a Xi :1106.6260 3 [g -qc] 11 No 2011
P essu e and olume in he i s law o black
hole he modynamics
B ian P. Dolan
Depa men o Ma hema ical Physics, Na ional Uni e si y o I eland,
Maynoo h, I eland
and
Dublin Ins i u e o Ad anced S udies, 10 Bu ling on Rd., Dublin, I eland
e-mail: bdolan@ hphys.nuim.ie
No embe 14, 2011
Abs ac
The mass o a black hole is in e p e ed, in e ms o he modynamic po en ials,
as being he en halpy, wi h he p essu e gi en by he cosmological cons an .
The olume is hen de ined as being he Legend e ans o m o he p essu e
and he esul ing ela ion be ween olume and p essu e is explo ed in he case
o posi i e p essu e. A i ial expansion is de eloped and a an de Waals
like c i ical poin de e mined. The i s law o black hole he modynamics
includes a PdV e m which modi ies he maximal e iciency o a Pen ose
p ocess. I is shown ha , in ou dimensional space- ime wi h a nega i e
cosmological cons an , an ex emal cha ged o a ing black hole can ha e an
e iciency o up o 75%, while o an elec ically neu al o a ing back hole
his igu e is educed o 52%, compa ed o he co esponding alues o 50%
and 29% espec i ely when he cosmological cons an is ze o.
PACS nos: 04.60.-m; 04.70.Dy
1 In oduc ion
The he modynamics o black holes is a ich and ascina ing a ea o esea ch
which con inues o yield su p ises. The i s law o black hole he modynam-
ics is usually w i en as
dM =TdS + ΩdJ + ΦdQ (1)
whe e T=κ
2πis he Hawking empe a u e o he black hole (wi h κ he
su ace g a i y), S=A
4 he en opy (wi h A he a ea in Planck uni s), Ω
he angula eloci y, J he angula momen um, Φ he elec os a ic po en ial
di e ence be ween in ini y and he ho izon, Q he elec ic cha ge and M
he mass. The mass is usually in e p e ed as he in e nal ene gy, in he
he modynamic sense, o he black hole, bu i was sugges ed in [1] ha i
is mo e co ec ly in e p e ed as he en halpy. In his con ex i is no able
ha he e is no PdV e m in (1), co esponding o a change in olume a
ambien p essu e P. When a cosmological cons an , Λ, is included he e is
a na u al candida e o a p essu e, P=−Λ
8π, and i was p oposed in [2]
ha he olume o he black hole be de ined as he he modynamic a iable
conjuga e o P. In e p e ing he mass as he en halpy, equa ion (1) should
hen be modi ied o
dM =TdS +V dP + ΩdJ + ΦdQ, (2)
whe e he he modynamic olume is de ined o be V=∂M
∂P S,J,Q, [1]. The
idea ha Λ should be hough o as a he modynamic a iable ha can be
a ied is no new and has been conside ed by a numbe o au ho s, [3]-[9].
Equa ion (2), wi h Q= 0, was s udied in [8], in he con ex o a ying Λ,
wi hou any pa icula physical in e p e a ion being gi en o he he mody-
namic conjuga e o Λ, o en deno ed Θ.
One may ques ion whe he i is app op ia e o iden i y Λ wi h a he mo-
dynamic p essu e. While a cosmological cons an gi es a p essu e e m in
Eins ein’s equa ions, a luid dynamical p essu e is no necessa ily he same as
a he modynamic p essu e. In equilib ium si ua ions howe e i is p esum-
ably co ec o iden i y he luid dynamical p essu e wi h he he modynamic
p essu e and we shall do so he e.
In gene al he en halpy, H, is he hea ene gy belo ed o chemis s, i is no
he in e nal ene gy, U, o he i s law o he modynamics. Tha dis inc ion
goes o he Legend e ans o m o he en halpy,
U=H−PV, (3)
1
whe e His a unc ion o S,P,Jand Qwhile U(S, V, J, Q) is a unc ion o
pu ely ex ensi e a iables. Then we ge he usual o m o he i s law,
dU =TdS −PdV + ΩdJ + ΦdQ. (4)
This equa ion was w i en down in [9], using Θ −Λ no a ion, bu i s conse-
quences we e no pu sued.
When Λ is non-ze o we should expec he P dV e m o con ibu e o
he mechanical ene gy ha can be ex ac ed om a black hole, by a Pen ose
p ocess o example. Fo a nega i e Λ (posi i e p essu e) he PdV e m gi es
a posi i e con ibu ion o dU i he black hole sh inks, and he P dV e m
educes he amoun o ene gy a ailable o ex ac ion as mechanical wo k W,
wi h dW =−dU, hence educing he e iciency. Howe e we shall show ha
he maximal e iciency ac ually inc eases, ela i e o he Λ = 0 case, when
Λ<0, because he maximal angula momen um o a black hole in AdS is
g ea e han ha o one wi h Λ = 0 and his can ou weigh he educ ion
in e iciency: he engine may no be as e icien a a gi en Jbu i can be
pushed o highe J. Con e sely one would expec ha , o a posi i e Λ mo e
ene gy becomes a ailable a a gi en J, ela i e o Λ = 0, as he black hole
sh inks.
O cou se he e a e no pis ons pushing agains a gas o a black hole, bu
a nega i e cosmological cons an con ibu es a nega i e ene gy densi y o
space- ime so a sh inking black-hole exposes nega i e ene gy, hus inc easing
he black hole’s in e nal ene gy and dec easing he amoun o ene gy a ail-
able o mechanical wo k. Con e sely a posi i e cosmological cons an would
p esumably elease ex a ene gy as he black hole sh inks, ha can be used
o do wo k, hence inc easing he e iciency a a gi en J.
The mos e icien way o ex ac ene gy om a black hole is in an isen-
opic p ocess, wi h dS = 0 and he a ea o he e en ho izon cons an .
So
dU ≥dUmin =−PdV + ΩdJ + ΦdQ. (5)
We shall see ha , o a o a ing black hole, i is possible o educe Vwhile
keeping Scons an . The maximum amoun o mechanical wo k ha can be
ex ac ed in passing om an ini ial s a e i o a inal s a e is
Wmax =−Z
i
dUmin.(6)
2
The e iciency is de ined o be he a io o he mechanical ene gy ex ac ed
o he ini ial hea ene gy (en halpy),
η=Wmax
Mi
,(7)
whe e he ini ial en halpy is iden i ied wi h he ini ial mass, Mi. I will be
shown ha , o Λ <0, his can be as high as 52% o a o a ing neu al black
hole and 75% o a cha ged black hole.
The olume o a black hole has only ecen ly been conside ed as a he mo-
dynamic a iable, [2, 13]. A he simples le el, he e is a na u al endency
o assume ha he a ea and he olume a e ela ed geome ically and a e
no independen . Fo a Schwa zschild black hole wi h adius h, o exam-
ple, he a ea is o cou se 4π 2
hand indeed he he modynamic olume wo ks
ou o be 4π
3 3
h, bu his seems co-inciden al and no pa icula signi icance
should be a ached o i . A a deepe le el i is no e en clea how o de ine
a olume geome ically, as he me ic is no s a ic in he in e io o a black
hole. Fo a Schwa zschild black hole which is no o a ing we shall see ha
he he modynamic olume and he a ea a e no independen : ixing S ixes
Vso dV = 0 in an isen opic p ocess, and he P dV e m does no con ibu e
o he i s law. Bu o a o a ing black hole he a ea o he e en ho izon
does no de e mine he he modynamic olume uniquely and i is possible
o a y he olume keeping he en opy cons an , by changing he angula
momen um and/o he cha ge.
The p ope ies o he he modynamic olume and i s con ibu ion o he
i s law o black hole he modynamics a e explo ed in de ail in his pape
o a o a ing cha ged black hole in ou dimensional space- ime wi h a neg-
a i e Λ. In sec ion 2 he modynamic po en ials and he equa ion o s a e
a e discussed and he Legend e ans o m om he en halpy o he in e nal
ene gy is gi en explici ly. In sec ion 3 he e iciency o a Pen ose ype p ocess
is analysed and sec ion 4 con ains a discussion and ou look. Two appendices
a e dedica ed o he echnicali ies o de i ing some esul s used in he ex
2 The in e nal ene gy
Including a p essu e e m in he i s law gi es
dU =TdS −PdV + ΩdJ + ΦdQ
3

whe e he in e nal ene gy, U(S, V, J, Q), is a unc ion o ex ensi e a iables.
The he modynamic olume Vis he conjuga e a iable o he p essu e and
is ob ained om he mass, which is iden i ied in [1] wi h he en halpy, M=
H(S, P, J, Q), by
V=∂H
∂P S,J,Q
.(8)
I was p oposed in [2] ha (8) be de ined o be he he modynamic olume
o he black hole.
The line elemen o a cha ged o a ing black hole in 4-dimensional an i-de
Si e space is [10]
ds2=−∆
ρ2d −asin2θ
Ξdφ2
+ρ2
∆d 2+ρ2
∆θ
dθ2+∆θsin2θ
ρ2ad − 2+a2
Ξdφ2
,
(9)
whe e
∆ = ( 2+a2)(L2+ 2)
L2−2m +q2,∆θ= 1 −a2
L2cos2θ,
ρ2= 2+a2cos2θ, Ξ = 1 −a2
L2,(10)
and he cosmological cons an is Λ = −3
L2, which is ela ed o he p essu e
by 1
L2=8πP
3.
The physical p ope ies o his space- ime a e well known [11], and he
i s law, applied o his me ic, was discussed in [12], bu wi hou a PdV
e m. The me ic pa ame e s mand qa e ela ed o he mass and cha ge by
M=m
Ξ2, Q =q
Ξ.(11)
The e en ho izon, +, lies a he la ges oo o ∆( ) = 0, so
M=( 2
++a2)(L2+ 2
+) + q2L2
2 +L2Ξ2,(12)
and he a ea o he e en ho izon is
A= 4π( 2
++a2)
Ξ.(13)
4
The empe a u e is
T=(L2+ 3 2
+) 2
+−a2(L2− 2
+)−q2L2
4πL2 +( 2
++a2).(14)
The angula momen um, J=aM and he ele an he modynamic an-
gula eloci y is
Ω = a(L2+ 2
+)
L2( 2
++a2).(15)
The elec os a ic po en ial is
Φ = q +
2
++a2.(16)
One can scale Lou om all he abo e exp essions by de ining dimen-
sionless a iables M=M/L, ¯a=a/L, ¯ += +/L, e c., bu we p e e o
keep Lexplici o expose mo e clea ly he ˆole o he p essu e, and make he
compa ison wi h he L→ ∞ limi clea .
Unde he assump ions made he e he he modynamic olume o he
Ke -Newman-AdS black hole wo ks ou o be
V=2π
3( 2
++a2)(2 2
+L2+a2L2− 2
+a2) + L2q2a2
L2Ξ2 +,(17)
which is a simple gene alisa ion o he Ke -AdS olume de i ed in [13]. A
di ec de i a ion o (17) om (12) is mos easily achie ed by i s w i ing he
mass as a unc ion o (S, P, J, Q), di e en ia ing wi h espec o P, and hen
ans o ming back o ( +, a, q, L).
When a= 0 he a ea (13) and olume (17) a e no independen and he
a ea de e mines he olume uniquely, bu when ais non-ze o he a ea and
he olume become independen .
Fo asymp o ically la space, wi h L→ ∞, one has
V=2π
3
( 2
++a2)(2 2
++a2) + q2a2
+
.(18)
While his educes o he na¨ı e esul , 4π 3
+
3, o he Schwa zschild black hole,
a geome ical in e p e a ion when ais non-ze o is no so clea .
A co ec desc ip ion o he he modynamics o he black hole, in e ms o
he modynamic po en ials, equi es eplacing he geome ic a iables ( +, L, a, q)
5
wi h he modynamic a iables (S, P, J, Q). The ele an exp ession o he
mass, and hence he en halpy, was de i ed in [9],
H(S, P, J, Q) := 1
2sS+πQ2+8P S2
32+ 4π21 + 8P S
3J2
πS .(19)
This gene alises he Ch is odoulou-Ru ini o mula [14] o he mass o a
o a ing black hole in e ms o i s i educible mass, Mi . The i educible
mass o a black hole wi h en opy Sis he mass o a Schwa zschild black
hole wi h he same en opy, M2
i =S
4π.
In e ms o he modynamic a iables he empe a u e is, [9],
T=∂H
∂S J,Q,P
=1
8πH "1 + πQ2
S+8PS
31−πQ2
S+ 8PS−4π2J
S2#
(20)
and he he modynamic olume is
V=∂H
∂P S,J,Q
=2
3πH SS+πQ2+8PS2
3+ 2π2J2,(21)
which is mani es ly posi i e.
The Legend e ans o m U=H−PV gi es he he mal ene gy, a unc ion
o pu ely ex ensi e a iables. The ans o m is e alua ed in an appendix o
be
U(S, V, J, Q) = π
S3
3V
4πS
2πS
π+Q2+J2(22)
−|J|(3V
4π2
−S
π3)1
2SQ2
π+J21
2
.
Dimensional analysis implies ha Uis only a unc ion o h ee independen
a iables, since U→λU when S→λ2S,V→λ3V,J→λ2Jand Q→λQ.
One mus be ca e ul aking he J→0 limi o hese po en ials. In his
limi he en halpy
H(S, P, 0, Q) = 1
2 S
π1 + πQ2
S+8SP
3(23)
6
is linea in Pand he Legend e ans o m is singula :
V=∂H
∂P S,Q
=4π
3S
π3/2
(24)
is independen o Pand so canno be in e ed o ob ain P(V). Con e sely,
when Jis ze o, he Legend e ans o m o (23) is
U=1
2 S
π1 + πQ2
S,(25)
which is independen o he olume and is no equal o he J→0 limi o
(22), unless a cons ain , (3V
4π)2=S
π3, is imposed. Indeed (25) gi es he
w ong J= 0 empe a u e (unless Q=P= 0, in which case U=Hand
T=∂U
∂S =1
4π +is he co ec Hawking empe a u e o a Schwa zschild black
hole). To ge he co ec empe a u e om (22) in he J→0 limi we mus
ake he pa ial de i a i e wi h espec o Sbe o e se ing J o ze o, and
ake no e o he ac ha (3V
4π)2=S
π3when J= 0. This cons ain can
be de i ed om (22) di ec ly by obse ing ha Uis no di e en iable wi h
espec o Ja J= 0 unless (3V
4π)2−(S
π)3J2+SQ2
π anishes he e. I
can also be seen di ec ly when a= 0 in (13) and (17).
To de i e he ela ion be ween he p essu e and he olume in gene al we
i s de ine := 3V
4πand s:= S
π. Then (22) becomes
U(s, , J, Q) = 1
s3n
2(s2+s Q2+ 2J2)−|J|p( 2−s3) (J2+s Q2)o(26)
wi h empe a u e
T=1
π
∂U
∂s =|J|{3J2(2 2−s3) + s Q2(5 2−2s3)}
2πs4p( 2−s3)(J2+s Q2)−
s2+ 2s Q2+ 6J2
2πs4
(27)
and p essu e
P=−3
4π
∂U
∂ =3 |J|
4πs3 J2+s Q2
2−s3−3
8πs3s2+s Q2+ 2J2.(28)
In he J→0 limi |J|and √ 2−s3mus anish oge he o ini e Tand
P. The equa ion o s a e, in he o m o he ela ion be ween he p essu e,
7
This allows us o e-exp ess Has a unc ion o V,
H=1
2sβ2−4αγ
V2−γ.(47)
We can immedia ely conclude ha
V2>4π
32S
π3
,(48)
in ag eemen wi h he obse a ion in [13].
I is now s aigh o wa d o de e mine
U=H−PV =H−HV 2
γ−βV
2γ=βV
2γ−p(V2−γ)(β2−4αγ)
2γ,(49)
which immedia ely gi es (22) in he ex .
Appendix 2
In his appendix he i ial expansion is de eloped. Fo simplici y we se
Q= 0, bu he same echniques can be applied o he case o non-ze o Q.
To de elop he expansion we use dimensionless a iables
y:=
J3/2, x := s
J, p := 8πP J
3, := 2πT J1/2,(50)
in e ms o which equa ions (27) and (28) can be w i en
=3(2y2−x3)
x4py2−x3−y(x2+ 6)
x4(51)
p=2y
x3py2−x3−x2+ 2
x3.(52)
When yand xa e la ge le y2≈x3, wi h y2−x3=z2, hen is ini e
p o ided z≈3
y2/3 in which case p≈2
3y1/3. Replacing xwi h zin equa ions
(51) and (52) gi es
3y2+z2=n y2−z24/3+yy2−z22/3+ 6 yoz, (53)
(y2−z2)zp = 2y−zn(y2−z2)2/3+ 2o
⇒p=2
z(y+z)−1
(y2−z2)1/3.(54)
14

Now we expand in powe s o u=1
y1/3. Le z=3u2
ζ o some ζ( , u), in e ms
o which (51) becomes
1 + 9u10ζ2
2=( 1−9u10ζ2
24/3
+u1−9u10ζ2
22/3
+ 6u5)ζ,
(55)
om which we can immedia ely conclude ha
ζ=
+u+ 6u5+ω( , u),(56)
whe e an expansion o ωin us a s a o de 10. Any desi ed o de can be
ob ained by u he expanding ω
ω=u10 ∞
X
n=0
an( )un,(57)
wi h he co-e icien s an( ) o be de e mined. Pu ing (57) and (56) in o
(55), and equa ing co-e icien s o powe s o u, gi es an i e a i e p ocedu e
o e alua ing he co-e icien s anwhich can hen be used o show ha :
ω= 7u10
2−16u11
3+ 27u12
4−40u13
5+ 55u14
6−36 2 + 3 4
7u15 +Ou16.(58)
Finally using his expansion in (54) gi es he equi ed i ial expansion
p=2 u
3−1
3u2+ 2 u6−u10 8u
−9u2
2+ 10u3
3−11u4
4+ 12u5
5(59)
−13 1 + 6 4
6u6+Ou17,
he i s h ee e ms o which a e used in he ex .
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16
0.003
8
0.002
0.001
6
042
0.005
10
0.004
Figu e 1: P-Vdiag am o J= 1 and Q= 0, plo ed using he app oxima ion
in equa ion (31). Pis plo ed as a unc ion o 1
3 o T= 0.02, 0.025, 0.3,
0.35, 0.04, 0.045 and 0.05. The c i ical poin o his alue o Jis Tc≈0.0413,
Pc≈0.00280 and c≈3.08. J > 0 causes he apid ise in Pa low alues o .
17
Figu e 2: CVas a unc ion o S/L2and J/L2, wi h ixed olume se o L3.
18
Figu e 3: CPas a unc ion o S/L2and J/L2, wi h ixed p essu e
se o 3
8π,i.e L= 1. CPdi e ges along cu e IV in igu e 4 and
anishes along cu e I.
19

1.41.210.80.60.40.20
0.2
0.15
0.1
0.05
0
43
2
2
1.5
1
1
0.5
0
0
J/L2
J/L2
S/L2
S/L2
I
III
II
V
IV
III
II
I
IV
Figu e 4: phase diag am o q= 0, plo ed in e ms o S/L2and J/L2.
The egion abo e I is o bidden, because T < 0;
in he egion abo e cu e II he 3-d Eins ein uni e se a in ini y o a es
as e han he speed o ligh ;
cu e III bounds he egion o local s abili y (an analysis o he Gibbs ee
ene gy shows ha he black hole is locally uns able abo e cu e III);
CPdi e ges on cu e IV;
in he egion abo e cu e V he black hole is uns able due o he Hawking-
Page phase ansi ion;
in egion below cu e V he black hole is s able.
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