Heat capacity of a self-gravitating spherical shell of radiations

2.9 MB·

by  Hyeong-Chan Kim

Heat capacity of a self-gravitating spherical shell of radiations
Hyeong-Chan Kim1,∗
1School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 380-702, Korea
We study the heat capacity of a static system of self-gravitating radiations analytically in the
context of general relativity. To avoid the complexity due to the conical singularity at the center,
weexcisethecentralpartandreplaceitwitharegularsphericallysymmetricdistributionofmatters
inwhichspecificationswearenotinterested. Weassumethatthemassinsidetheinnerboundaryand
thelocationsoftheinnerandtheouterboundariesaregiven. Then,wederiveaformularelatingthe
variationsofphysicalparametersattheouterboundarywiththoseattheinnerboundary. Because
there is only one free variation at the inner boundary, the variations at the outer boundary are
dependent on each other, which determines the heat capacity. To get an analytic form for the heat
capacity,weusethethermodynamicidentityδS =βδM ,whichisderivedfromthevariational
rad rad
relation of the entropy formula. Even if the radius of the inner boundary of the shell goes to zero,
the heat capacity does not go to the form of the regular sphere.
7
1
PACSnumbers: 04.20.Jb,04.40.Nr,04.40.Dg
0
Keywords: self-gravitatingradiation,gravity,horizon,heatcapacity
2

n
a
J
I. INTRODUCTION
0
2
In 1980, Landau and Lifshitz [1] pointed out that system bound by long range forces might exhibit negative heat

]
capacity even though the specific heat of each volume element is positive. Since then, many such examples were
c
q found, e.g., mainly stars and blackholes. A self-gravitating isothermal sphere also belongs to this class, which can
- be regarded as a model of a small dense nucleus of stellar systems [2]. Based on general relativity, the model was
r
g dealtbySorkin, Wald, andJiu[3]in1981asasphericallysymmetricsolutionwhichmaximizesentropy. Schmidtand
[ Homann[4]calledthegeometrya‘photonstar’. Theheatcapacityandstabilityofthesolutionwerefurtheranalyzed

in Refs. [5–8]. Thereafter, the system has drawn attentions repeatedly in relation to the entropy bound [9, 10],
1
blackhole thermodynamics [11], maximum entropy principle [12–14], holograpic principle [15–17], and conjecture
v
7 excluding blackhole firewalls [18]. A system of self-gravitating radiations in an Anti-de Sitter spacetime was also
9 pursued[19–21]. Studiesonthesystemsofself-gravitatingperfectfluidsareundergoing[22]. Aninterestingextension
8 of the self-gravitating system was presented in Ref. [4, 23] where a conical singularity was included at the center as
5 an independent mass source from the radiation. Some of the singular solutions were argued to have an interesting
0
geometry,whichissimilartoaneventhorizoninthesensethat1−2m(r)/rhasaminimumvalueclosetozero. Analytic
.
1 approximation was tried to understand the situation that a blackhole is in equilibrium with the radiations [24]. It
0 was also argued that the thermodynamics of a black hole in equilibrium implies the breakdown of Einstein equations
7 on a macroscopic near-horizon shell [25]. The geometrical details of solutions having conical singularity were dealt in
1
Ref. [26].
:
v Let us consider a static spherically symmetric system of self-gravitating radiations confined in a spherical shell
i bounded by two boundaries located at r = r and r = r in the context of general relativity. For a generic time
X − +
symmetric data, the initial value constraint equations become simply (3)R = 16πρ. As described in Ref. [3], this
r
a determines the spatial metric to be the form hijdxidxj =[1−2m(r)/r]−1dr2+r2dΩ2, where
(cid:90) r
m(r)=M +4π ρ(r(cid:48))r(cid:48)2dr(cid:48). (1)
−
r−
Here M represents the mass inside the inner boundary at r . At present, we do not assume anything for the nature
− −
of M except for the spherical symmetry. Therefore, it can take negative value. The mass of the radiations in the
−
shell is
M =M −M ; M ≡ lim m(r), (2)
rad + − +
r→∞
∗Electronicaddress: [email protected]
2
where M denotes the total mass of the solution. We neglect the energy density of the confining shell and assume
+
thatnomatterslieoutsideoftheouterboundary. Weassumethattheradiationdoesnotinteractwithmattersinside.
The energy density of the radiation at the outer surface of the shell with local temperature T is
+
ρ(r )=σT4, (3)
+ +
where σ is the Stefan-Boltzmann constant. The local heat capacity (LHC) for fixed volume with respect to the local
temperature is
(cid:18) (cid:19)
∂M
C ≡ rad . (4)
local ∂T
+ r±,M−
In fact, the condition of fixed volume is not transparent because the metric g contributes to the volume. In this
rr
work, we simply use the terminology to represent that the areal radius of the inner and outer boundaries do not
change. Direct analytic calculation of the heat capacity is impossible because it requires to solve the corresponding
equationofmotionsanalytically,whichwassolvedonlynumericallyinthepreviousliteratures. However,onecanfind
a detour through the variation of entropy.
Formally, the entropy of the radiation with equation of state, ρ(r) = 3p(r), can be obtained by integrating its
entropy density over the volume, [3]
(cid:90) r+ 4(4πσ)1/4r1/2[m(cid:48)(r)]3/4
S = L(r)dr; L≡ , (5)
rad 3 χ(r)
r−
(cid:112)
where χ(r)≡ 1−2m(r)/r. The variation of S with respect to a small change of m(r) gives
rad
(cid:90) r+(cid:20)∂L d ∂L (cid:21) (cid:20) ∂L (cid:21)r+
δS = − δmdr+ δm
rad ∂m dr∂m(cid:48) ∂m(cid:48)
r− r−
(cid:90) r+ δS
= radδmdr+β δM −β δM , (6)
δm + + − −
r−
where
(cid:18) ∂L (cid:19) (cid:20)r1/2(cid:16) 4πσ (cid:17)1/4(cid:21)
β ≡ = . (7)
± ∂m(cid:48) χ m(cid:48)(r)
r→r± r→r±
Noting the relation of mass with the surface energy density in Eqs. (1) and (3), the local temperature T is related
+
with β ≡β as
+
β−1 =χ T . (8)
+ +
One may introduce the metric component g so that the local temperature at r is given by the Doppler-shifted
tt
temperature as,
(cid:112) (cid:112) (cid:16)ρ(r)(cid:17)1/4
−g (r)T(r)= −g (r) =β−1. (9)
tt tt σ
This result can also be obtained by solving the Einstein’s equation directly. This equation indicates that β−1 is the
global temperature measured at the asymptotic region. On the other hand, β is not directly related with the local
−
temperature T by the relation in Eq. (9) but is related by
−
(cid:114)r
β−1 = + χ T . (10)
− r − −
−
Given the temperature β−1, the heat capacity for fixed volume of the shell is defined by
(cid:16)∂M (cid:17) (cid:16)∂M (cid:17) (cid:16)∂M (cid:17)
C = rad = + − − . (11)
V ∂β−1 r+ ∂β−1 r+ ∂β−1 r+
At the present case, the second term vanishes because M is held.
−
3
Introducing scale invariant variables u and v as
2m(r) dm(r)
u≡ , v ≡ =4πr2ρ(r)=4πσr2T(r)4, (12)
r dr
the variational equation δS /δm=0 becomes a first order differential equation for u and v,
rad
dv 2v(1−2u−2v/3)
=f(u,v)≡ . (13)
du (1−u)(2v−u)
ThisequationisequivalenttothegeneralrelativisticTolman-Oppenheimer-Volkoffequationofhydrostaticequilibrium
for a radiation. The allowed range of (u,v) is u < 1 and v ≥ 0, where each inequality represents the fact that the
spacetimes is static and the energy density of radiation is non-negative, respectively. Integrating this equation on
the (u,v) plane, solution curves were found in Refs. [3, 24]. Any solution curve will be parallel to the u-axis when it
crosses the line
2v
P : 2u+ =1, (14)
3
and is parallel to the v-axis when it crosses the line
H : u=2v. (15)
The solution curve eventually converges to the point R≡(3/7,3/14) where the two lines P and H meet. A specific
solution curve C is characterized by
ν
2m(r )
ν ≡1−u =1− H , (16)
H r
H
the orthogonal distance of C from the line u=1 on the (u,v) plane [26]. Here the subscript represents the point
ν H
whereC crossesH,whichisthepositionoftheapproximatehorizondefinedbyasurfacethatresemblesanapparent
ν
horizon [26]. We quote the name ‘approximate horizon’ from Ref. [25]. The value of ν varies from zero, the solution
correspondingtotheformationofaneventhorizon,toν ≈0.50735,theeverywhereregularsolution. Agivensolution
r
curve is parameterized by a scale invariant variable
r
ξ ≡log . (17)
r
H
Therefore, the physically relevant region of (u,v) plane can be equivalently coordinated by using the set (ν,ξ). Now,
aspecificspheresolutionofradiationscanbecharacterizedbychoosingaboundarypointonacurveC afterpicking
ν
the radius of the boundary r .
+
Agivensphericalshellofradiationcanbedenotedbyfourdifferentnumbers,(ν,rH,eξ+,eξ−),representingaspecific
solution curve, the radius of the approximate horizon for the solution curve, and the positions of the inner and the
outerboundariesrelativetotheapproximatehorizon,respectively. Thescaleinvariantvariablesattheouterboundary
are related with the total mass, local temperature, and the radius as
2M
r =r eξ+, u ≡u (ξ )= +, v ≡v (ξ )=4πr2ρ(r )=4πσr2T4, (18)
+ H + ν + r + ν + + + + +
+
where we put the subscript ν to u and v to represent the specific solution curve C . The scale invariant values at the
ν
inner boundary are given by
2M
r =r eξ−, u ≡u (ξ )= −, v ≡v (ξ )=4πr2ρ(r ). (19)
− H − ν − r − ν − − −
−
In this work, the value of u and r are held. On the other hand, the value of v should be determined by tracing
− − −
in the solution curve C from the data at the outer boundary.
ν
Even though the static solution of the self-gravitating radiations were studied well, its stability needs additional
study. Toachievethispurposewestudyitsheatcapacity. InSec. II,wefirstderivetherelationbetweenthevariations
of(u,v)andthoseof(ν,ξ). Byusingthefactthatδν andδξ arethesameif(u ,v )and(u ,v )areononesolution
+ + − −
curve C , we relate the variations of physical parameters at the outer boundary with those at the inner boundary. In
ν
Sec. III, we calculate the (local) heat capacity for fixed volume from the variational equation of entropy. We show
thatthegeneralheatcapacityislocatedinthemiddleofthetwoextremeforms, thatoftheregularsolutionandthat
of other extreme. In Sec. IV, we study various limiting behaviors of the heat capacity. We summarize the results in
Sec. V.
4
II. VARIATIONS OF THE SCALE INVARIANT VARIABLES
The difficulty in calculating the heat capacity of spherical shell of matters lays on the fact that the physical
parameters at the inner boundary are dependent on those at the outer boundary, which exact relation needs analytic
solutions. Rather than searching for an exact analytic solution, we find a variational relation between them. Because
δr = 0 = δM , we have δu = 0 leaving only δv be dependent on the variations at the outer boundary. We
− − − −
study how to relate the variations at the outer boundary with those at the inner boundary in a general setting. To
do this, we find the variations δν and δξ corresponding to the variations (δu ,δv ). Then, we use (i) The variation
+ +
δν is independent of the position of (u ,v ) if it is on the same solution curve C with (u ,v ). (ii) The variation
− − ν + +
δξ = δr/r −δr /r is also independent of the position of (u ,v ) if r = r are held. (iii) ν and ξ defines an
H H − − ±
orthogonal coordinates system which is equivalent to (u,v) physically.
A. Generic variations
Let us consider the variation δξ and δν, which represent the variations parallel to and orthogonal to the solution
curve C , respectively. The two variations are orthogonal to each other and define most general changes of boundary
ν
points, (u,v) ≡ (u ,v ), on C . An important point here is that ν is independent of the choice of the boundary
± ± ν
point on C by definition and the variation of ξ = log(r /r ) is required to be dependent only on the change of
ν ± ± H
r because r are held. Therefore, we have δξ = δr /r = δξ . In this subsection, we omit the subscript ± for
H ± + H H −
notational simplicity.
From Eqs. (12) and (17), we get (∂u/∂ξ) = 2v−u. Therefore, by using Eq. (13), the tangent along the solution
curve is given by
∂ ∂ ∂
=(2v−u) +(2v−u)f(u,v) . (20)
∂ξ ∂u ∂v
On the other hand, the derivative orthogonal to the solution curve C can be written as
ν
∂ ∂ ∂
=−B(u,v)f(u,v) +B(u,v) , (21)
∂ν ∂u ∂v
where we use ∂/∂ξ ⊥∂/∂ν and B is a local function of (u,v) defined by
(cid:18) (cid:19)
∂v
B(u,v)≡ . (22)
∂ν
ξ
TheexplicitfunctionalformofB willbedeterminedlaterinthissectioninEq.(35)fromconsistency. FromEqs.(20)
and (21), we determine δu and δv in terms of δν and δξ as,
δu=(2v−u)δξ−Bfδν, δv =(2v−u)fδξ+Bδν. (23)
Inverting Eq. (23), the variation δν and δξ are given by
−fδu+δv δu+fδv
δν = , δξ = . (24)
B(1+f2) (2v−u)(1+f2)
Let us see the results at the point r = r where u = 2v . At that point, δξ and δν are parallel to δv and δu,
H H H
respectively. Therefore, (δu/δξ) =0, (δv/δν) =0. In addition, from Eq. (23),
r=rH r=rH
(cid:20) (cid:21)
14v /3−1
δv =[(2v−u)f(u,v)] δξ = H v δξ, δu =− lim B(u,v)f(u,v) δν. (25)
H r→rH 1−2vH H H r→rH
Fromthefirstequation, onenotesthatδv divergesasv →1/2, whichcorrespondstothelimitofforminganevent
H H
horizon. The second equation, by using Eq. (16), determines the normalization of B to be
lim f(u,v)B(u,v)=1. (26)
r→rH
Later in this section, we finalize the function B from this normalization condition. Because f(u,v) diverges on H,
the value of B will vanish there.
5
B. Variations at the inner and the outer boundaries
By using the fact that δν and δξ are independent of the position on a given solution curve C , we relate the
ν
variations at the outer boundary with those at the inner boundary. From Eqs. (23) and (24), the variation of u can
−
be written by the variations at the outer boundary as
δu = −B f δν+(2v −u )δξ
− − − − −
(cid:18) (cid:19) (cid:18) (cid:19)
f f B 1 2v −u f B f 2v −u
= + − − + − − δu + − − − + + − − δv , (27)
1+f2 B f f 2v −u + 1+f2 B f 2v −u +
+ + + − + + + + − + +
where B and f stand for B(u ,v ) and f(u ,v ), respectively. Using the definition of the LHC in Eq. (4) after
± ± ± ± ± ±
dividing Eq. (27) by δM , we get
+
(cid:18) (cid:19) (cid:18) (cid:19)
2r v B 1 2r v f 2v −u
f − + +C−1 − =− + + +C−1 + − −, (28)
+ T local B f T local f 2v −u
+ + + + − + +
where we use (∂u /∂M ) = 0 because r and M are held. In a similar manner, the variation of v can be
− + r±,M− − − −
written by means of the variations at the outer boundary as
δv = B δν+(2v −u )f δξ
− − − − −
(cid:18) (cid:19) (cid:18) (cid:19)
f B (2v −u )f 1 B (2v −u )f f
= + − − + − − − δu + − + − − − + δv . (29)
1+f2 B (2v −u )f + 1+f2 B 2v −u +
+ + + + + + + + +
Using Eq. (4) after dividing Eq. (29) by δM , we get
+
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
∂v 2 1 2r v B 2v −u 1 2r v
− = − f − + +C−1 − +f f − − + + +C−1 . (30)
∂M r 1+f2 + T local B + −2v −u f T local
+ r±,M− + + + + + + + +
Putting Eq. (28) to Eq. (30), one gets
(cid:18) ∂v (cid:19) 2 f +f−12v −u (cid:20) 1 2r v (cid:21)
− = − − − − + + +C−1 . (31)
∂M r f +f−1 2v −u f T local
+ r±,M− + + + + + + +
Once we get the LHC we can obtain the function B from Eq. (28) in addition to the relation between the outer
boundary and the inner boundary through Eq. (31).
Even though C will be identified later in Eq. (44), we use the result here to determine the function B. Equa-
local
tion (28) gives
(cid:114)r f B (2v −u )A
− − − = − − −, (32)
r f B (2v −u )A
+ + + + + +
where A ≡A(u ,v ) and
± ± ±
v3/4 f v3/4χ F
A(u,v)≡ = . (33)
χ (2v−u)2(1+f2) 2v−uF2+G2
In the second equality, we use f =F/G with
(cid:0) 2v(cid:1)
F ≡2v 1−2u− , G≡(1−u)(2v−u). (34)
3
Note that the function (2v −u)A(u,v) is regular on the whole range of physical interest except for the point R,
which corresponds to the asymptotic infinity r →∞ of all solution curves. B(u,v) must be a local function of (u,v).
Therefore, Eq. (32) determines B(u,v) up to a proportionality constant which is a function of ν only,
(cid:114)r (2v−u)A(u,v) (cid:114)r v3/4χG
B(u,v)=α H =α H . (35)
ν r f(u,v) ν r F2+G2
Here ν and r/r =eξ are implicitly dependent on u and v. It goes to zero as expected in Eq. (26). B(u,v) diverges
H
on R. The proportionality constant α can be fixed by using Eq. (26), after choosing (u ,v ) = (u ,v ) and
ν + + H H
(u ,v )=(u,v), to be
− −
1 23/4(7ν−4)(1−ν)1/4
α = lim = . (36)
ν r→rH (2v−u)A 3 ν1/2
Note that α is negative definite because ν is restricted to be 0<ν ≤ν <4/7.
ν r
6
III. CALCULATION OF THE HEAT CAPACITY
Once the equation of motion in Eq. (13) holds, the variational equation (6) of the entropy presents the first law of
thermodynamics as
δS =βδM −β δM =βδM +(β−β )δM . (37)
rad + − − rad − −
When δM =0, one can identify β ≡β as the inverse temperature measured at the asymptotic region. In a general
− +
situation, δM could be dependent on the choice of the values at the outside and the radius of the inner boundary as
−
∂M ∂M ∂M ∂M
δM = −δM + −δβ+ −δr + −δr . (38)
− ∂M + ∂β ∂r + ∂r −
+ + −
To obtain heat capacities other than that conserve the mass inside the inner boundary, δM should be dealt in this
−
way.
Fortunately,theintegrationinEq.(5)canbeexecutedtogiveananalyticformfortheentropyoftheradiation[3,6],
r3/2 (cid:18)4πσ(cid:19)1/4(cid:16)2v (cid:17) r β (cid:16)2v (cid:17)
S ≡S −S , S (u ,v ,r )= ± ± +u = ± ± ± +u . (39)
rad + − ± ± ± ± 3χ v 3 ± 3 3 ±
± ±
Note that S does not represent the entropy of the objects inside r . For later convenience, we put the derivative of
± ±
S as
±
1 2v r 2−u +2v /3 r 2v −u
β−1dS = ( ± +u )dr + ± ± ± du + ± ± ±dv . (40)
± ± 2 3 ± ± 6 1−u ± 12 v ±
± ±
If we consider on-shell change [du and dv are related by Eq. (13)], we get the first law of thermodynamics, dM =
β−1dS−p(4πr2)dr.
A. Local heat capacity
Now, let us calculate the LHC for fixed volume in Eq. (4). Direct calculation of the heat capacity needs to solve
the equation of motion (13) from r to r , which is impossible analytically. On the other hand, the entropy has an
− +
exact analytic expression. Therefore, it would be better to use Eq. (37) to obtain the heat capacity. When M is
−
held, Eq. (37) becomes
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
∂S ∂S ∂S
0= rad −β = + −β− − , (41)
∂M ∂M ∂M
rad r±,M− + r+ + r±,M−
where S is given in Eq. (39). Because r and M are held, S and S are local functions of (u ,v ) and v ,
rad ± − + − + + −
respectively.
Before dealing complex general cases, let us review how the heat capacity for a self-gravitating radiation sphere
with regular center was calculated in Ref. [5] by choosing ν = ν and r = 0. Because S = 0, the last term in the
r − −
right-hand side of Eq. (41) vanishes. Noting r is held, by using Eqs. (18) and (40), the right-hand side of Eq. (41)
+
becomes
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
∂S 2 ∂S 4v ∂T ∂S
+ −β = + + + + + −β
∂M r ∂u T ∂M ∂v
+ r+ + + r+,v+ + + r+ + r+,u+
(cid:34) (cid:18) (cid:19) (cid:35)
βr (2v −u ) 2f 4v ∂T
= + + + − + + + + . (42)
12v r T ∂M
+ + + + r+
Foraregularsolution,thereremainsonlyonefreedegreeoffreedominthephysicalparametersattheouterboundary
because the size r is held. Therefore, the variations δu and δv must be dependent on each other, which relation
+ + +
determines the LHC. Now, the LHC for the regular solution is given after setting Eq. (42) to zero:
(cid:18) (cid:19)
∂M 2r v
Creg = + = + +. (43)
local ∂T T f
+ r+ + +
7
The LHC for self-gravitating regular sphere of radiations diverges when the solution curve intersects P and vanishes
when the solution curve intersects H.
To obtain the LHC for a general case with r (cid:54)= 0, the effect of S should also be taken into account. Equating
− −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Eq. (41) by using Eqs. (31), (39), (40), (42) and using ∂S− = ∂v− ∂S− , we get
∂M+ r±,M− ∂M+ r± ∂v− r−,u−
2r v 1−A (cid:114)r A
C = + + , A≡ − +, (44)
local T f 1+f−2A r A
+ + + + −
where we use Eqs. (8), (9), (10), and (33) to simplify the form of the equation. In the limit r →r , the LHC goes
− +
1.0 1v.0
+ C >0
local
P
0.8 0.8
P
0.6 0.6
H
5
0.4 H 0.4 0
1
•R •
0.2 0 0.2 R
C <0
local
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 u+1.0
FIG.1: T C /r forthelimitofA→0(L)andA→∞(R).Oneachfigure,singularitiesoftheLHCareplottedaswiggly
+ local +
line and marked by a red character.
to zero as expected. The LHC is formally the same as that of the regular sphere in Eq. (43) when A→0, which limit
is shown in the left panel of Fig. 1. It is singular on P and vanishes on H. When viewed from the clockwise direction
centered on R, the LHC takes positive values from H to P and negative values elsewhere. This limit happens when
A → ∞ which is achieved if the inner boundary is located on the approximate horizon, i.e., r = r . To outward
− − H
seeming, the limit r → 0 also leads to the result. However, as will be shown in subSec. IVB, this is not the case
−
exceptfortheregularsolutionbecauseA behavesnontrivially. TheoppositelimitA→∞isachievedwhenA →0.
− −
This limit happens if the inner boundary is located on the line P. This case is displayed in the right panel of Fig. 1.
In this case, the LHC becomes
2r v f
CP ≈− + + +. (45)
local T
+
ThisissingularonthelineH andvanishesonthelineP. Whenviewedfromtheclockwisedirection, ittakespositive
values from P to H and negative values elsewhere.
Writing the LHC (44) explicitly in terms of (u ,v ),
+ +
C = 2r+v+G+(F+2 +G2+)− A−(vr+3+//4rχ−3)1/2F+. (46)
local T+ F (F2 +G2)+ v+3/4χ3 G
+ + + A−(r+/r−)1/2 +
Note that the denominator vanishes on the curve
(cid:16) 2v (cid:17) v−1/4χ5
S : 2 1−2u − + (F2 +G2)+ + (2v −u )=0. (47)
+ 3 + + A (r /r )1/2 + +
− + −
Onthiscurve, theLHCissingular. ThecurveS passesthepointRalongthelineH becauseF →0andG →0at
+ +
R leaving the last term in Eq. (47) as the first nontrivial corrections. Equation (47) indicates that S must be around
P for A (r /r )1/2 (cid:29) 1 and around H for A (r /r )1/2 (cid:28) 1. These behaviors are manifest in the left and right
− + − − + −
panel of Fig. 2, respectively. As F or G are larger, i.e. v increases, the singular curve gradually approaches the
+ + +
8
S S v S
1.4 1.4 1.4+ Clocal >0
1.2 1.2 1.2
N
1.0 1.0 1.0
2
0.8 0.8 0.8
2
0.6 0.6 0.6 1
0.4 0.4 0.4 C <0 0
N local
0.2 0 0.2 0 0.2
N
0.0 0.0 0.0
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 u+1.0
FIG. 2: T C /(r ) with respect to the location of the outer boundary. The heat capacity for A r1/2 = 20,1, and 1/20,
+ local + − +
respectively from the left.
lineP. CombiningthepicturesinFig.1withthoseinFig.2,onemayfindthatthesingularcurveS graduallychange
from the line P to the line H as A (r /r )1/2 decreases. The numerator of Eq. (46) vanishes on the curve
− + −
√
2 1−u v7/4(cid:16) 2v (cid:17)
N :(2v −u )(F2 +G2)− + 1−2u − + =0. (48)
+ + + + A (r /r )1/2 + 3
− + −
The LHC changes signature on the null curve N. The null curve passes the point R along the line P because
F ,G →0atRleavingthelastterminEq.(48)asthefirstnontrivialcorrections. Equation(48)indicatesthatthe
+ +
curve N must overlaps with H for A (r /r )1/2 (cid:29) 1 and with P for A (r /r )1/2 (cid:28) 1, respectively. Combining
− + − − + −
the pictures in Fig. 1 with those in Fig. 2, one may notice the the null curve gradually changes from the line H to the
line P as A (r /r )1/2 decreases. When viewed from the clockwise direction centered at R, the LHC takes positive
− + −
values from N to S and negative values elsewhere.
B. Heat capacity
The heat capacity (11) with respect to the global temperature β−1 can be obtained based on the value of LHC. At
the present case, the second term in Eq. (11) vanishes because M is held. Let us calculate the first term in the right
−
hand side. Varying Eq. (8), we get
M T T
δβ−1 = + +δr − + δM +χδT , (49)
χ2r2 + r χ + +
+ +
where we regard β as a function of T , M , and r . Generally, the three variations δT , δM , and δr are
+ + + + + +
independent. However,ifthestateinsidetheinnerboundaryisinvariantunderthechangesofthephysicalparameters
at the outer boundary, i.e. δr = 0 = δM , only two of the three variations will be independent. If the size of the
− −
shell is held, δr = 0, only one independent variation remains. In this case, the variations δT and δM must be
+ + +
related. The heat capacity is given by dividing both sides of Eq. (49) with δβ−1,
(cid:16)∂M (cid:17) r χ2 1
C = rad = + . (50)
V ∂β−1 r±,M− β−1 (r+χ2/T+)Cl−oc1al−1
For the case of a regular sphere, inserting the result in Eq. (43) to Eq. (50), the heat capacity becomes
(cid:16)∂M (cid:17) r χ2 2v −u
Creg = + =− + + + . (51)
V ∂β−1 r+ β−1 8v+/3−1+u+
The heat capacity changes sign on the line H and is singular on the line
8v
Q: + +u =1. (52)
3 +
9
In a general case, inserting Eq. (44) to Eq. (50), the heat capacity for the shell is given by
r χ2 1−A
C = + . (53)
V β−1 χ2f /(2v )−1+(χ2/(2v f )+1)A
+ + + +
InthelimitA→0,theheatcapacityreproducesthatoftheregularsphereinEq.(51). OntheoppositelimitA→∞,
1.0 1v.0
+
SP
0.8 0.8
P
0.6 0.6
0.4 H 0.4 0
Q
−0.2
0.2 0.5 0.2
0
0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 u+1.0
FIG. 3: T C /r for the limit of A→0 (L) and A→∞ (R).
+ local +
we have
r χ2 4v2(1−2u −2v /3)
CP ≡ lim C =− + + + . (54)
V A→∞ V β−1 (1−u+)2(2v+−u+)+4v+2(1−2u+−2v+/3)
As shown the the right panel of Fig. 3, the heat capacity in this limit vanishes on the line P and is singular on the
curve
SP :(1−u )2(2v −u )+4v2(1−2u −2v /3)=0. (55)
+ + + + + +
The curve passes the point R. It overlaps with the line H for v (cid:28)1 and approaches the line
+
2(cid:18) 22/3 √ (cid:19)
1−u =βv , β = −1− √ +(8+6 2)1/3 ≈0.5062 (56)
+ + 3 (4+3 2)1/3
for v (cid:29)1. These behaviors are manifest in the right panel of Fig. 3.
+
Writing the heat capacity (53) explicitly in terms of (u ,v ), we get
+ +
C = r+χ2 (2v+−u+)(F+2 +G2+)− A−(vr+3+//4χr−+)1/2F+ . (57)
V β−1 (cid:0)1−u − 8v+(cid:1)(F2 +G2)+ v+3/4χ+ (cid:2)F +χ2G /(2v )(cid:3)
+ 3 + + A−(r+/r−)1/2 + + + +
The denominator of Eq. (57) vanishes on the curve given by
(cid:16) 8v (cid:17) v−1/4χ (cid:16) (cid:17)
S(cid:48) : 1−u − + (F2 +G2)+ + + χ2G +2v F =0. (58)
+ 3 + + 2A (r /r )1/2 + + + +
− + −
On this curve, the heat capacity is singular. The curve passes R along the curve SP because F → 0 and G → 0
+ +
at R leaving the last term in Eq. (58) as the first nontrivial corrections. Equation (58) indicates that the singular
curve S(cid:48) must be around the line Q and the curve SP for A (r /r )1/2 (cid:29) 1 and A (r /r )1/2 (cid:28) 1, respectively.
− + − − + −
These behaviors are manifest in the left and right panels of Fig. 4, respectively. As F or G are larger, i.e., χ or v
+ + +
increases, the singular curve gradually approaches the line Q. Combining the pictures in Figs. 3 and 4, one may find
the the singular curve gradually change from the line Q to the curve SP as A (r /r )1/2 decreases.
− + −
The numerator of Eq. (57) vanishes on the curve
v7/4χ (cid:16) 2v (cid:17)
N(cid:48) :(2v −u )(F2 +G2)− + 1−2u − + =0. (59)
+ + + + A (r /r )1/2 + 3
− + −
10
1.4 1.4 1v.4+ Clocal >0
N(cid:48) S(cid:48)
1.2 1.2 1.2
1.0 1.0 1.0
−1
0.8 0.8 0.8
S(cid:48)
0.6 0.6 0.6
S(cid:48)
0.4 0.4 0.4 C <0 −0.2
local
N(cid:48)
0.2 1 0 0.2 0 0.2
N(cid:48) 0
0.0 0.0 0.0
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 u+1.0
FIG. 4: β−1C /(r ) with respect to the location of the outer boundary. The heat capacity for A r1/2 = 20,1, and 1/20,
+ V + − +
respectively from the left.
TheheatcapacitychangessignonN(cid:48). ThecurvepassesthepointRalongthelineP becauseF →0andG →0at
+ +
R leaving the last term in Eq. (59) as the first nontrivial corrections. Equation (59) indicates that the curve N(cid:48) must
be around the line H for A (r /r )1/2 (cid:29)1 and around P for A (r /r )1/2 (cid:28)1, respectively. These behaviors are
− + − − + −
manifest in the left and right panels of Fig. 4, respectively. Combining the pictures in Fig. 3 and Fig. 4, one may find
the the curve N(cid:48) gradually changes from the line H to the line P as A (r /r )1/2 decreases. When viewed from the
− + −
clockwisedirectioncenteredatR,theheatcapacitytakespositivevaluesfromN(cid:48) toS(cid:48) andnegativevalueselsewhere.
IV. VARIOUS LIMITS
Because the functional forms of the LHC and the heat capacity are complicated, we present various physically
interesting limits to improve our understanding on the system.
A. Thin shell limit
We first consider the thin shell limit, δr ≡ r −r (cid:28) r ≈ r, δu ≡ u −u (cid:28) u , and δv ≡ v −v (cid:28) v . By
+ − + + − + + − +
using
rδu
δr =rδξ = , δv =f(u,v)δu,
2v−u
the heat capacity and the LHC in this limit take the form,
χ−1C C v (cid:20) δlogA(cid:21)
V = local ≈ 1−2r
δr δr T(f +f−1) δr
2v(2v−u)FG(cid:20)4v2−4v−u2+28uv/3 2
≈ −
3T(F2+G2) (1−u)(2v−u)2 1−2u−2v/3
2−16v3+4(7−13u)v2+2(u−1)(5u−3)v−9u(u−1)(2u−1) (cid:21)
+ . (60)
3 F2+G2
Interestingly,theheatcapacityisregularoverallphysicallyallowedvaluesof(u,v)(cid:54)=R. Eventhoughtheytakeboth
signatures, their values change smoothly.
B. r (cid:28)r approximation
− H
Let us next consider the ‘almost sphere’ case which excises only the central singularity by using the limit r →0.
−
We are interested in the solution other than the regular one, ν (cid:54)= ν . Solving Eqs. (12) and (13) around the center
r