Pressure and volume in the first law of black hole thermodynamics

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 . Re e ences [1] D. Kas o , S. Ray and J. T aschen, Class. Quan um G a . 26 (2009) 195011, [a Xi :0904.2765 [hep- h]]. [2] B.P. Dolan, Class. Quan um G a . 28 (2011) 125020, [a Xi :1008.5023[g -qc]]. 15 [3] M. Henneaux and C. Tei elboim, Phys. Le . 143B (1984) 415; M. Henneaux and C. Tei elboim, Phys. Le . 222B (1989) 195. [4] C. Tei elboim, Phys. Le . 158B (1985) 293. [5] S. Wang, S-Q. Wu, F. Xie and L. Dan, Chin. Phys. Le .,23 (2006) 1096, [a Xi :hep- h/0601147]. [6] Y. Sekiwa, Phys. Re . D73 (2006) 084009, [a Xi :hep- h/0602269]. [7] E.A. La a˜naga Rubio, S ingy Gene aliza ion o he Fi s Law o The - modynamics o Ro a ing BTZ Black Hole wi h a Cosmological Cons an as S a e Pa ame e , [a Xi :0711.0012 [g -qc]]. [8] S. Wang, S-Q. Wu, F. Xie and L. Dan, Chin. Phys. Le .,23 (2006) 1096, [a Xi :hep- h/0601147]; S. Wang, The modynamics o high dimensional Schwa zschild de Si e space imes: a iable cosmological cons an , [a Xi :g -qc/0606109]. [9] M.M. Calda elli, G. Cognola and D. Klemm, Class. Quan um G a . 17 (2000) 399, [a Xi :hep- h/9908022]. [10] B. Ca e , Comm. Ma h. Phys. 10 (1968) 280. [11] M. Henneaux and C. Tei elboim, Comm. Ma h. Phys. 98 (1985) 391. [12] G.W. Gibbons, M.J. Pe y and C.N. Pope, Class. Quan um G a . 22 (2005) 1503, [a Xi :hep- h/0408217]. [13] M. C e ic, G.W. Gibbons, D. Kubiznak and C.N. Pope, Black Hole En halpy and an En opy Inequali y o he The modynamic Volume, [a Xi :1012.2888[g -qc]]. [14] D. Ch is odoulou, Phys. Re . Le . 25 (1970) 1596; D. Ch is odoulou and R. Ru ini, Phys. Re . D4 (1971) 3552. [15] S.W. Hawking, C.J. Hun e and M.M. Taylo -Robinson, Phys. Re . D59 (1999) 0640055, [a Xi :hep- h/9811056]. [16] R.M. Wald, Gene al Rela i i y, Uni e si y o Chicago P ess (1984). [17] See e.g. page 44 in L.D. Landau and E.M. Li schi z, S a is ical Physics. Pa 1, 3 d ed. (1980) Else ie . 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. 20