}=G$. Thus, for
the shell’s matter equations of state given in Eqs. (49) and (50) with in
addition $a=1$ and $\gamma=4\pi$, one finds that the shell at radius $R$ and
with area $A$, has black hole features, it has precisely the Bekenstein-
Hawking entropy, as given in Eq. (54). So, thermodynamically, this spacetime
being not a black hole spacetime, rather it is a shell spacetime, actually
mimics thermodynamically the corresponding black hole spacetime, i.e., the
black hole that has the same gravitational radius $r_{+}$. Indeed, for any
radius $R$ greater than the shell’s gravitational radius $r_{+}$, $R\geq
r_{+}$, the shell’s entropy is always the same, it is the Bekenstein-Hawking
entropy.

####  III.9.2 Black hole limit and Smarr formula

To get a shell which not only has black hole features but is almost a black
hole, i.e., a quasiblack hole, we have to take the precise limit of the shell
radius $R$ going into the shell gravitational radius $r_{+}$, $R\rightarrow
r_{+}$. In this case, in order to not have divergences in the quantum state of
the matter and to maintain thermal equilibrium, the temperature of the shell
must be precisely the Hawking temperature,
$T_{+}(r_{+},r_{-})=\frac{d-3}{4\pi}\frac{r_{+}^{d-3}-r_{-}^{d-3}}{r_{+}^{d-2}}$,
in which case the entropy of the shell at its own gravitational radius has to
be Bekenstein-Hawking entropy, $S_{+}=\frac{1}{4}\frac{A_{+}}{A_{p}}$, see Eq.
(54). When the shell is at its own gravitational radius, the shell spacetime
is in a quasiblack hole state, the gravitational radius being now a
quasihorizon radius. This limit can be thought of as a sequence of quasistatic
thermodynamic equilibrium states of the shell that reach the equilibrium state
of the black hole. Note that the pressure in Eq. (30) diverges in this limit.
In a sense this means that all degrees of freedom are excited in this limit
and the entropy is maximal. Clearly the shell formalism, that provides an
exact solution for its dynamics and its thermodynamics, yields in the
appropriate limit the black hole features, notably, the Bekenstein-Hawking
entropy of a black hole. The quasiblack hole formalism, different from the
shell formalism and with some correspondence to the membrane paradigm
formalism, deals with generic matter systems on the verge of becoming a black
hole and is also able to bring out all the thermodynamic properties of black
holes Lemos:2010 ; Lemos:2011 ; lemoszaslavskii .

The extremal electrically charged black hole merits a complete investigation.
Here, we mention some important points connected to the entropy and
thermodynamics of an extremal Reissner-Nordström shell solution in
$d$-dimensions and the corresponding Reissner-Nordström black hole. The
extremal Reissner-Nordström spacetime obeys the relation $r_{+}=r_{-}$, and so
for a reduced equation of state of the form given in Eq. (49) one has that the
extremal charged shell case has zero temperature, whereas the reduced electric
potential still has the form given also in Eq. (50), and so both are well
defined in the extremal case. On the other hand, the entropy of such a shell
is a subtle issue. If from a nonextremal shell, with $R>r_{+}$ we take the
limit $r_{+}=r_{-}$, then one obtains by continuity directly that the entropy
for the shell is given by Eq. (51). On the other hand, if we start with an
extremal shell a priori then the entropy of the shell is some function of
$A_{+}$, $S(A_{+})$, that is not specified, i.e., one is free to choose it
Lemos:2015c . At the black hole limit in the extremal case, i.e., when the
radius $R$ of the shell approaches its gravitational radius $r_{+}=r_{-}$, and
the reduced equations of state are given in Eq. (53), the situation is even
more subtle. Besides the two possible cases similar to the two shell cases
just mentioned, namely, the shell is nonextremal and is then put to its
gravitational radius, and the shell is extremal and is then put to its
extremal radius, there is a third case when the shell is turning to being
extremal and simultaneously it is approaching its own gravitational radius
lemoshellextremal12 . The first case gives the Bekenstein-Hawking entropy for
the shell, $S_{+}=\frac{1}{4}\frac{A_{+}}{A_{p}}$, the second case gives that
the entropy is some unspecified function of $A_{+}$,
$S_{+}=S_{+}\left(\frac{A_{+}}{A_{p}}\right)$, and the third case gives again
the Bekenstein-Hawking entropy for the shell,
$S_{+}=\frac{1}{4}\frac{A_{+}}{A_{p}}$. Thus, if we take the entropy of an
extremal shell at its own gravitational radius as representative of the
entropy of an extremal black hole, then the entropy of an extremal black hole
depends on its past, specifically, on the way it was formed, see also
Lemos:2011  where the quasiblack hole approach is applied.

Now, we turn to the Smarr formula. It will be derived from the Euler relation
for the shell, Eq. (52), in the black hole limit. The shell with black hole
properties has $a=1$ and is given by Eqs. (53) and (54). Multiplying the Euler
relation, Eq. (52), by the factor $k$, one obtains for a shell with black hole
properties $\frac{d-3}{d-2}kM=T_{+}S_{+}-kpA+\frac{d-3}{d-2}k\Phi_{+}Q$, where
$\Phi_{+}$ is $\Phi$ defined in Eq. (46) with black hole characteristics,
i.e., $\Phi_{+}(r_{+},r_{-},R)=Q\frac{r_{+}^{-(d-3)}-R^{-(d-3)}}{k}$. One can
now take the black hole limit, $R=r_{+}$. Then the redshift function is zero,
$k=0$. This means that $kM=0$. One also has
$k\Phi_{+}(r_{+},r_{-},R)=Q\left({r_{+}^{-(d-3)}-R^{-(d-3)}}\right)_{R=r_{+}}=0$.
So, the nonzero terms are $T_{+}S_{+}$ and $-kpA$. Then using Eqs. (26) and
(30) for $-kpA$, we obtain putting $R=r_{+}$,
$0=T_{+}S_{+}-\frac{1}{2\mu}\frac{d-3}{d-2}(r_{+}^{d-3}-r_{-}^{d-3})$ or, upon
rearrangements,
$0=T_{+}S_{+}-\frac{1}{2\mu}\frac{d-3}{d-2}(r_{+}^{d-3}+r_{-}^{d-3})+\frac{1}{\mu}\frac{d-3}{d-2}r_{-}^{d-3}$.
From Eqs. (14) and (17) one has
$\frac{1}{2\mu}\left(r_{+}^{d-3}+r_{-}^{d-3}\right)=m$. The last term can be
written as $\frac{1}{\mu}\frac{d-3}{d-2}r_{+}^{-(d-3)}(r_{+}r_{-})^{d-3}$, and
from Eq. (17) this is
$\frac{d-3}{d-2}r_{+}^{-(d-3)}Q^{2}=\frac{d-3}{d-2}[Qr_{+}^{-(d-3)}]Q=\frac{d-3}{d-2}\phi_{+}Q$,
where the black hole potential $\phi_{+}$ is naturally defined as
$\phi_{+}=Qr_{+}^{-(d-3)}$. Then, the Euler relation becomes
$0=T_{+}S_{+}-\frac{d-3}{d-2}m+\frac{d-3}{d-2}\phi_{+}Q$, i.e.,

$\displaystyle m=\frac{d-2}{d-3}T_{+}S_{+}+\phi_{+}Q\,\,,$ (55)

which is the Smarr formula for a $d$-dimensional Reissner-Nordström, i.e., a
Reissner-Nordström-Tangherlini black hole. This relation is the integral
version of the first law of thermodynamics for black holes, which can be
picked up from the first law formula $dS=bdm-b\phi dQ$ or, swapping places,
$dm=T_{0}dS+\phi dQ$, with $T_{0}=\frac{1}{b}$, found after Eq. (47) for thin
shells, when applied to black holes. For the extremal case, $r_{+}=r_{-}$, one
has $T_{+}=0$ and
$\phi_{+}Q=Qr_{+}^{-(d-3)}\,Q=\frac{Q}{\sqrt{q}}=\frac{Q}{\sqrt{\mu}}$, where
Eq. (17) has been used, and the equality $\mu=q$ in our convention of units
has been applied. Thus, for the extremal case, the Smarr formula given in Eq.
(55) turns into $\sqrt{\mu}m=Q$ as it should be. Now, in four dimensions,
$d=4$, Eq. (55) gives $m=2T_{+}S_{+}+\phi_{+}Q$ which is the original Smarr
formula for a four-dimensional Reissner-Nordström black hole. Still in $d=4$,
the extremal case gives $\sqrt{G}m=q$, or $m=q$ if one puts $G=1$, which is
the usual mass formula for an extremal four-dimensional Reissner-Nordström
black hole.

##  IV Intrinsic thermodynamic stability for the given equations of state

###  IV.1 Stability conditions

The shell with generic equations of state given in Eqs. (39)-(41) will have
its thermodynamic equilibrium state for some entropy $S(M,A,Q)$ found from the
first law of thermodynamics Eq. (36). We now look at the intrinsic
thermodynamic stability of the shell, see Callen’s thermodynamics book for the
formalism.

In general, a system in thermodynamic equilibrium is susceptible to
perturbations. Let the system with entropy $S$ be split into two subsystems.
Then, the fluctuations of the matter in the boundary between the subsystems
will allow exchanges in the thermodynamic variables, in this case $(M,A,Q)$.
The entropy of the system after those exchanges, $S+\Delta S$, will be the sum
of the entropy of the two subsystems. By the second law of thermodynamics, if
$S+\Delta S\leq S$, i.e., $\Delta S\leq 0$, then the system will stay in
equilibrium, hence the system is stable. Otherwise, the system will evolve
away from equilibrium, building up inhomogeneities, and therefore the
equilibrium is unstable. For very small fluctuations, the conditions of
intrinsic stability are given by $dS(M,A,Q)=0$ and $d^{2}S(M,A,Q)\leq 0$,
i.e., $S$ is a maximum with the Hessian of $S$ being seminegative definite.
Notice that in general the quantities $(M,A,Q)$ do not have a relation between
themselves. However, in our case there is a relation between those quantities
since first, $S$ is solely a function of $r_{+}$, and so the equilibrium
configurations are given by $r_{+}(M,A,Q)$, and second, since the condition
$dS(M,A,Q)=0$ holds it implies that $r_{+}(M,A,Q)={\rm constant}$, so
$(M,A,Q)$ are tied by a relation between themselves.

The stability conditions with respect to the second derivatives of $S$,
denominated by $S_{h_{i}h_{j}}=\frac{\partial^{2}S}{\partial h_{i}\partial
h_{j}}$, are

$\displaystyle S_{MM}\leq 0\,,\,\,S_{AA}\leq 0\,,\,\,S_{QQ}\leq 0\,,$
$\displaystyle S_{MM}S_{AA}-S_{MA}^{2}\geq 0\,,$ $\displaystyle
S_{MM}S_{QQ}-S_{MQ}^{2}\geq 0\,,$ $\displaystyle S_{QQ}S_{AA}-S_{QA}^{2}\geq
0\,,$ $\displaystyle(S_{MM}S_{AQ}-S_{MA}S_{MQ})^{2}$
$\displaystyle-(S_{AA}S_{MM}-S_{AM}^{2})(S_{QQ}S_{MM}-S_{QM}^{2})\leq 0\,,$
(56)

which have to be employed with care for each appropriate physical situation as
it is detailed below. Note that there is a freedom on the choice of sufficient
conditions for each physical situation, which depends on the order of the
variables that one chooses. Here, we are choosing the order $h_{1}=M$,
$h_{2}=A$ and $h_{3}=Q$. The derivation of these conditions and the
explanation of the redundancy of these conditions are present in the Appendix
B.

###  IV.2 Entropy and equations of state

Now, we apply the formalism above. For that, we rewrite Eq. (51) for the
entropy as

$\displaystyle S(M,A,Q)=\frac{\gamma}{16\pi G}A_{+}^{a}\,,$ (57)

to emphasize that we are dealing with the variables $M$, $A$, and $Q$.
Clearly, $A_{+}$ is a function of $M$, $A$, and $Q$, since $A_{+}$ is a
function of $r_{+}$, see Eq. (16), which is a function of $M$, $A$, and $Q$
through Eqs. (15), (33), and (26).

There are three equations of state that must be provided, one for the
temperature, one for the pressure, and one for the electric potential. These
already have been found in the previous section.

For the temperature, or better, for the inverse temperature, one has Eq. (45),
and using the specific choice of the reduced equation of state given in Eq.
(49), one finds the explicit form of the generic equation given in Eq. (39),
namely,

$\displaystyle\beta(M,A,Q)=\frac{a\gamma\Omega^{a-1}}{d-3}\frac{r_{+}^{a(d-2)}}{r_{+}^{d-3}-r_{-}^{d-3}}k\,,$
(58)

where clearly it is a function of $M$, $A$, and $Q$.

For the thermodynamic pressure, the equation of state is given by Einstein
equations, so it is also a dynamic pressure. Then, Eq. (30) is indeed the
explicit form of the generic equation given in Eq. (40), namely,

$\displaystyle
p(M,A,Q)=\frac{1}{2\mu\Omega}\frac{d-3}{d-2}\Bigg{[}(1-k)^{2}R^{2(d-3)}-qQ^{2}\Bigg{]}\frac{1}{R^{2d-5}k},$
(59)

where clearly it is a function of $M$, $A$, and $Q$.

For the potential, one has Eq. (46), and using the specific choice for the
reduced equation of state given in Eq. (50) one finds the explicit form of the
generic equation given in Eq. (41), namely,

$\displaystyle\Phi(M,A,Q)=Q\left(\frac{1}{r_{+}^{d-3}}-\frac{1}{R^{d-3}}\right)\frac{1}{k}\,,$
(60)

where clearly it is a function of $M$, $A$, and $Q$.

###  IV.3 First and second derivatives of the entropy

For the equations of state we use, the final form of the entropy of the shell
is given in Eq. (57). Then, the first derivatives of the entropy can be
computed either directly, or more easily through the first law given in Eq.
(36) together with Eqs. (58)-(60). They are

$\displaystyle
S_{M}=\frac{a\gamma\Omega^{a-1}}{d-3}\frac{r_{+}^{a(d-2)}}{r_{+}^{d-3}-r_{-}^{d-3}}k\,,$
$\displaystyle
S_{A}=\frac{a\gamma\Omega^{a-2}r_{+}^{a(d-2)}\Big{[}(1-k)^{2}R^{2(d-3)}-qQ^{2}\Big{]}}{2\mu(d-2)R^{2d-5}(r_{+}^{d-3}-r_{-}^{d-3})}\,,$
$\displaystyle
S_{Q}=-\frac{a\gamma\Omega^{a-1}Q}{{(d-3)}}\left(\frac{r_{+}^{3-d}-R^{3-d}}{r_{+}^{d-3}-r_{-}^{d-3}}\right)r_{+}^{a(d-2)}\,.$
(61)

For the calculation of the second derivatives of the entropy, it is useful to
consider that $\partialderivative{r_{\pm}}{M}=\pm
2\mu\frac{r_{\pm}\,k}{(d-3)(r_{+}^{d-3}-r_{-}^{d-3})}$,
$\partialderivative{r_{\pm}}{R}=\pm\mu\frac{r_{\pm}}{r_{+}^{d-3}-r_{-}^{d-3}}\frac{\mu
M^{2}-Q^{2}}{R^{d-2}}$, and
$\partialderivative{r_{\pm}}{Q}=\mp\frac{2qQr_{\pm}\left(r_{\pm}^{3-d}-R^{3-d}\right)}{(d-3)(r_{+}^{d-3}-r_{-}^{d-3})}$.
The components of the Hessian are then

$\displaystyle S_{MM}=\frac{a\gamma\Omega^{a-2}8\pi
Gr_{+}^{a(d-2)}}{(d-3)(d-2)(r_{+}^{d-3}-r^{d-3}_{-})R^{d-3}}\,S_{1},$
$\displaystyle
S_{AA}=\frac{a\gamma\Omega^{a-3}r_{+}^{a(d-2)}}{2\mu(d-2)^{2}(r_{+}^{d-3}-r_{-}^{d-3})R^{d-1}}\,S_{2}\,,$
$\displaystyle
S_{QQ}=\frac{a\gamma\Omega^{a-1}r_{+}^{a(d-2)}(1-x)}{{(d-3)}(r_{+}^{d-3}-r_{-}^{d-3})r_{+}^{d-3}}\,S_{3}\,,$
$\displaystyle
S_{MA}=\frac{a\gamma\Omega^{a-2}r_{+}^{a(d-2)}}{(d-2)(r_{+}^{d-3}-r_{-}^{d-3})R^{d-2}}\,S_{12}\,,$
$\displaystyle S_{MQ}=-\frac{2\mu
a\gamma\Omega^{a-1}r_{+}^{a(d-2)}Qk}{(d-3)^{2}(r_{+}^{d-3}-r_{-}^{d-3})r_{+}^{2d-6}}\,S_{13}\,,$
$\displaystyle
S_{AQ}=-\frac{a\gamma\Omega^{a-2}r_{+}^{a(d-2)}Q}{{(d-2)}(r_{+}^{d-3}-r_{-}^{d-3})r_{+}^{d-3}R^{d-2}}\,S_{23}\,,$
(62)

where

$\displaystyle S_{1}=\frac{2k^{2}\mathcal{G}}{(d-3)x}-1\,,$ $\displaystyle
S_{2}=\mathcal{F}\left[\frac{\mathcal{F}\mathcal{G}}{x}-2d+5\right]\,,$
$\displaystyle
S_{3}=-1+\frac{2y}{d-3}\left[\mathcal{G}(1-x)-\frac{2(d-3)}{1-y}\right]\,,$
$\displaystyle S_{12}=1-k+\frac{k\mathcal{G}}{x(d-3)}\mathcal{F}\,,$
$\displaystyle S_{13}=\mathcal{G}(1-x)-\frac{(d-3)}{1-y}\,,$ $\displaystyle
S_{23}=x+\frac{\mathcal{F}}{x(d-3)}\Bigg{[}\mathcal{G}(1-x)-\frac{(d-3)}{1-y}\Bigg{]}\,,$

with the auxiliary functions $\mathcal{G}$, $\mathcal{F}$, and $k$, being
given by

$\displaystyle\mathcal{G}=\frac{1}{1-y}\Bigg{[}a(d-2)-(d-3)\frac{1+y}{1-y}\Bigg{]}\,,$
$\displaystyle\mathcal{F}=2-2k{-}x(1-y)\,,$ $\displaystyle
k=\sqrt{(1-x)(1-xy)}\,,$ (64)

and we have made use of the definitions

$\displaystyle x=\frac{r_{+}^{d-3}}{R^{d-3}}\,,\quad\quad
y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}\,.$ (65)

The set of inequalities in Eq. (56) with the entropy equation given in Eq.
(57) and the equations of state given in Eqs. (58)-(60) can be written as
restricting conditions in terms of the functions given in Eq.
(LABEL:eq:Sfunctions). The conditions will then restrict the parameter space
described by the points $(d,a,x,y)$ constrained by

$\displaystyle d\geq 4\,,\quad a>0\,,\quad 0<x<1\,,\quad 0<y<1\,.$ (66)

Here, $d\geq 4$ since for lower $d$ there is no proper Reissner-Nordström
solution, $a>0$ because in the no black hole limit, $A_{+}=0$, i.e.,
$r_{+}=0$, the entropy expression, Eq. (57), should not diverge, $0<x<1$
because the shell has to be in the limits between no shell, $x=0$, and the
black hole state, $x=1$, $0<y<1$ because the electric charge state of the
shell considered here can run from an uncharged one, $y=0$, to an extremally
charged one, $y=1$, overcharged shells are not treated here since the
equations of state, Eqs. (58)-(60), do not apply to overcharged shells.

In what follows we deal with the algebraic conditions that arise from the
conditions given in Eq. (62) together with the auxiliary functions Eqs.
(LABEL:eq:Sfunctions)-(65). In the Appendix C we make plots to help in the
understanding of these conditions.

###  IV.4 Mass fluctuations only

A shell with only mass fluctuations will have the stability condition given by
$S_{MM}\leq 0$, see Eq. (56). For the equations of state we are using, and
with the help of Eq. (62), one has that $S_{MM}\leq 0$ can be written as

$\displaystyle S_{1}\leq 0\,.$ (67)

Then, from Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle
a\leq\frac{x(d-3)(1-y)}{2(d-2)k^{2}}+\frac{(d-3)}{(d-2)}\frac{(1+y)}{(1-y)}\,,$
(68)

where Eqs. (64) and (65) have been used. From a quick analysis, the right-hand
side tends to infinity at the points $x=1$ or $y=1$. It has its minimum value
at $(x,y)=(0,0)$, corresponding to $a=\frac{d-3}{d-2}$. A detailed analysis of
Eq. (68) can be seen in Fig. 1 which is itself split into four plots (a), (b),
(c), and (d). It is interesting to comment on the case of the shell with
thermodynamic black hole features, i.e., the case with $a=1$. For an uncharged
shell, $y=0$, the range of $x$ for thermodynamic stability is given by
$\frac{2}{d-1}<x<1$, in agreement with Lemos:2010 . Increasing the value of
$y$ will also increase the range of $x$ for thermodynamic stable
configurations, i.e., if the shell has more electric charge then a higher
radius $R$ is allowed for stability. The stability is guaranteed in the full
range of $x$ if $y\geq\frac{1}{2d-5}$, see also Fig. 1(d) top for this $a=1$
case. It is also interesting to see the stability with respect to the
variables $\frac{M}{R^{d-3}}$ and $\frac{Q}{R^{d-3}}$. We do this below and
one can also refer to Fig. 1(d) bottom.

###  IV.5 Area fluctuations only

A shell with only area fluctuations will have the stability condition given by
$S_{AA}\leq 0$, see Eq. (56). For the equations of state we are using, and
with the help of Eq. (62), one has that $S_{AA}\leq 0$ can be written as

$\displaystyle S_{2}\leq 0\,.$ (69)

Then, from Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle
a\leq\frac{(2d-5)x(1-y)}{(d-2)\mathcal{F}}+\frac{(d-3)}{(d-2)}\frac{(1+y)}{(1-y)}\,,$
(70)

where Eqs. (64) and (65) have been used, and employed the fact that the
multiplication factor $\mathcal{F}$ is always positive for $0<x<1$ and
$0<y<1$, it is also proportional to $M-m$. The right-hand side of Eq. (70) has
the minimum at $(x=1,y=0)$, with the value $a=3-\frac{2}{d-2}$. The function
then increases in the direction of $x\rightarrow 0$ or $y\rightarrow 1$, where
it tends to infinity. A detailed analysis of Eq. (70) can be seen in Fig. 2
which is itself split into three plots (a), (b), and (c). The case of the
shell with thermodynamic black hole features, i.e., the case with $a=1$, does
not need a more detailed analysis since all the configurations with $a=1$ are
below the surface of marginal stability, therefore they are stable to these
thermodynamic perturbations.

###  IV.6 Charge fluctuations only

A shell with only electric charge fluctuations will have the stability
condition given by $S_{QQ}\leq 0$, see Eq. (56). For the equations of state we
are using, and with the help of Eq. (62), one has that $S_{QQ}\leq 0$ can be
written as

$\displaystyle S_{3}\leq 0\,.$ (71)

Then, from Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle a\leq\frac{(d-3)(1-y)}{2(d-2)y(1-x)}+\frac{2(d-3)}{(d-2)(1-x)}$
$\displaystyle+\frac{(d-3)}{(d-2)}\frac{(1+y)}{(1-y)}\,,$ (72)

where Eqs. (64) and (65) have been used. The right-hand side of Eq. (72)
describes a concave surface, faced to $a\rightarrow+\infty$. The minimum,
restricted to the parameter space, resides in
$\left(x=0,y=\frac{1}{3}\right)$, where its value is $a=5\frac{d-3}{d-2}$. It
diverges to infinity at the axes $x=1$, $y=0$ and $y=1$. A detailed analysis
of Eq. (72) can be seen in Fig. 3 which is itself split into three plots (a),
(b), and (c). The case of the shell with thermodynamic black hole features,
i.e., the case with $a=1$, does not need a more detailed analysis since all
the configurations with $a=1$ are below the surface of marginal stability,
therefore they are stable to these thermodynamic perturbations.

###  IV.7 Mass and area fluctuations together

A shell with mass and area fluctuations will have the stability conditions
given by $S_{MM}\leq 0$, $S_{AA}\leq 0$, and $S_{MM}S_{AA}-S_{MA}^{2}\geq 0$,
see Eq. (56). Note, however, that there is redundancy on this system of
inequations, see Appendix B. The sufficient conditions can be chosen to be
$S_{MM}\leq 0$ and $S_{MM}S_{AA}-S_{MA}^{2}\geq 0$. For the equations of state
we are using, one has that $S_{MM}\leq 0$ yields Eq. (68), and with the help
of Eq. (62) one has that $S_{MM}S_{AA}-S_{MA}^{2}\geq 0$ can be written as

$\displaystyle S_{4}=-\frac{1}{2(d-3)}S_{1}S_{2}+S_{12}^{2}\leq 0\,.$ (73)

Then, from Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle
a\leq\frac{(1-y)x\Big{(}(d-\frac{5}{2})\mathcal{F}-(d-3)(1-k)^{2}\Big{)}}{(d-2)\mathcal{F}\Big{(}\frac{k^{2}}{d-3}+2k+\frac{\mathcal{F}}{2}\Big{)}}$
$\displaystyle+\frac{(d-3)}{(d-2)}\frac{(1+y)}{(1-y)}\,,$ (74)

where Eqs. (64) and (65) have been used. The right-hand side of Eq. (74) is
minimum at $x=1$, where $a=1$. From there toward $x=0$, the function will bend
into $a=\frac{d-3}{d-2}\frac{1+y}{1-y}$. At $y=1$, it will tend to infinity.
When compared with Eq. (68), the right-hand side of Eq. (74) is always lower
and thus Eq. (74) is sufficient to describe the stability region, in this
case. A detailed analysis of Eq. (74) can be seen in Fig. 4 which is itself
split into four plots (a), (b), (c), and (d). The case of the shell with
thermodynamic black hole features, i.e., the case with $a=1$, shows that
increasing the value of $y$ will decrease the range of $x$ for thermodynamic
stable configurations, i.e., if the shell has more electric charge then it
needs to have lower $R$ for stability, see also Fig. 1(d) for this $a=1$ case.

###  IV.8 Mass and charge fluctuations together

A shell with mass and charge fluctuations will have the stability conditions
given by $S_{MM}\leq 0$, $S_{QQ}\leq 0$, and $S_{MM}S_{QQ}-S_{MQ}^{2}\geq 0$,
see Eq. (56). Note, however, that there is redundancy on this system of
inequations, see Appendix B. The sufficient conditions can be chosen to be
$S_{MM}\leq 0$ and $S_{MM}S_{QQ}-S_{MQ}^{2}\geq 0$. For the equations of state
we are using, one has that $S_{MM}\leq 0$ yields Eq. (68), and with the help
of Eq. (62) one has that $S_{MM}S_{QQ}-S_{MQ}^{2}\geq 0$ can be written as

$\displaystyle S_{5}=-x(1-x)S_{1}S_{3}+\frac{4yk^{2}}{(d-3)^{2}}S_{13}^{2}\leq
0\,.$ (75)

Then, from Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle a\leq\frac{(d-3)}{2(d-2)}\,\frac{2-x(1+y)}{1-x}\,,$ (76)

where Eqs. (64) and (65) have been used. The condition given in Eq. (76) is
sufficient to describe the stability, since the right hand side of it is lower
than the condition given by Eq. (68), in the respective parameter space. The
inequality is quite simple enough for analytical treatment. The function set
by the right-hand side at $x=0$ or $y=1$ takes the value $a=\frac{d-3}{d-2}$.
At $x=1$, the function diverges to infinity. Thus, the function bends from a
constant value to $a=\frac{d-3}{2(d-2)}\,\frac{2-x}{1-x}$, going from $y=1$ to
$y=0$. A detailed analysis of Eq. (76) can be seen in Fig. 5 which is itself
split into four plots (a), (b), (c), and (d). The case of the shell with
thermodynamic black hole features, i.e., the case with $a=1$, shows that
increasing the value of $y$ will decrease the range of $x$ for thermodynamic
stable configurations, i.e., if the shell has more electric charge, then it
must have lower radius $R$ for stability, see also Fig. 1(d) for this $a=1$
case.

###  IV.9 Area and charge fluctuations together

A shell with area and charge fluctuations will have the stability conditions
given by $S_{AA}\leq 0$, $S_{QQ}\leq 0$, and $S_{AA}S_{QQ}-S_{AQ}^{2}\geq 0$,
see Eq. (56). Note, however, that there is redundancy on this system of
inequations, see Appendix B. The sufficient conditions can be chosen to be
$S_{AA}\leq 0$ and $S_{AA}S_{QQ}-S_{AQ}^{2}\geq 0$. For the equations of state
we are using, one has that $S_{AA}\leq 0$ yields Eq. (70), and with the help
of Eq. (62) one has that $S_{AA}S_{QQ}-S_{AQ}^{2}\geq 0$ can be written as

$\displaystyle S_{6}=-\frac{(1-x)}{2(d-3)}S_{2}S_{3}+xyS_{23}\leq 0\,.$ (77)

Then, from Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle
a\leq\frac{\frac{(1-x)\mathcal{F}(2d-5)}{2(d-3)}(1+3y)-x^{3}y(1-y)+2\mathcal{F}xy-\frac{y\mathcal{F}^{2}}{x(1-y)}}{(d-2)(1-x)\Big{(}\frac{\mathcal{F}^{2}}{2x(d-3)}+\frac{2d-5}{(d-3)^{2}}y(1-x)\mathcal{F}+\frac{2\mathcal{F}xy}{(d-3)}\Big{)}}$
$\displaystyle+\frac{(d-3)}{(d-2)}\,\frac{1+y}{1-y}\,.$ (78)

where Eqs. (64) and (65) have been used. The condition given in Eq. (78) is
sufficient to describe the stability, since the right hand side of it is lower
than the condition given by Eq. (70), in the respective parameter space. At
$y=0$, the function set by the right-hand side intersects $S_{2}$. The
function then grows without bound at $(x=0,y=0)$ or $y=1$. In the limit of
$x\rightarrow 1$, the function approaches the value of
$a=\frac{8+6y-3d(1+y)}{(d-2)(1+3y)}$. At $x=0$, the right-hand side approaches
$S_{3}$ from below. A detailed analysis of Eq. (78) can be seen in Fig. 6
which is itself split into three plots (a), (b), and (c). The case of the
shell with thermodynamic black hole features, i.e., the case with $a=1$, does
not need a more detailed analysis since all the configurations with $a=1$ are
below the surface of marginal stability, therefore they are stable to these
thermodynamic perturbations.

###  IV.10 Mass, area, and charge fluctuations altogether

A shell with mass, area, and charge fluctuations will have the stability
conditions given by all the inequalities in Eq. (56). Note, however, that
there is redundancy on this system of inequations, see Appendix B. The
sufficient conditions can be chosen to be $S_{MM}\leq 0$,
$S_{MM}S_{AA}-S_{MA}^{2}\geq 0$, and
$(S_{MM}S_{AQ}-S_{MA}S_{MQ})^{2}-(S_{AA}S_{MM}-S_{AM}^{2})(S_{QQ}S_{MM}-S_{QM}^{2})\leq
0$. For the equations of state we are using, one has that $S_{MM}\leq 0$
yields Eq. (68), $S_{MM}S_{AA}-S_{MA}^{2}\geq 0$ yields Eq. (74), and with the
help of Eq. (62) one has that
$(S_{MM}S_{AQ}-S_{MA}S_{MQ})^{2}-(S_{AA}S_{MM}-S_{AM}^{2})(S_{QQ}S_{MM}-S_{QM}^{2})\leq
0$ can be written as

$\displaystyle
S_{7}=\left(xS_{1}S_{23}-\frac{2k}{d-3}S_{12}S_{13}\right)^{2}y-S_{4}S_{5}\leq
0\,.$ (79)

Then, even though Eq. (79) appears to be a polynomial on $a$ of degree 4, from
Eq. (LABEL:eq:Sfunctions) this inequality can be rearranged as

$\displaystyle
a\leq\frac{d-3}{d-2}\left(\frac{4-4k+x^{2}(d(1-y)^{2}+C)}{4-4k+x^{2}d(1-y)^{2}+xD}\right)\,,$
(80)

where $C=2x(1+y)(k-2)-2-2(y-4)y$, and $D=4k-2y-6-x\left(1+y(3y-8)\right)$, and
Eqs. (64) and (65) have been used. The right-hand side of Eq. (80) when
compared with the conditions in Eqs. (68) and (74) assumes always lower
values, in the respective parameter space, therefore Eq. (80) is the
sufficient condition of stability. The equality in the condition given in Eq.
(80) has its lowest value of $a=\frac{d-3}{d-2}$ at $x=0$, for every $y$. It
then increases toward $x=1$, where the limit gives $a=1$. At the limit of
$y=1$, the equality is given by the lowest value of $a=\frac{d-3}{d-2}$ for
every $x$ except $x=1$ where the limit gives $a=1$. Thus, the condition for
stability in Eq. (80) implies that every configuration with
$a\leq\frac{d-3}{d-2}$ is stable. On the other hand, for $\frac{d-3}{d-2}<a<1$
the stability region decreases with increasing $y$, being zero in the limit of
$y=1$. This means that shells with more electric charge will have less
configurations of stability. The space of stable configurations in the $a-d$
plane can also be made and is similar to the analysis made for the uncharged
case in Andre:2019 . A detailed analysis of Eq. (80) can be seen in Fig. 7
which is itself split into three plots (a), (b), and (c). The case of the
shell with thermodynamic black hole features, i.e., the case with $a=1$, does
not need a more detailed analysis since all the configurations with $a=1$ are
above the surface of marginal stability, hence unstable, except for the points
with $x=1$ which lie on the limit of the surface, hence marginally stable. In
the black hole limit, i.e., not only $a=1$ but also $x=1$, the configurations
for every value of $y$ are marginally stable.

###  IV.11 Further comments on the behavior of intrinsic stability with $a$

We now make some important comments, leftovers from the previous sections.

When discussing mass fluctuations only, Sec. IV.4, we mentioned that one can
make a corresponding stability analysis in terms of the variables
$\frac{M}{R^{d-3}}$ and $\frac{Q}{R^{d-3}}$, instead of
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$ and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$ of Eq.
(65). The analysis in $\frac{M}{R^{d-3}}$ and $\frac{Q}{R^{d-3}}$ yields some
interesting insight. The condition given in Eq. (68) becomes

$\displaystyle
a\leq\frac{\mu}{2}\frac{(d-3)}{(d-2)}\frac{\left(\frac{Q^{2}}{R^{2(d-3)}}+\frac{\mu
M^{2}}{R^{2(d-3)}}-\frac{2M}{R^{d-3}}\right)}{\left(1-\frac{\mu
M}{R^{d-3}}\right)^{2}}$ $\displaystyle\times\frac{\left(\frac{2\mu
M}{R^{d-3}}+\frac{\mu(Q^{2}-M^{2}\mu^{2})}{R^{2(d-3)}}-2\right)}{\sqrt{\frac{\mu(\mu
M^{2}-Q^{2})}{R^{2(d-3)}}\left(\left(2-\frac{\mu
M}{R^{d-3}}\right)^{2}-\frac{\mu Q^{2}}{R^{2(d-3)}}\right)}}\,.$ (81)

The possible physical values of $\frac{M}{R^{d-3}}$ and $\frac{Q}{R^{d-3}}$
are restricted by the condition of subextremality, i.e., $\sqrt{\mu}M>Q$, and
the condition of no trapped surfaces, i.e., $\frac{r_{+}}{R}<1$. One finds
from Eq. (81) that for small values of $\frac{M}{R^{d-3}}$, the shell needs
some minimum charge $Q$, or correspondingly a minimum value of
$\frac{Q}{R^{d-3}}$, for it to be stable. When $\frac{M}{R^{d-3}}$ has a value
that corresponds to $x=\frac{2}{d-1}$, the minimum charge for the shell to be
stable reaches zero, which means $y=0$. For higher $\frac{M}{R^{d-3}}$, the
region of stability is restricted by the physically possible values, namely,
$\frac{\sqrt{\mu}M}{R^{d-3}}>\frac{Q}{R^{d-3}}$ and $\frac{r_{+}}{R}<1$. Thus,
in brief, for $\frac{M}{R^{d-3}}$ small, thermodynamic stability exists only
for sufficiently large electric charge. For $\frac{M}{R^{d-3}}$ having a value
such that $x=\frac{2}{d-1}$ when $y=0$, i.e., $Q=0$, the shell is marginally
stable. Here, it is important to note that the value of $x=\frac{2}{d-1}$ for
$Q=0$ means that the shell is at the photonic orbit . For higher values of
$x$, maintaining $Q=0$, the shell is inside the photonic orbit, and it is
stable. This means that for $\frac{M}{R^{d-3}}$ yielding values higher than
$x=\frac{2}{d-1}$ when $y=0$, i.e, $Q=0$, the shell is thermodynamically
stable, see also Andre:2019 . This latter behavior, i.e., the behavior for
$\frac{M}{R^{d-3}}$ yielding values of $x$ equal or higher than
$x=\frac{2}{d-1}$, is precisely the same behavior of the large black hole in
the canonical ensemble found by York York:1986 ; can  and generalized to $d$
dimensions in andrelemos1 ; andrelemos2 . For higher values of
$\frac{M}{R^{d-3}}$ increasing the electrically charge $Q$, and so essentially
increasing $\frac{Q}{R^{d-3}}$, does not alter the stability, the solutions
are all thermodynamically stable. The result can be interpreted heuristically.
To understand it, note that the reduced inverse temperature $b$, can be
envisaged as a length scale, a thermal one. The inverse temperature $b$ here
is the one given in Eq. (49). For small $\frac{M}{R^{d-3}}$ and $Q=0$ one has
that the shells have radii higher than the photonic orbit and are
thermodynamically unstable. What happens is that the thermal length $b$ being
proportional to $M$, still in the uncharged case, is smaller than or of the
order of the radius of the shell, and thus the shell loses energy and mass
along these thermal lengths. Losing mass, means that the thermal length $b$
decreases, and so the process is a runaway process and thus unstable. If the
charge $Q$ increases, or more correctly if the ratio $\frac{Q}{R^{d-3}}$
increases, the thermal length $b$ gets correspondingly higher, and for a
certain sufficiently high electric charge $Q$, or better for a sufficiently
high $\frac{Q}{R^{d-3}}$, $b$ is now sufficiently greater than the radius of
the shell, so that it is not possible to lose energy anymore. Thus, the
electric shell is stable for charges higher than this minimum electric charge.
For higher electric charge, i.e., higher $\frac{Q}{R^{d-3}}$, such that one is
near the extremal limit, one has that $b$ is proportional to
$\frac{1}{\sqrt{M-Q}}$ and so it is indeed divergingly larger than $R$. For
$Q=0$ and a value of $\frac{M}{R^{d-3}}$ such that $x=\frac{2}{d-1}$, one has
a shell with radius equal to the photonic orbit. In this case the thermal
length $b$, as the calculations show, is barely sufficiently to not allow
thermal loss from the shell, and so maintain the shell in thermodynamic
equilibrium. For higher $\frac{M}{R^{d-3}}$ and $Q=0$, the shell is inside the
photonic orbit, and the thermal length $b$ is now sufficiently large
relatively to the radius of the shell to not allow thermal loss from the
shell, and this holds even truer for higher $Q$, i.e., higher
$\frac{Q}{R^{d-3}}$, where $b$ gets even larger and thus the shell in all
these cases is thermodynamically stable. The comments made here for generic
dimensions $d$, apply to the $d=4$ electric charged case studied in
Lemos:2015a  and are exemplified for $d=5$ in Fig. 1(d) bottom.

Another important point, is that in Sec. IV.10 we have pointed out that for
mass, area, and charge fluctuations altogether, shells with more electric
charge will have less configurations of stability. This behavior differs from
the case of mass fluctuations only of Sec. IV.4 that was also commented in the
previous paragraph, where, for certain configurations, more electric charge
aids to the stability. There is no contradiction between the two cases. The
mass, area, and charge fluctuations altogether is much more restrictive than
the mass fluctuations only case, in the sense that stable points in the former
fluctuations are also stable points in the latter fluctuations, but the
converse is not true.

There is still another point worth noting. In the case of one or two fixed
quantities, Secs. IV.4-IV.9, there are shell configurations with $a\geq 1$
that are stable. But one notices that the higher the $a$ the higher the
entropy $S$ since it goes with a power of $a$. For instance, for the area
fluctuations only of Sec. IV.5, we have seen that Eq. (70) has its minimum at
$\frac{r_{+}}{R}=1$ for zero charge, i.e., $(x=1,y=0)$, with the value for $a$
given by $a=3-\frac{2}{d-2}$. Since values of $a=3-\frac{2}{d-2}$ are always
greater than one, this could mean that a shell with lower $a$ would tend to
settle into a shell with higher $a$ since the latter would have higher
entropy. One could think that a change of $a$ could be achieved by some
rearrangement of the material on the shells and in this way higher entropies
could be attained. However, the stability analysis performed is for fixed $a$,
since the very exponent $a$ gives a precise temperature equation of state for
the matter, and to treat changes in the exponent $a$ one would have perhaps to
envisage some type of phase transition.

Having worked out the thermodynamic stability criterion for all types of
fluctuation in Secs. IV.4-IV.10 through the parameter $a$, it begs now the
question of what is the physical reason for the behavior of the intrinsic
stability with $a$ itself. We now turn into this point.

##  V Intrinsic thermodynamic stability in laboratory variables

###  V.1 The rational to introduce laboratory variables

It begs now the question of what is the physical reason for the behavior of
intrinsic stability with $a$. In order to understand the physical meaning of
the intrinsic thermodynamic stability associated to this self-gravitating thin
shell, we rewrite the stability conditions with variables that can be measured
in the laboratory. One of these variables that we are going to define gives a
good example of the way the stability conditions get clearer when written in
terms of thermodynamic coefficients. The heat capacity at constant area and
charge, $C_{A,Q}$, can be defined as $C_{A,Q}^{-1}=\left(\frac{\partial
T}{\partial M}\right)$. This variable is important since we also have
$S_{MM}=-\beta^{2}C_{A,Q}^{-1}\leq 0$, and so the stability condition for
changes in proper mass only is that the heat capacity $C_{A,Q}$ is positive.
The aim is to generalize this reasoning for two types of fluctuations which
seem the most interesting, namely, the mass and charge fluctuations studied in
Sec. IV.8, and mass, area, and charge fluctuations studied in Sec. IV.10.

###  V.2 Laboratory variables for mass and charge fluctuations together

Here we discuss the new laboratory variables for mass and charge fluctuations,
see Sec. IV.8. It emerges that the heat capacities at fixed area play an
important role when treating mass and charge fluctuations together. There are
two such heat capacities, namely, the heat capacity at constant area and
electric charge $C_{A,Q}$, and the heat capacity at constant area and electric
potential $C_{A,\Phi}$.

The three equations of state $T(M,A,Q)$, $p(M,A,Q)$, and $\Phi(M,A,Q)$, given
in Eqs. (58)-(60) are to be rewritten in laboratory variables, which are also
called thermodynamic coefficients. In fact, for mass and charge fluctuations,
one only needs two equations of state, the ones for the temperature $T(M,A,Q)$
and for the thermodynamic electric potential $\Phi(M,A,Q)$. Since the area is
kept fixed here we do not need to use the equation of state for the pressure,
$p(M,A,Q)$. The equations for $T(M,A,Q)$ and $\Phi(M,A,Q)$ will allow us to
establish the stability conditions for mass and charge fluctuations in the new
variables.

For the equation of state for temperature, $T(M,A,Q)$, we want to define the
laboratory variables in terms of the derivatives of $S(T,A,Q)$. For that, note
that one is able to write the differential $dS(T,A,Q)$ as
$dS=\left(\frac{\partial S}{\partial T}\right)_{A,Q}dT+\left(\frac{\partial
S}{\partial A}\right)_{T,Q}dA+\left(\frac{\partial S}{\partial
Q}\right)_{T,A}dQ$. Now, the heat capacity $C_{A,Q}$ is defined as
$C_{A,Q}^{-1}=\left(\frac{\partial T}{\partial M}\right)_{A,Q}$ which is
equivalent to $\frac{1}{T}C_{A,Q}=\left(\frac{\partial S}{\partial
T}\right)_{A,Q}$. The latent heat capacity at constant temperature and charge,
$\lambda_{T,Q}$, is defined as $\lambda_{T,Q}=\left(\frac{\partial S}{\partial
A}\right)_{T,Q}$. The latent heat capacity at constant temperature and area,
$\lambda_{T,A}$, is defined as $\lambda_{T,A}=\left(\frac{\partial S}{\partial
Q}\right)_{T,A}$. One can then change the equality for $dS$ into an equality
for $dT$, such that $T=T(S,A,Q)$, to obtain
$dT=\frac{T}{C_{A,Q}}dS-T\frac{\lambda_{T,Q}}{C_{A,Q}}dA-\frac{T\lambda_{T,A}}{C_{A,Q}}dQ$.
One can now use the first law, Eq. (36), i.e., $TdS=dM+pdA-\Phi dQ$,
substitute for the variation of the entropy $dS$ above and put the equation
found as an equality for $dT$, namely,

$\displaystyle
dT=\frac{1}{C_{A,Q}}dM-\frac{T\lambda_{T,Q}-p}{C_{A,Q}}dA-\frac{T\lambda_{T,A}+\Phi}{C_{A,Q}}dQ\,,$
(82)

So, $dT$ is written in terms of the laboratory variables, namely, the heat
capacity $C_{A,Q}$, the latent heat capacity at constant temperature and
charge $\lambda_{T,Q}$, and the latent heat capacity at constant temperature
and area $\lambda_{T,A}$. For the equation of state for the thermodynamic
electric potential $\Phi(M,A,Q)$, we define the laboratory variables with
respect to $\Phi(S,A,Q)$ so that, with the aid of $S(M,A,Q)$, we obtain
$\Phi(M,A,Q)$. Now, note that one is able to write the differential
$d\Phi(S,A,Q)$ as $d\Phi=\left(\frac{\partial\Phi}{\partial
S}\right)_{A,Q}dS+\left(\frac{\partial\Phi}{\partial
A}\right)_{S,Q}dA+\left(\frac{\partial\Phi}{\partial Q}\right)_{S,A}dQ$. We
define the adiabatic electric susceptibility, $\chi_{S,A}$, as
$\frac{1}{\chi_{S,A}}=\left(\frac{\partial\Phi}{\partial Q}\right)_{S,A}$, and
define the electric pressure at constant entropy and charge, $P_{S,Q}$, as
$P_{S,Q}=\left(\frac{\partial\Phi}{\partial A}\right)_{S,Q}$. The remaining
derivative of $\Phi$ is given by the Maxwell relation
$\left(\frac{\partial\Phi}{\partial S}\right)_{A,Q}=\left(\frac{\partial
T}{\partial Q}\right)_{S,A}=-\frac{T\lambda_{T,A}}{C_{A,Q}}$, which was
calculated using the definitions given above for $C_{A,Q}$, $\lambda_{T,Q}$,
and $\lambda_{T,A}$, and swapping the equality for $dS(T,A,Q)$ into an
equality for $d\Phi(S,A,Q)$, i.e.,
$d\Phi=-\frac{T\lambda_{T,A}}{C_{A,Q}}dS+P_{S,Q}dA+\frac{1}{\chi_{S,A}}dQ$.
One can now use the first law, Eq. (36), i.e., $TdS=dM+pdA-\Phi dQ$,
substitute for the variation of the entropy $dS$ above and the equation found
as an equality for $d\Phi$, namely,

$\displaystyle
d\Phi=-\frac{\lambda_{T,A}}{C_{A,Q}}dM+\left(P_{S,Q}-p\frac{\lambda_{T,A}}{C_{A,Q}}\right)dA$
$\displaystyle+\left(\frac{1}{\chi_{S,A}}+\frac{\Phi\lambda_{T,A}}{C_{A,Q}}\right)dQ\,.$
(83)

So, $d\Phi$ is written in terms of the laboratory variables, namely, the heat
capacity, $C_{A,Q}$, the latent heat capacity at constant temperature and
area, $\lambda_{T,A}$, the electric pressure at constant entropy and charge,
$P_{S,Q}$, and the adiabatic electric susceptibility, $\chi_{S,A}$. Finally,
it is useful to define also the heat capacity at constant area and constant
electric potential, $C_{A,\Phi}$, defined by $C_{A,\Phi}=T\left(\frac{\partial
S}{\partial T}\right)_{A,\Phi}$ which can be written in terms of the
coefficients in Eqs. (82) and (83) as
$C_{A,\Phi}=C_{A,Q}\left(1-\frac{T\lambda_{T,A}^{2}}{C_{A,Q}}\chi_{S,A}\right)^{\hskip-4.26773pt-1}$.

The intrinsic thermodynamic stability of a thin shell for mass and charge
fluctuations together can be determined by considering the two sufficient
stability conditions which can be taken from Eq. (56), yielding $S_{MM}\leq 0$
and $S_{MM}S_{QQ}-S_{MQ}^{2}\geq 0$. The first condition is almost immediate
since with the definition in Eq. (82) we obtain
$S_{MM}=-\beta^{2}\frac{1}{C_{A,Q}}$ and so it implies that $C_{A,Q}\geq 0$.
The second condition requires some more care and some more algebra, yielding
in the end $S_{MM}S_{QQ}-S_{MQ}^{2}=\
\beta^{2}\frac{1}{C_{A,\Phi}\chi_{S,A}}$, and so it implies
$C_{A,\Phi}\,\,\chi_{S,A}\geq 0$. The stability conditions in the laboratory
variables can then be written as

$\displaystyle C_{A,Q}\geq 0\,,$ $\displaystyle C_{A,\Phi}\,\,\chi_{S,A}\geq
0\,.$ (84)

For the specific equations of state we used, Eqs. (58)-(60), the coefficient
$\chi_{S,A}$ is always positive. Hence, from the two equations given in Eq.
(V.2) , the stability conditions become $C_{A,Q}\geq 0$ and $C_{A,\Phi}\geq
0$, respectively. Moreover, we have found in Sec. IV.8 that for these
equations of state the condition $S_{MM}S_{QQ}-S_{MQ}^{2}\geq 0$ is the
sufficient condition for stability. Therefore, the thin shell considered is
thermodynamic stable for mass and charge fluctuations together if

$\displaystyle C_{A,\Phi}\geq 0\,.$ (85)

This occurs precisely when Eq. (76) is satisfied, i.e., Eq. (85) is equivalent
to Eq. (76), as for a thin shell with the equations of state given in Eqs.
(58)-(60). Note that, when there is equality in Eq. (76), one must be careful
in regard to the value of the heat capacity at constant area and constant
electric potential, $C_{A,\Phi}$. If one performs the limit to the equality as
a succession of stable configurations, by starting from a configuration with
the exponent $a$ satisfying Eq. (85) and then increasing $a$, the coefficient
$C_{A,\Phi}$ becomes infinite and positive. If, on the contrary, one performs
the limit to the equality as a succession of unstable configurations, by
starting from a configuration with the exponent $a$ not satisfying Eq. (85)
and then decreasing $a$, the coefficient $C_{A,\Phi}$ becomes infinite and
negative.

The details of all the calculations presented in this section are shown in
Appendix D.1.

###  V.3 Laboratory variables for mass, area, and charge fluctuations
altogether

Here we discuss the new laboratory variables for mass, area, and charge
fluctuations, see Sec. IV.10. The analysis in Sec. V.2 highlights the
importance of the specific heat capacities $C_{A,Q}$ and $C_{A,\Phi}$ in the
intrinsic stability of thermodynamic systems with fixed area. Nevertheless,
there are other important quantities playing a role in the intrinsic stability
for the case of mass, area and charge fluctuations. In this case the important
laboratory quantities are the heat capacity at constant area and electric
charge $C_{A,Q}$ again, the expansion coefficient at constant temperature and
electric charge $\kappa_{T,Q}$, and the electric susceptibility at constant
pressure and temperature $\chi_{p,T}$.

The three equations of state $T(M,A,Q)$, $p(M,A,Q)$, and $\Phi(M,A,Q)$ are to
be rewritten in laboratory variables. This will allow us to establish the
stability conditions in these new variables for mass, area, and charge
fluctuations, and so no fixed quantities. We now want to define the laboratory
variables in terms of the derivatives of $S(T,p,Q)$, $A(T,p,Q)$, and
$\Phi(T,p,Q)$, for convenience, since these variables will simplify the
considered stability conditions. Notice that the three functions $S(T,p,Q)$,
$A(T,p,Q)$ and $\Phi(T,p,Q)$ are the derivatives of the Gibbs potential, i.e.,
$dG=-SdT+Adp+\Phi dQ$. Let us start with the area $dA$ for which the
coefficients have a direct physical meaning. We write $dA=\left(\frac{\partial
A}{\partial T}\right)_{p,Q}dT+\left(\frac{\partial A}{\partial
p}\right)_{T,Q}dp+\left(\frac{\partial A}{\partial Q}\right)_{T,p}dQ$, such
that $\alpha_{p,Q}=\frac{1}{A}\left(\frac{\partial A}{\partial
T}\right)_{p,Q}$ is the expansion coefficient,
$\kappa_{T,Q}=-\frac{1}{A}\left(\frac{\partial A}{\partial p}\right)_{T,Q}$ is
the isothermal compressibility, and
$\kappa_{p,T}=-\frac{1}{A}\left(\frac{\partial A}{\partial Q}\right)_{T,p}$ is
the electric compressibility. Now, $dS$ is written as $dS=\left(\frac{\partial
S}{\partial T}\right)_{p,Q}dT+\left(\frac{\partial S}{\partial
p}\right)_{T,Q}dp+\left(\frac{\partial S}{\partial Q}\right)_{T,p}dQ$, where
$\left(\frac{\partial S}{\partial T}\right)_{p,Q}$ can be written in terms of
previously defined coefficients, specifically, $\left(\frac{\partial
S}{\partial
T}\right)_{p,Q}=\frac{C_{A,Q}}{T}+A\frac{\alpha_{p,Q}^{2}}{\kappa_{T,Q}}$,
$\left(\frac{\partial S}{\partial p}\right)_{T,Q}$ can be written using the
Maxwell relation $\left(\frac{\partial S}{\partial
p}\right)_{T,Q}=-\left(\frac{\partial A}{\partial
T}\right)_{T,Q}=A\alpha_{p,Q}$, and $\lambda_{p,T}=\left(\frac{\partial
S}{\partial Q}\right)_{p,T}$ is a new coefficient, the latent heat capacity.
Finally, $d\Phi$ is written as $d\Phi=\left(\frac{\partial\Phi}{\partial
T}\right)_{p,Q}dT+\left(\frac{\partial\Phi}{\partial
p}\right)_{T,Q}dp+\left(\frac{\partial\Phi}{\partial Q}\right)_{T,p}dQ$, where
two of the derivatives are written using Maxwell relations, i.e.,
$\left(\frac{\partial\Phi}{\partial T}\right)_{p,Q}=-\left(\frac{\partial
S}{\partial Q}\right)_{p,T}=\lambda_{p,T}$ and
$\left(\frac{\partial\Phi}{\partial p}\right)_{T,Q}=\left(\frac{\partial
A}{\partial Q}\right)_{p,T}=-A\kappa_{p,T}$ as defined above, and
$\frac{1}{\chi_{p,T}}=\left(\frac{\partial\Phi}{\partial Q}\right)_{p,T}$ is a
new coefficient, the isothermal electric susceptibility. With the
differentials $dA(T,p,Q)$, $dS(T,p,Q)$ defined above in terms of physical
coefficients, we are able to invert the system composed by these two
differentials in order to obtain $dT(S,A,Q)$ and $dp(S,A,Q)$. Then, using Eq.
(36), i.e., $TdS=dM+pdA-\Phi dQ$, we are able to obtain the differentials of
the two equations of state in the desired form, i.e., $dT(M,A,Q)$ and
$dp(M,A,Q)$. Inserting $dT(M,A,Q)$ and $dp(M,A,Q)$ into $d\Phi(T,p,Q)$, we
find the differential of the remaining equation of state, $d\Phi(M,A,Q)$.
Thus, these differentials are written in terms of the defined laboratory
variables and the differentials of $dM$, $dA$, and $dQ$, as

$\displaystyle dT=$
$\displaystyle\frac{dM}{C_{A,Q}}+\left(\frac{p}{C_{A,Q}}-T\frac{\alpha_{p,Q}}{C_{A,Q}\kappa_{T,Q}}\right)dA$
$\displaystyle-\left(\frac{\Phi}{C_{A,Q}}+T\frac{\lambda_{p,T}}{C_{A,Q}}+A\frac{\alpha_{p,T}\kappa_{p,T}}{\kappa_{T,Q}C_{A,Q}}\right)dQ\,,$
(86) $\displaystyle dp=$
$\displaystyle\frac{\alpha_{p,Q}}{C_{A,Q}\kappa_{T,Q}}dM$
$\displaystyle-\left[\frac{1}{A\kappa_{T,Q}}-\frac{\alpha_{p,Q}}{C_{A,Q}\kappa_{T,Q}}\left(p-T\frac{\alpha_{p,Q}}{\kappa_{T,Q}}\right)\right]dA$
$\displaystyle-\left(\frac{\kappa_{p,T}}{\kappa_{T,Q}}-\frac{\alpha_{p,Q}}{C_{A,Q}\kappa_{T,Q}}\mathcal{C}\right)dQ\,,$
(87) $\displaystyle d\Phi=$
$\displaystyle-\mathcal{B}dM+\left[\frac{\kappa_{p,T}}{\kappa_{T,Q}}-\left(p-T\frac{\alpha_{p,Q}}{\kappa_{T,Q}}\right)\mathcal{B}\right]dA$
$\displaystyle+\left(\mathcal{B}\mathcal{C}+\frac{1}{\chi_{p,T}}+A\frac{\kappa_{p,T}^{2}}{\kappa_{T,Q}}\right)dQ\,,$
(88)

where $\mathcal{B}$ is defined as
$\mathcal{B}=A\frac{\kappa_{p,T}\alpha_{p,Q}}{C_{A,Q}\kappa_{T,Q}}+\frac{\lambda_{p,T}}{C_{A,Q}}$,
and $\mathcal{C}$ is defined as
$\mathcal{C}=TA\frac{\kappa_{p,T}\alpha_{p,Q}}{\kappa_{T,Q}}+T\lambda_{p,T}+\Phi$.
With the differentials $dT(M,A,Q)$, $dp(M,A,Q)$, and $d\Phi(M,A,Q)$ in Eqs.
(86)-(88), and the first law of thermodynamics, Eq. (36), i.e.,
$TdS=dM+pdA-\Phi dQ$, the second derivatives of the entropy that enter into
the thermodynamic stability problem and are given in Eq. (62) can be
calculated directly.

The intrinsic thermodynamic stability of a thin shell for generic mass, area,
and charge fluctuations, is given by the three sufficient stability conditions
which can be taken from Eq. (56), yielding $S_{MM}\leq 0$,
$S_{MM}S_{AA}-S_{MA}^{2}\geq 0$, and
$(S_{MM}S_{AQ}-S_{MA}S_{MQ})^{2}-(S_{AA}S_{MM}-S_{AM}^{2})(S_{QQ}S_{MM}-S_{QM}^{2})\leq
0$. Now, having the second derivatives of the entropy written in terms of the
laboratory variables defined in this section, one finds that $S_{MM}\leq 0$ is
equivalent to $\beta^{2}\frac{1}{C_{A,Q}}\geq 0$, $S_{MM}S_{AA}-S_{MA}^{2}\geq
0$ is equivalent to $\beta^{3}\frac{1}{A\kappa_{T,Q}C_{A,Q}}\geq 0$, and
$(S_{MM}S_{AQ}-S_{MA}S_{MQ})^{2}-(S_{AA}S_{MM}-S_{AM}^{2})(S_{QQ}S_{MM}-S_{QM}^{2})\leq
0$ is equivalent to
$\beta^{6}\frac{1}{AC_{A,Q}^{2}\kappa_{T,Q}\,\,\chi_{p,T}}\geq 0$. We then
have that the stability conditions for generic mass, area, and charge
fluctuations are given by the following three equations,

$\displaystyle C_{A,Q}\geq 0,\,$ $\displaystyle\kappa_{T,Q}\geq 0\,,$
$\displaystyle\chi_{p,T}\geq 0\,,$ (89)

i.e., all three laboratory quantities have to be positive, specifically, the
heat capacity $C_{A,Q}$ which is related to changes in temperature, the
isothermal compressibility $\kappa_{T,Q}$ which is related to changes in the
pressure, and the isothermal electric susceptibility $\chi_{p,T}$ which is
related to changes in the electric charge, have to be positive, with the case
of marginal stability corresponding to these physical variables going to
infinity.

For the specific equations of state we use, Eqs. (58)-(60), for $T$, $p$, and
$\Phi$, respectively, one finds that the most restrictive condition is for
$\chi_{p,T}$,

$\displaystyle\chi_{p,T}\geq 0\,,$ (90)

so the positivity of the isothermal electric susceptibility $\chi_{p,T}$ is
the sufficient condition in the case of the thin shell we are considering,
with marginal stability happening when this quantity is infinite, and with the
unstable configurations having a negative electric susceptibility and thus
departing from equilibrium. Making the connection to Sec. IV.10, Eq. (90) is
equivalent to Eq. (80) and one finds that for $a<\frac{d-3}{d-2}$ the
isothermal electric susceptibility $\chi_{p,T}$ will always be positive, for
$\frac{d-3}{d-2}<a<1$ it will be positive for some values of
$(r_{+},r_{-},R)$, for $a=1$ and $R>r_{+}$ it will be negative with the case
$R=r_{+}$ having to be treated with care, and for $a>1$ it will always be
negative. The shell with black hole features, namely, $a=1$ and $R=r_{+}$, is
thermodynamic unstable if the shell approaches its own gravitational radius
$r_{+}$, $R\to r_{+}$, since in this case $\chi_{p,T}\rightarrow-\infty$. But
there is the possibility, of having a configuration with $R=r_{+}$ that is
created from the start, i.e., a configuration not belonging to a sequence of
quasistatic configurations that has its radius $R$ decreased up to $r_{+}$. In
this case the stability depends on whether the exponent $a$ of the equation of
state approaches $a=1$ from below or from above. If the exponent $a$ of the
equation of state approaches $a=1$ from below then the $R=r_{+}$ configuration
is marginally stable with $\chi_{p,T}\rightarrow+\infty$, which means that
changes in the electric charge of the configuration will not have any impact
on the electric potential. If the exponent $a$ of the equation of state
approaches $a=1$ from above then the $R=r_{+}$ configuration is unstable,
$\chi_{p,T}\rightarrow-\infty$. Moreover, in the region of
$\frac{d-3}{d-2}<a<1$, shells with more electric charge show more difficulty
in having positive isothermal electric susceptibility $\chi_{p,T}$, a property
that can be deduced from Fig. 7 by the decreasing amount of stable
configurations with electric charge.

The details of all the calculations presented in this section are shown in
Appendix D.2.

##  VI Conclusions

We have used the thin shell formalism to determine the mechanics of a static
charged spherical thin shell in $d$ dimensions in general relativity and
studied its thermodynamics by imposing the first law. The fact that the rest
mass density and so the rest mass behaves as a thermal quasilocal energy and
that the pressure is determined just by general relativity indicates there is
a relation between general relativity and thermodynamics as one equation of
state of the shell becomes fixed. We computed the entropy in terms of the
thermodynamic quantities of the shell, namely, the rest mass $M$, the area $A$
and the electric charge $Q$ of the shell. The derivatives of the entropy are
directly related to the temperature and the electric potential. Indeed, with
the first law of thermodynamics and general relativity alone, we were able to
restrict the expressions for the equations of state for the temperature and
the thermodynamic electric potential. These equations of state in turn imply
that the entropy $S$ depends solely on the two natural radius of the Reissner-
Nordström shell spacetime, the gravitational radius $r_{+}$ and the Cauchy
radius $r_{-}$, which in turn depend on $M$, $A$ and $Q$. That the entropy $S$
of the $d$ dimensional shell spacetime does not depend on the radius of the
shell, $R$, is a remarkable fact, which nevertheless has been found for other
thin shell spacetimes.

To calculate an exact expression for the entropy of the shell, one still needs
full expressions for the temperature and the thermodynamic electric potential
equations of state. We used a power law in $r_{+}$ with exponent $a$ for the
temperature, and opted for the characteristic electric potential of a
Reissner-Nordström shell spacetime, to obtain that the entropy $S$ of the
shell is proportional to $A_{+}^{a}$, where $A_{+}$ is the gravitational area
corresponding to $r_{+}$. Shells with such entropy are of great interest as it
is possible to obtain the black hole limit and recover the thermodynamics of
black holes.

We have then studied the thermodynamic intrinsic stability of thin shells with
such an entropy equation. The shell is stable if the Hessian of the entropy is
negative semidefinite. We analyzed the Hessian for seven possible types of
fluctuations that can occur in the shell. Fluctuations of the shell with one
free and two fixed thermodynamic quantities are of three types, fluctuations
of the shell with two free and one fixed quantities are also of three types,
and fluctuations of the shell with three free quantities, i.e., no fixed
quantities, are of one type. The most important and general type of
fluctuations are the ones with no fixed quantities. In the case of our entropy
equation, we have found that only one condition is sufficient for the shell to
be stable. This condition establishes that for $0<a\leq\frac{d-3}{d-2}$ all
the configurations of the shell are thermodynamic stable, for
$\frac{d-3}{d-2}<a<1$ stability depends on the mass and electric charge, for
$a=1$ the configurations are unstable, unless the shell is at its own
gravitational radius, i.e., at the black hole limit, in which case it is
marginally stable, and that for $1<a<\infty$ all configurations are unstable.

The physical interpretation for the stability is analyzed through new
thermodynamic variables that can be measured in the laboratory. One finds
that, generically, stable shells have positive heat capacity, positive
isothermal compressibility, and positive isothermal electric susceptibility.
For the specific equations of state that we gave, and so for our entropy
equation, it is found that every shell with positive electric susceptibility
has positive heat capacity and positive isothermal compressibility, which is
an interesting property of the shells with the considered entropy. In
addition, marginal stability means that the electric susceptibility is
infinite, which seems to be the case in the black hole limit. Shells with
negative electric susceptibility depart from the initial state and they should
rearrange themselves until the susceptibility becomes positive or the shell
breaks down.

This work has derived some thermodynamic properties for electrically charged
spherical matter shells in higher dimensions and complements a set of works in
the thermodynamics of thin shells. Still, more future work should be done in
the investigation of the link between thermodynamics and general relativity,
and hopefully contribute to the understanding of black hole physics and with
it trying to grasp gravitation at the tiniest possible scales.

## Acknowledgments

We thank financial support from Fundação para a Ciência e Tecnologia - FCT
through the project No. UIDB/00099/2020.

##  Appendix A Temperature as a power law in the ADM mass and corresponding
thermodynamic electric potential as alternative to the equations of state of
Sec. III

The reduced equations of state for the inverse temperature and thermodynamic
electric potential given in Eqs. (49) and (50) of Sec. III are a good choice
for reduced equations of state as they yield naturally the black hole
equations of state. But there are alternatives to these equations of state.

In $d=4$, an alternative was provided in Lemos:2015a , where the reduced
inverse temperature was chosen to be
$b(r_{+},r_{-})=2a(r_{+}+r_{-})^{\alpha}$, for some constant $a$ and exponent
$\alpha$. Thus, $b$, which represents the inverse temperature at infinity, is
given as a power law depending on the ADM mass, since in $d=4$, one has
$r_{+}+r_{-}=2Gm$. The solution for the reduced thermodynamic electric
potential $c(r_{+},r_{-})$ is then
$c(r_{+},r_{-})=2\gamma\frac{(r_{+}r_{-})^{\delta}}{(r_{+}+r_{-})^{\alpha}}$
for some $\gamma$, $\delta$, and so is a power law in the charge $Q$ combined
with a power law in $m$ Lemos:2015a . For this $d=4$ case, it is possible
analytically to study its stability.

Motivated by this choice for a reduced equation of state for $d=4$, we
consider for a reduced equation of state for the temperature in generic $d$
dimensions, the following expression

$\displaystyle b(r_{+},r_{-})=a(r_{+}^{d-3}+r_{-}^{d-3})^{\alpha}\,,$ (91)

where $a$ and $\alpha$ are arbitrary real numbers, and such a choice means
that $b$ is indeed a power law in the $d$-dimensional ADM mass $m$. With the
chosen reduced equation of state for the inverse temperature, the condition in
Eq. (48) can be used to restrict the form of $c(r_{+},r_{-})$, which is given
in this case by $\frac{\partial c}{\partial
r_{-}}\frac{r_{-}}{d-3}-\frac{\partial c}{\partial
r_{+}}\frac{r_{+}}{d-3}=\alpha
c\frac{r_{+}^{d-3}-r_{-}^{d-3}}{r_{+}^{d-3}+r_{-}^{d-3}}$. The general
solution for $c(r_{+},r_{-})$ is

$\displaystyle
c(r_{+},r_{-})=\frac{f(r_{+}r_{-})}{(r_{+}^{d-3}+r_{-}^{d-3})^{\alpha}}\,,$
(92)

where $f(r_{+}r_{-})$ is an arbitrary function depending only on the charge of
the shell. This generalizes to $d$ dimensions the $c(r_{+},r_{-})$ obtained in
Lemos:2015a . The choices for the reduced equations of state provided, namely,
Eqs. (91) and (92), permit to treat overcharged shells, i.e., shells with
$Q>m$. The thermodynamic stability can be performed, but we will not do it
here.

##  Appendix B Thermodynamic stability: Generic and specific considerations to
be added to Sec. IV

###  B.1 Seminegative definite Hessian

Here we establish the rules that lead to the stability conditions in Eq. (56)
of Sec. IV. A thermodynamic system is in equilibrium if it reaches a maximum
of entropy, i.e., $dS=0$, and it obeys $d^{2}S\leq 0$.

Let us assume that the system is in the state $dS=0$. To understand what
$d^{2}S\leq 0$ leads to, it is important to write the Hessian matrix of the
entropy, $S_{ij}$, i.e., $S_{ij}=\frac{\partial^{2}S}{\partial h_{i}\partial
h_{j}}$, $h_{i}$ being a set of unfixed independent parameters. Then
$d^{2}S\leq 0$ means that $S_{ij}$ has to be semidefinite negative, and so for
any arbitrary vector $\bm{v}$, one has

$\sum_{ij}S_{ij}v_{i}v_{j}\leq 0\,.$ (93)

###  B.2 1-parameter case

In the 1-parameter case, the entropy $S$ is a function of one unfixed
parameters $h_{1}$, i.e., the system is allowed to change very small amounts
in $h_{1}$. The thermodynamic stability condition, Eq. (93), turns into
$S_{h_{1}h_{1}}v_{1}^{2}\leq 0$, for an arbitrary vector $\bm{v}=(v_{1})$, and
so one finds

$\displaystyle S_{h_{1}h_{1}}\leq 0\,.$ (94)

###  B.3 2-parameter case

In the 2-parameter case, the entropy $S$ is a function of two unfixed
parameters $h_{1}$ and $h_{2}$. Thus, the generic stability condition, Eq.
(93), states
$S_{h_{1}h_{1}}v_{1}^{2}+2S_{h_{1}h_{2}}v_{1}v_{2}+S_{h_{2}h_{2}}v_{2}^{2}\leq
0$, for an arbitrary vector $\bm{v}=(v_{1},v_{2})$. One can choose the vector
$\bm{v}=(v_{1},0)$, which yields $S_{h_{1}h_{1}}v_{1}^{2}\leq 0$, for any
$v_{1}$, and so a necessary condition is

$\displaystyle S_{h_{1}h_{1}}\leq 0\,.$ (95)

One can choose the vector $\bm{v}=(0,v_{2})$, which for any $v_{2}$, yields
$S_{h_{2}h_{2}}v_{2}^{2}\leq 0$, and so another necessary condition is

$\displaystyle S_{h_{2}h_{2}}\leq 0\,.$ (96)

The third condition comes from completing the square of the stability
condition for this 2-parameter case, yielding
$\frac{(S_{h_{1}h_{1}}v_{1}+S_{h_{1}h_{2}}v_{2})^{2}}{S_{h_{1}h_{1}}}+\Big{(}S_{h_{2}h_{2}}-\frac{S_{h_{1}h_{2}}^{2}}{S_{h_{1}h_{1}}}\Big{)}v_{2}^{2}\leq
0$. Thus, since $S_{h_{1}h_{1}}\leq 0$, the third necessary condition is

$\displaystyle S_{h_{1}h_{1}}S_{h_{2}h_{2}}-S_{h_{1}h_{2}}^{2}\geq 0\,.$ (97)

###  B.4 3-parameter case

In the 3-parameter case, the entropy $S$ is a function of three unfixed
parameters $h_{1}$, $h_{2}$, and $h_{3}$. Thus, the generic stability
condition, Eq. (93), states
$S_{h_{1}h_{1}}v_{1}^{2}+S_{h_{2}h_{2}}v_{2}^{2}+S_{h_{3}h_{3}}v_{3}^{2}+2S_{h_{1}h_{2}}v_{1}v_{2}+2S_{h_{1}h_{3}}v_{1}v_{3}+2S_{h_{3}h_{2}}v_{2}v_{3}\leq
0$, for an arbitrary vector $\bm{v}=(v_{1},v_{2},v_{3})$. Analogous to the
2-parameter case above, one can set components of $\bm{v}$ to be zero. This
will give the 2-parameter conditions for each pair of parameters. Thus, one
has

$\displaystyle S_{h_{i}h_{i}}\leq 0\,,$ (98) $\displaystyle
S_{h_{i}h_{i}}S_{h_{j}h_{j}}-S_{h_{i}h_{j}}^{2}\geq 0\,,\,\,i\neq j\,,$ (99)

for $i,j=1,2,3$, i.e., there are six conditions. The seventh condition can be
obtained by multiplying the generic stability condition $S_{h_{1}h_{1}}$ and
once again completing the square in the following way,
$(S_{h_{1}h_{1}}v_{1}+S_{h_{1}h_{2}}v_{2}+S_{h_{1}h_{3}}v_{3})^{2}+(S_{h_{2}h_{2}}S_{h_{1}h_{1}}-S_{h_{2}h_{1}}^{2})v_{2}^{2}+(S_{h_{3}h_{3}}S_{h_{1}h_{1}}-S_{h_{1}h_{3}}^{2})v_{3}^{2}+2(S_{h_{3}h_{2}}S_{h_{1}h_{1}}-S_{h_{1}h_{2}}S_{h_{1}h_{3}})v_{3}v_{2}\geq
0$. Now one can again perform the square completion but it is more simple to
consider the roots of the polynomial $P(v_{2},v_{3})$ composed by the second,
third and fourth terms. Since $P(v_{2},v_{3})\geq 0$ and its second
derivatives are positive, the polynomial must have only one root or no roots.
Thus, the following inequality holds

$\displaystyle(S_{h_{3}h_{2}}S_{h_{1}h_{1}}-S_{h_{1}h_{2}}S_{h_{1}h_{3}})^{2}$
$\displaystyle-(S_{h_{2}h_{2}}S_{h_{1}h_{1}}-S_{h_{2}h_{1}}^{2})(S_{h_{3}h_{3}}S_{h_{1}h_{1}}-S_{h_{1}h_{3}}^{2})\leq
0\,.$ (100)

###  B.5 Redundancy

By the method above, it seems one needs 3 conditions for the 2-parameter case
and 7 conditions for the 3-parameter case. Since the number of conditions is
higher than the number of eigenvalues of the Hessian, there are redundant
conditions. For any specific choice of $h_{i}$, the sufficient conditions for
the 2-parameter case are Eq. (94) and Eq. (97). For the 3-parameter case, the
sufficient conditions are Eq. (94), Eq. (97) and Eq. (100). Another method to
determine the conditions is to consider only the pivots of the Hessian.

###  B.6 The case studied

There is still the freedom to choose the order of parameters in the
construction of the Hessian. For example, one can pick $h_{1}=M$, $h_{2}=A$
and $h_{3}=Q$, which has been our choice, but of course any permutation of
parameters is allowed.

For the static thin shell system we studied, the first condition for
stability, $dS=0$, imposes that $r_{+}={\rm const}$ case, as the entropy is
proportional to a power of $r_{+}$, $S=\frac{\gamma}{16\pi G}A_{+}^{a}$, see
Eq. (51), with $r_{+}$ itself being a function of $M$, $A$, and $Q$. The other
condition $d^{2}S\leq 0$ leads to Eqs. (94)-(100) which are essential the set
of equations given in Eq. (56) with the choice $h_{1}=M$, $h_{2}=A$ and
$h_{3}=Q$. The stability analysis itself is then performed through Eqs.
(62)-(80).

##  Appendix C Graphics relevant to Sec. IV

This appendix is dedicated to the display of the plots of the marginal
stability of the shell with the entropy given by Eq. (51) that were discussed
in Sec. IV. Each figure has plots of the marginal condition for each seven
types of fluctuations, which are all the possible combinations of mass, area
and charge fluctuations. The correspondence is the following: Fig. 1
corresponds to Sec. IV.4 and to Sec. IV.11, Fig. 2 to Sec. IV.5, Fig. 3 to
Sec. IV.6, Fig. 4 to Sec. IV.7, Fig. 5 to Sec. IV.8, Fig. 6 to Sec. IV.9, and
Fig. 7 to Sec. IV.10. The surfaces of marginal stability are 3-surfaces lying
in $\mathbb{R}^{4}$ with coordinates $(x,y,a,d)$, with
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$ and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$, where
$r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius of the
Reissner-Nordström shell spacetime, $R$ is the radius of the shell, $a$ is the
exponent of the equation of state of the shell that appears in Eq. (49), and
$d$ is the number of spacetime dimensions. Each point in $\mathbb{R}^{4}$
corresponds to a configuration of the shell for a fixed value of $r_{+}$. The
subset of $\mathbb{R}^{4}$ that corresponds to physical configurations is
described by $x\in\,\,]0,1[\,$, $y\in\,\,]0,1[\,$, $a\in\,\,[0,\infty[\,$ and
$d\in\,\,[4,\infty[\,$. The intersection of the 3-surface with this subset
separates the subset of physical configurations that are stable from the ones
that are unstable.

For reasons of presentation, we display cuts of the 3-surface of marginal
stability. In each figure, we display a plot of the 3-surface as $a(x)$ for
different values of $y$ with $d=5$, as $a(y)$ for different values of $x$ with
$d=5$ and as $a(d)$ for different values of $x$ and $y$. We choose $d=5$ since
it is the closest generalization of the 4-dimensional case and has
implications in holography and unified theories. Also, we only display the
curve of marginal stability with $a=1$ in three cases: the case of mass
fluctuations in Fig. 1, the case of mass and area fluctuations in Fig. 4 and
the case of mass and charge fluctuations in Fig. 5. The plot of the 3-surface
with $a=1$ has physical importance since these configurations correspond to a
shell with black hole features, in particular, a shell with the same entropy
as a black hole with same mass and charge. However, displaying this plot for
the other cases would have no interest for the following reasons. For the
cases of Figs. 2, 3 and 6, there would be simply no curve. All the
configurations with $a=1$ are below the 3-surface and therefore are stable.
For the case of Fig. 7, the plot would be of a straight line in $x=1$, hence
only the configurations with $x=1$ are marginally stable with the others being
unstable.

From the analysis in Sec. IV, we must note that at least Fig. 1, Fig. 4 and
Fig. 7 have a physical interpretation. The surface of marginal stability in
Fig. 1 describes configurations with an infinite heat capacity, whereas stable
configurations have positive heat capacity and unstable ones have negative
heat capacity. The analogous happens in Fig. 4 and Fig. 7, that describe
respectively infinite isothermal compressibility and isothermal electric
susceptibility.

(a) $d=5$, $y$ fixed

(b) $d=5$, $x$ fixed

(c) $(x,y)$ fixed

(d) $a=1$

Figure 1:  Thermodynamic stability of the shell considering only mass
fluctuations is described, see Sec. IV.4 and Eqs. (67), (68) and (81). Plots
of $S_{1}\left(d,a,x,y\right)=0$ with $d$ being the number of spacetime
dimensions, $a$ being the exponent of the equation of state of the shell,
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$, and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$,
where $r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius
of the Reissner-Nordström shell spacetime, respectively, and $R$ is the radius
of the shell, are shown. The equality $S_{1}\left(d,a,x,y\right)=0$ describes
marginal stability of the shell considering only mass fluctuations. (a) The
function is plotted as $a(x)$ with $d=5$ and for fixed values of $y$, with
points with lower $a$ than the function corresponding to stable
configurations; (b) The function is plotted as $a(y)$ with $d=5$ and for fixed
values of $x$, with points with lower $a$ corresponding to stable
configurations; (c) The function is plotted as $a(d)$ for fixed values of the
pair $(x,y)$, with points with lower $a$ corresponding to stable
configurations; (d) In the top plot, for $a=1$ with fixed values of $d$ the
function is plotted as $y(x)$, with configurations with higher $x$ than the
curve being stable; in the bottom plot, one uses the the function in the form
$S_{1}\left(d=5,a=1,\frac{M}{R},\frac{Q}{R}\right)=0$, where $M$ is the rest
mass of the shell and $Q$ is the charge of the shell, see Sec. IV.11 and Eq.
(81), to plot, for $a=1$ with $\mu=1$ and $d=5$, the function $Q(M)$ in the
blue curve, the region of stability in the yellow line filled area, $Q=M$ in
the black line, and $r_{+}=R$ in the red curve.

(a) $d=5$, $y$ fixed

(b) $d=5$, $x$ fixed

(c) $(x,y)$ fixed

Figure 2:  Thermodynamic stability of the shell considering only area
fluctuations is described, see Sec. IV.5 and Eqs. (69) and (70). Plots of
$S_{2}\left(d,a,x,y\right)=0$, with $d$ being the number of spacetime
dimensions, $a$ being the exponent of the equation of state of the shell,
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$, and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$,
where $r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius
of the Reissner-Nordström shell spacetime, respectively, and $R$ is the radius
of the shell, are shown. The equality $S_{2}\left(d,a,x,y\right)=0$ describes
marginal stability of the shell considering area fluctuations only. (a) The
function is plotted as $a(x)$ with $d=5$ and for fixed values of $y$; (b) The
function is plotted as $a(y)$ with $d=5$ and for fixed values of $x$; (c) The
function is plotted as $a(d)$ for fixed values of the pair $(x,y)$. Points
with lower $a$ than the given function correspond to stable configurations in
all three plots. All the configurations with $a=1$ are below the surface of
marginal stability, therefore they are stable so there is no need for a plot
(d).

(a) $d=5$, $y$ fixed

(b) $d=5$, $x$ fixed

(c) $(x,y)$ fixed

Figure 3:  Thermodynamic stability of the shell considering only charge
fluctuations is described, see Sec. IV.6 and Eqs. (71) and (72). Plots of
$S_{3}\left(d,a,x,y\right)=0$, with $d$ being the number of spacetime
dimensions, $a$ being the exponent of the equation of state of the shell,
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$, and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$,
where $r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius
of the Reissner-Nordström shell spacetime, respectively, and $R$ is the radius
of the shell, are shown. The equality $S_{3}\left(d,a,x,y\right)=0$ describes
marginal stability of the shell considering charge fluctuations only. (a) The
function is plotted as $a(x)$ with $d=5$ and for fixed values of $y$; (b) The
function is plotted as $a(y)$ with $d=5$ and for fixed values of $x$; (c) The
function is plotted as $a(d)$ for fixed values of the pair $(x,y)$. Points
with lower $a$ than the given function correspond to stable configurations in
all three plots. All the configurations with $a=1$ are below the surface of
marginal stability, therefore they are stable so there is no need for a plot
(d).

(a) $d=5$, $y$ fixed

(b) $d=5$, $x$ fixed

(c) $(x,y)$ fixed

(d) $a=1$

Figure 4:  Thermodynamic stability of the shell considering mass and area
fluctuations together is described, see Sec. IV.7 and Eqs. (73) and (74).
Plots of $S_{4}\left(d,a,x,y\right)=0$, with $d$ being the number of spacetime
dimensions, $a$ being the exponent of the equation of state of the shell,
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$, and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$,
where $r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius
of the Reissner-Nordström shell spacetime, respectively, and $R$ is the radius
of the shell, are shown. The equality $S_{4}\left(d,a,x,y\right)=0$ describes
marginal stability of the shell considering together mass and area
fluctuations. (a) The function is plotted as $a(x)$ with $d=5$ and for fixed
values of $y$, with points with lower $a$ than the function corresponding to
stable configurations; (b) The function is plotted as $a(y)$ with $d=5$ and
for fixed values of $x$, with points with lower $a$ corresponding to stable
configurations; (c) The function is plotted as $a(d)$ for fixed values of the
pair $(x,y)$, with points with lower $a$ corresponding to stable
configurations; (d) The function is plotted as $y(x)$ for $a=1$ with fixed
values of $d$, with configurations with higher $x$ than the curve being
stable.

(a) $d=5$, $y$ fixed

(b) $d=5$, $x$ fixed

(c) $(x,y)$ fixed

(d) $a=1$

Figure 5:  Thermodynamic stability of the shell considering mass and charge
fluctuations together is described, see Sec. IV.8 and Eqs. (75) and (76).
Plots of $S_{5}\left(d,a,x,y\right)=0$, with $d$ being the number of spacetime
dimensions, $a$ being the exponent of the equation of state of the shell,
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$, and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$,
where $r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius
of the Reissner-Nordström shell spacetime, respectively, and $R$ is the radius
of the shell, are shown. The equality $S_{5}\left(d,a,x,y\right)=0$ describes
marginal stability of the shell considering together mass and charge
fluctuations. (a) The function is plotted as $a(x)$ with $d=5$ and for fixed
values of $y$, with points with lower $a$ than the function corresponding to
stable configurations; (b) The function is plotted as $a(y)$ with $d=5$ and
for fixed values of $x$, with points with lower $a$ corresponding to stable
configurations; (c) The function is plotted as $a(d)$ for fixed values of the
pair $(x,y)$, with points with lower $a$ corresponding to stable
configurations; (d) The function is plotted as $y(x)$ for $a=1$ with fixed
values of $d$, with configurations with higher $x$ than the curve being
stable.

(a) $d=5$, $y$ fixed

(b) $d=5$, $x$ fixed

(c) $(x,y)$ fixed

Figure 6:  Thermodynamic stability of the shell considering area and charge
fluctuations together is described, see Sec. IV.9 and Eqs. (77) and (78).
Plots of $S_{6}\left(d,a,x,y\right)=0$, with $d$ being the number of spacetime
dimensions, $a$ being the exponent of the equation of state of the shell,
$x=\frac{r_{+}^{d-3}}{R^{d-3}}$, and $y=\frac{r_{-}^{d-3}}{r_{+}^{d-3}}$,
where $r_{+}$ and $r_{-}$ are the gravitational radius and the Cauchy radius
of the Reissner-Nordström shell spacetime, respectively, and $R$ is the radius
of the shell, are shown. The equality $S_{6}\left(d,a,x,y\right)=0$ describes
marginal stability of the shell considering together area and charge