Interpreting the cosmological constant as a pressure, whose thermodynamically
conjugate variable is a volume, modifies the first law of black hole
thermodynamics. Properties of the resulting thermodynamic volume are
investigated: the compressibility and the speed of sound of the black hole are
derived in the case of non-positive cosmological constant. The adiabatic
compressibility vanishes for a non-rotating black hole and is maximal in the
extremal case --- comparable with, but still less than, that of a cold neutron
star. A speed of sound $v_s$ is associated with the adiabatic compressibility,
which is is equal to $c$ for a non-rotating black hole and decreases as the
angular momentum is increased. An extremal black hole has $v_s^2=0.9 \,c^2$
when the cosmological constant vanishes, and more generally $v_s$ is bounded
below by $c/ {\sqrt 2}$.Comment: 8 pages, 1 figure, uses revtex4, references added in v

B. Lukács

Brian P. Dolan

English

arXiv

Dolan, Brian P

Heriot Watt Pure

Compressibility of rotating black holes

Interpreting the cosmological constant as a pressure, whose thermodynamically conjugate variable is a volume, modifies the first law of black hole thermodynamics. Properties of the resulting thermodynamic volume are investigated: the compressibility and the speed of sound of the black hole are derived in the case of nonpositive cosmological constant. The adiabatic compressibility vanishes for a nonrotating black hole and is maximal in the extremal case—comparable with, but still less than, that of a cold neutron star. A speed of sound vs is associated with the adiabatic compressibility, which is equal to c for a nonrotating black hole and decreases as the angular momentum is increased. An extremal black hole has v2s=0.9  c2 when the cosmological constant vanishes, and more generally vs is bounded below by c/√2

Dolan, Brian P.

Maynooth University ePrints and eTheses Archive

MURAL - Maynooth University Research Archive Library

Compressibility of rotating black holesBrian P. DolanDepartment of Mathematical Physics, National University of Ireland, Maynooth, Irelandand Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin, Ireland(Received 2 September 2011; published 16 December 2011)Interpreting the cosmological constant as a pressure, whose thermodynamically conjugate variable is avolume, modifies the first law of black hole thermodynamics. Properties of the resulting thermodynamicvolume are investigated: the compressibility and the speed of sound of the black hole are derived in thecase of nonpositive cosmological constant. The adiabatic compressibility vanishes for a nonrotating blackhole and is maximal in the extremal case—comparable with, but still less than, that of a cold neutron star.A speed of sound vs is associated with the adiabatic compressibility, which is equal to c for a nonrotatingblack hole and decreases as the angular momentum is increased. An extremal black hole has v2s ¼ 0:9 c2when the cosmological constant vanishes, and more generally vs is bounded below by c=ffiffiffi2p.DOI: 10.1103/PhysRevD.84.127503 PACS numbers: 04.70.DyI. INTRODUCTIONIt has been suggested by a number of authors [1–8] that,for a black hole of massM when a cosmological constant isincluded, the first law of thermodynamics should readdM ¼ TdSþdJþdQþd; (1)where S, J, andQ are the entropy, angular momentum, andcharge, respectively, while  is a thermodynamic variableconjugate to the cosmological constant . Since  isassociated with a pressure,  ¼ 8P in units with G ¼c ¼ 1, one interpretation of the d term is that it isrelated to the familiar term PdV term in the first law,with  proportional to the volume. This in turn requiresinterpreting the black hole mass as corresponding to thethermodynamic enthalpy, H ¼ MðS; P; J;QÞ, rather thanthe more usual interpretation as the thermal energy U[9,10]. One then hasdM ¼ TdSþdJþdQþ VdP; (2)where the thermodynamic volume,V ¼ @M@PS;J;Q¼ 8; (3)is the Legendre transform of the pressure. In full generality,the first law of black hole thermodynamics now readsdU ¼ TdSþdJ þdQ PdV; (4)where U ¼ H PV. In this work we shall set Q ¼ 0 forsimplicity; our results are easily extended to include theQ  0 case.Of course if  is truly constant, P cannot be varied andthe last term in (2) is always zero, though the last term in(4) need not be zero and can have physical consequences[11]. But it is commonly supposed, in inflationary models,for example, that  has at some time in the past beensignificantly different from its current value and so mustvary.Unfortunately the thermodynamics of black holes with> 0 are not well understood: such a system has twodifferent temperatures in general, one associated with theblack hole horizon and one with the cosmological horizon,and in addition there is no asymptotic regime in which aninvariant mass can be defined. For these reasons we shallprimarily restrict ourselves to the better understood < 0case in this work. This is sufficient to access the  ¼ 0limit.The thermodynamics of anti–de Sitter black holes wasexamined in detail in the seminal paper of Hawking andPage [7] and there has been a large body of work on thistopic since then. The mass of a charged AdS-Kerr blackhole, with cosmological constant  ¼ 8P, was ex-pressed as a function of S, J, and Q in [8]. When Q ¼ 0the expression isMðS; P; JÞ ¼ 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ 8PS3 ÞfS2ð1þ 8PS3 Þ þ 42J2gSs; (5)where G ¼ c ¼ 1 and S is 14 the area of the event horizon.From this the temperature can be evaluated,T¼@M@SP;J¼ 18M1þ8PS3ð1þ8PSÞ42J2S2; (6)with ℏ ¼ 1. The temperature vanishes for extremal blackholes, with maximum angular momentumJ2max ¼S221þ 8PS3ð1þ 8PSÞ: (7)The stability properties of Kerr-AdS black holes werestudied in [8], where it was shown that there is a criticalpoint near J ¼ 0:0236 (the authors of [8] used units inwhich P ¼ 38 ).*bdolan@thphys.nuim.iePHYSICAL REVIEW D 84, 127503 (2011)1550-7998=2011=84(12)=127503(3) 127503-1  2011 American Physical SocietyThe thermodynamic volume (3),V ¼ @M@PS;J¼ 23MS21þ 8PS3þ 22J2; (8)is always greater than the naive geometric resultV ¼ 43 ðSÞ3=2, as observed in [12].The Legendre transform of (5),UðS;V;JÞ¼S38<:3V4S222þJ2J2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3V42S3s 9=;;(9)was studied in [11], where the difference between thespecific heat at constant pressureCP ¼ T @S@TP;J¼ T 1@2H@S2jP;J(10)and specific heat at constant volumeCV ¼ T @S@TV;J¼ T 1@2U@S2jV;J(11)was also explored. Explicit expressions confirm thatCPCV 1. CP diverges along the curve in the J–S planeshown in Fig. 1 (plotted using dimensionless variablesJP and SP). CP is negative in the region below this curveand the critical point corresponds to the maximum valueof JP which is determined numerically to lie atJP ¼ 0:002 86 . . . and SP ¼ 0:0820 . . . . When J ¼ 0,CP ¼ 2Sð1þ 8PSÞ8PS 1 ; (12)which is negative when P ¼ 0, the famous negative heatcapacity of Schwarzschild black holes refers to CP. Thisinstability is reflected in the isothermal compressibilityT ¼  1V@V@PT;J¼ 48Sf9j6 þ 3ð5þ 8pÞð3þ 8pÞj4 þ 6ð3þ 8pÞ3j2 þ ð3þ 8pÞ4gð6þ 16pþ 3j2Þf9ð9þ 32pÞj4 þ 6ð3þ 16pÞð3þ 8pÞ2j2 þ ð8p 1Þð3þ 8pÞ3g ; (13)where j ¼ 2J=S and p ¼ PS. T diverges when thedenominator vanishes, which is the same locus of pointson which CP diverges and is shown below in the JP–SPplane appropriate for constant P processes.For a nonrotating black holeTjJ¼0 ¼ 24S8PS 1 (14)is negative for PS < 18 .Perhaps a more useful quantity to consider is the adia-batic compressibility,S ¼  1V@V@PS;J: (15)Evaluating S from (8) shows that it is manifestly positiveand regular,S ¼ 36Sj4ð3þ 8pÞð3þ 8pþ 3j2Þð6þ 16pþ 3j2Þ : (16)Thus increasing the pressure, keeping the area and an-gular momentum constant, causes the volume of the blackhole to decrease, while at the same time the mass mustincrease according to (5). S vanishes when J ¼ 0, sononrotating black holes are adiabatically incompressible,but it increases with J attaining a maximum value in theextremal case (7) whenSjT¼0 ¼ 2Sð1þ 8pÞ2ð3þ 8pÞ2ð1þ 4pÞ : (17)A speed of sound, vs, can also be associated with theblack hole, in the usual thermodynamic sense thatv2s ¼@@PS;J¼1þS¼1þ 9j4ð6þ16pþ3j2Þ2 ; (18)where  ¼ MV is a density. This is not the sound associatedwith any kind of surface wave on the event horizon, animpossibility due to the no-hair theorem, but rather it isassociated with a breathing mode for the black hole as thevolume changes with the pressure, keeping the area con-stant. Clearly 0  v2s  1 (this is also true when Q  0).The speed of sound is unity for incompressible nonrotatingblack holes and is lowest for the extremal casev2s jT¼0 ¼ 1þ1þ 8p3þ 8p2; (19)giving v2s ¼ 0:9 when P ¼ 0 and v2s achieving a minimumvalue of 12 when PS! 1.These results show that the equation of state is very stifffor adiabatic variations of nonrotating black holes andsoftens as J increases. For comparison, the adiabatic com-pressibility of a degenerate gas of N relativistic neutrons ina volume V at zero temperature follows from the degener-acy pressurePdeg ¼ ð32Þ1=3 cℏ4VN4=3 ) S ¼ 34Pdeg : (20)For a neutron star NV  1045 m3 and S1034 kgms2.With zero cosmological constant the black hole adiabaticcompressibility at zero temperature is given by (17) withP ¼ 0,BRIEF REPORTS PHYSICAL REVIEW D 84, 127503 (2011)127503-2SjT¼P¼0 ¼ 2S9 ¼4M2G39c8; (21)where the relevant factors of c and G are included. Puttingin the numbersSjT¼0 ¼ 2:6 1038MM2m s2 kg1; (22)which is still 2 orders of magnitude less than that of aneutron star even for M  10 M. We conclude that thezero temperature black hole equation of state is stiffer thanthat of a neutron star.A subtlety with this analysis of adiabatic compressibilityis that, if a black hole is extremal and the pressure goesdown, keeping the entropy and the angular momentumfixed, then the extremal value of the angular momentum(7) goes down and the black hole ends up with an angularmomentum above the new extremal value, giving a nakedsingularity. One interpretation of this would be that cosmiccensorship simply does not allow such a process to occurand, at zero T, the entropy must increase, if the pressure isdecreased, in such a way as to avoid the extremal angularmomentum going down. If the extremal angular momen-tum is kept constant then, from (7),@S@PT¼0¼  16S2ð1þ 4pÞ3þ 48pþ 128p2 : (23)With this constraint the zero temperature compressibility isincreased above the adiabatic value; in fact it becomesidentical to the zero temperature isothermal compressibility,T¼0 ¼ 2Sð11þ 80pþ 128p2Þð1þ 4pÞð3þ 48pþ 128p2Þ : (24)At zero pressure this is a factor of 33 larger than (21), hencecosmic censorship effectively makes the zero temperatureequation of state softer.Although the analysis presented here is strictly onlyvalid for P  0 nothing obviously goes wrong for P< 0,provided PS   18 , and it may be reasonable to analyti-cally continue these thermodynamic relations to negativepressures [13], provided the resulting positive cosmologi-cal constant is not too large.It has been suggested that mini black holes could havebeen created in the early Universe. If this is the case, andthey are created with a range of angular momenta, it isinteresting to speculate that a collection of such compress-ible black holes might affect the speed of sound through thesurrounding medium, in the same way that a suspension ofcompressible spheres would affect the speed of sound in afluid.[1] M. Henneaux and C. Teitelboim, Phys. Lett. 143B, 415(1984); 222, 195 (1989).[2] C. Teitelboim, Phys. Lett. 158B, 293(1985).[3] S. Wang, S.-Q. Wu, F. Xie, and L. Dan, Chin. Phys. Lett.23, 1096 (2006).[4] Y. Sekiwa, Phys. Rev. D 73, 084009(2006).[5] E. A. Larran˜aga Rubio, arXiv:0711.0012.[6] S. Wang, arXiv:gr-qc/0606109.[7] S.W. Hawking and D.N. Page, Commun. Math. Phys. 87,577 (1983).[8] M.M. Caldarelli, G. Cognola, and D. Klemm, ClassicalQuantum Gravity 17, 399 (2000).[9] D. Kastor, S. Ray, and J. Traschen, Classical QuantumGravity 26, 195011 (2009).[10] B. P. Dolan, Classical Quantum Gravity 28, 125020(2011).[11] B. P. Dolan, Classical Quantum Gravity 28, 235017(2011).[12] M. Cvetic, G.W. Gibbons, D. Kubiznak, and C.N. Pope,Phys. Rev. D 84, 024037 (2011).[13] B. Luka´cs and K. Martina´s, Acta Phys. Pol. B 21, 177(1990).SP0.140.120.10.080.060.040.02JP00.010.0080.0060.0040.0020FIG. 1 (color online). Region in the JP–SP plane in which CPis negative. CP diverges on the lower curve and is negative belowit. The upper curve shows Jmax, i.e. T ¼ 0, on which CPvanishes.BRIEF REPORTS PHYSICAL REVIEW D 84, 127503 (2011)127503-3

Interpreting the cosmological constant as a pressure, whose thermodynamically conjugate variable is a volume, modifies the first law of black hole thermodynamics. Properties of the resulting thermodynamic volume are investigated: the compressibility and the speed of sound of the black hole are derived in the case of non-positive cosmological constant. The adiabatic compressibility vanishes for a non-rotating black hole and is maximal in the extremal case - comparable with, but still less than, that of a cold neutron star. A speed of sound vs is associated with the adiabatic compressibility, which is is equal to c for a non-rotating black hole and decreases as the angular momentum is increased. An extremal black hole has v_s^2=0.9c^2 when the cosmological constant vanishes, and more generally v_s is bounded below by c/√2

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arXiv:1109.0198v2  [gr-qc]  8 Dec 2011Compressibility of rotating black holesBrian P. Dolan∗Department of Mathematical Physics,National University of Ireland,Maynooth, IrelandandDublin Institute for Advanced Studies,10 Burlington Rd., Dublin, IrelandInterpreting the cosmological constant as a pressure, whose thermodynamicallyconjugate variable is a volume, modifies the first law of black hole thermodynamicsby adding a PdV term, giving a complete form of the first law. Properties of thethermodynamic volume are investigated: the compressibility and the speed of soundof the black hole are derived in the case of non-positive cosmological constant. Theadiabatic compressibility vanishes for a non-rotating black hole and is maximal inthe extremal case — comparable with, but still less than, that of a cold neutronstar. A speed of sound vs is associated with the adiabatic compressibility, which isis equal to c for a non-rotating black hole and decreases as the angular momentumis increased. An extremal black hole has v2s = 0.9 c2 when the cosmological constantvanishes, and more generally vs is bounded below by c/√2.PACS numbers: PACS nos: 04.60.-m; 04.70.DyI. INTRODUCTIONThe first law of black hole thermodynamics has had a long and subtle history, startingwith Bekenstein’s proposal [1] that the entropy should be proportional to the area followedby Hawking’s seminal work on black hole-temperature [2]. Notably absent in the discussion∗Electronic address: bdolan@thphys.nuim.ie2of the first law has been a PdV term, associated with changes in volume. It is shown herethat, for rotating black holes, a thermodynamic volume can be defined which is independentof the area allowing a complete form of the first law to be formulated.It has been suggested by a number of authors [3]-[8] that, for a black hole of mass Mwhen a cosmological constant is included, the first law of thermodynamics should readdM = TdS + ΩdJ + ΦdQ +ΘdΛ, (1)where S, J and Q are the entropy, angular momentum and charge respectively while Θ isa thermodynamic variable conjugate to the cosmological constant Λ. Since Λ is associatedwith a pressure, Λ = −8piP in units with G = c = 1, one interpretation of the Θ dΛ termis that it is related to the familiar term PdV term in the first law, with Θ proportional tothe volume. This in turn requires interpreting the black hole mass as corresponding to thethermodynamic enthalpy, H = M(S, P, J,Q), rather than the more usual interpretation asthe thermal energy U , [9, 10]. One then hasdM = TdS + ΩdJ + ΦdQ+ V dP, (2)where the thermodynamic volume,V =∂M∂P∣∣∣∣S,J,Q= −8piΘ, (3)is the Legendre transform of the pressure. In full generality, the first law of black holethermodynamics now readsdU = TdS + ΩdJ + ΦdQ− PdV, (4)where U = H − PV . In this work we shall set Q = 0 for simplicity, our results are easilyextended to include the Q 6= 0 case.Of course if Λ is truly constant, P cannot be varied and the last term in (2) is alwayszero, though the last term in (4) need not be zero and can have physical consequences [12].But it is commonly supposed, in inflationary models for example, that Λ has at some timein the past been significantly different from its current value and so must vary.Unfortunately the thermodynamics of black holes with Λ > 0 are not well understood:such a system has two different temperatures in general, one associated with the blackhole horizon and one with the cosmological horizon, and in addition there is no asymptotic3regime in which an invariant mass can be defined. For these reasons we shall primarilyrestrict ourselves to the better understood Λ < 0 case in this work. This is sufficient toaccess the Λ = 0 limit.The mass of a charged AdS-Kerr black hole, with cosmological constant Λ = −8piP , wasexpressed as a function of S, J and Q in [8]. When Q = 0 the expression isM(S, P, J) =12√(1 + 8PS3) {S2(1 + 8PS3)+ 4pi2J2}piS, (5)where G = c = 1 and S is 14the area of the event horizon. From this the temperature canbe evaluated,T =∂M∂S∣∣∣∣P,J=18piM{(1 +8PS3)(1 + 8PS)− 4pi2J2S2}, (6)with h¯ = 1. The temperature vanishes for extremal black holes, with maximum angularmomentumJ2max =(S2pi)2(1 +8PS3)(1 + 8PS) . (7)The stability properties of Kerr-AdS black holes were studied in [8], where it was shownthat there is a critical point near J = 0.0236 (the authors of [8] used units in which P = 38pi).The thermodynamic volume (3),V =∂M∂P∣∣∣∣S,J=23piM{S2(1 +8PS3)+ 2pi2J2}, (8)is always greater than the na¨ıve geometric resultV = 4pi3(Spi) 32 , as observed in [11].The Legendre transform of (5),U(S, V, J) =(piS)3(3V4pi)(S22pi2+ J2)− J2√(3V4pi)2−(Spi)3 , (9)was studied in [12], where the difference between the specific heat at constant pressureCP = T∂S∂T∣∣∣∣P,J= T1∂2H∂S2∣∣P,J(10)and specific heat at constant volumeCV = T∂S∂T∣∣∣∣V,J= T1∂2U∂S2∣∣V,J(11)4was also explored. Explicit expressions confirm that CPCV≥ 1. CP diverges along the curve inthe J −S plane shown in figure 1 (plotted using dimensionless variables JP and SP ). CP isnegative in the region below this curve and the critical point corresponds to the maximumvalue of JP which is determined numerically to lie at JP = 0.00286 . . . and SP = 0.0820 . . . .When J = 0,CP =2S(1 + 8PS)8PS − 1 (12)which is negative when P = 0, the famous negative heat capacity of Schwarzschild blackholes refers to CP .This instability is reflected in the isothermal compressibilityβT = − 1V(∂V∂P)T,J(13)=48S{9 j6 + 3 (5 + 8 p) (3 + 8 p) j4 + 6 (3 + 8 p)3 j2 + (3 + 8 p)4}(6 + 16 p+ 3 j2){9 (9 + 32 p) j4 + 6 (3 + 16 p) (3 + 8 p)2 j2 + (8 p− 1) (3 + 8 p)3} ,where j = 2piJ/S and p = PS. βT diverges when the denominator vanishes, which isthe same locus of points on which CP diverges and is shown below in the JP − SP planeappropriate for constant P processes,SP0.140.120.10.080.060.040.02JP00.010.0080.0060.0040.0020Figure 1: region in the JP − SP plane in which CP isnegative. CP diverges on the lower curve and is negat-ive below it. The upper curve shows Jmax, i.e. T = 0,on which CP vanishes.For a non-rotating black holeβT |J=0 =24S8PS − 1 (14)5is negative for PS < 18.Perhaps a more useful quantity to consider is the adiabatic compressibility,βS = − 1V(∂V∂P)S,J. (15)Evaluating βS from (8) shows that it is manifestly positive and regular,βS =36Sj4(3 + 8 p) (3 + 8 p+ 3 j2) (6 + 16 p+ 3 j2). (16)Thus increasing the pressure, keeping the area and angular momentum constant, causesthe volume of the black hole to decrease, while at the same time the mass must increaseaccording to (5). βS vanishes when J = 0, so non-rotating black holes are adiabaticallyincompressible, but it increases with J attaining a maximum value in the extremal case, (7),whenβS|T=0 =2S (1 + 8 p)2(3 + 8 p)2 (1 + 4 p). (17)A speed of sound, vs, can also be associated with the black hole, in the usual thermody-namic sense thatv−2s =∂ρ∂P∣∣∣∣S,J= 1 + ρ βS = 1 +9 j4(6 + 16 p+ 3 j2)2, (18)where ρ = MVis a density. This is not the sound associated with any kind of surface wave onthe event horizon, an impossibility due to the no hair theorem, but rather it is associatedwith a breathing mode for the black hole as the volume changes with the pressure, keepingthe area constant. Clearly 0 ≤ v2s ≤ 1 (this is also true when Q 6= 0). The speed of soundis unity for incompressible non-rotating black holes and is lowest for the extremal casev−2s∣∣T=0= 1 +(1 + 8 p3 + 8 p)2, (19)giving v2s = 0.9 when P = 0 and v2s achieving a minimum value of 1/2 when PS →∞.These results show that the equation of state is very stiff for adiabatic variations of non-rotating black holes and softens as J increases. For comparison, the adiabatic compressibilityof a degenerate gas of N relativistic neutrons in a volume V at zero temperature followsfrom the degeneracy pressurePdeg = (3pi2)13ch¯4(VN)− 43⇒ βS = 34Pdeg. (20)6For a neutron star NV≈ 1045 m−3 and βS ≈ 10−34 kg ms−2. With zero cosmological constantthe black hole adiabatic compressibility at zero temperature is given by (17) with P = 0,βS|T=P=0 =2S9=4piM2G39 c8, (21)where the relevant factors of c and G are included. Putting in the numbersβS|T=0 = 2.6× 10−38(MM⊙)2ms2 kg−1, (22)which is still two orders of magnitude less than that of a neutron star even for M ≈ 10M⊙.We conclude that the zero temperature black hole equation of state is stiffer than that of aneutron star.A subtlety with this analysis of adiabatic compressibility is that, if a black hole is extremaland the pressure goes down, keeping the entropy and the angular momentum fixed, thenthe extremal value of the angular momentum (7) goes down and the black hole ends upwith an angular momentum above the new extremal value, giving a naked singularity. Oneinterpretation of this would be that cosmic censorship simply does not allow such a processto occur and, at zero T , the entropy must increase, if the pressure is decreased, in sucha way as to avoid the extremal angular momentum going down. If the extremal angularmomentum is kept constant then, from (7),∂S∂P∣∣∣∣T=0= − 16S2(1 + 4 p)3 + 48 p+ 128 p2. (23)With this constraint the zero temperature compressibility is increased above the adiabaticvalue, in fact it becomes identical to the zero temperature isothermal compressibility,βT=0 =2S (11 + 80 p+ 128 p2)(1 + 4 p) (3 + 48 p+ 128 p2). (24)At zero pressure this is a factor of 33 larger than (21), hence cosmic censorship effectivelymakes the zero temperature equation of state softer.Although the analysis presented here is strictly only valid for P ≥ 0 nothing obviouslygoes wrong for P < 0, provided PS ≥ −18, and it may be reasonable to analytically continuethese thermodynamic relations to negative pressures, [13], provided the resulting positivecosmological constant is not too large.It has been suggested that mini black holes could have been created in the early universe,[14]. If this is the case, and they are created with a range of angular momenta, it is interesting7to speculate that a collection of such compressible black holes might affect the speed of soundthrough the surrounding medium, in the same way that a suspension of compressible sphereswould affect the speed of sound in a fluid.In summary it has been argued that a PdV term should be included to complete thefirst law of black hole thermodynamics since the thermodynamic volume is independent ofthe area for rotating black holes. The famous instability of black holes due to negativeheat capacity is associated with the heat capacity at constant pressure rather than the heatcapacity at constant volume and compressibilities, together with the speed of sound, havebeen calculated.[1] J.D. Bekenstein, Lett. Nuovo. Cimento 4 (1972) 737;J.D. Bekenstein, Phys. Rev. D7 (1973) 2333.[2] S.W. Hawking, Nature 248 (1974) 30;S.W. Hawking, Comm. Math. Phys. 43 (1975) 199;S.W. Hawking, Phys. Rev. D13 (1976) 191.[3] M. Henneaux and C. Teitelboim, Phys. Lett. 143B (1984) 415;M. Henneaux and C. Teitelboim, Phys. Lett. 222B (1989) 195.[4] C. Teitelboim, Phys. Lett. 158B (1985) 293.[5] Y. Sekiwa, Phys. Rev. D73 (2006) 084009, [arXiv:hep-th/0602269].[6] E.A. Larran˜aga Rubio, Stringy Generalization of the First Law of Thermodynamics for Ro-tating BTZ Black Hole with a Cosmological Constant as State Parameter,[arXiv:0711.0012 [gr-qc]].[7] S. Wang, S-Q. Wu, F. Xie and L. Dan, Chin. Phys. Lett., 23 (2006) 1096,[arXiv:hep-th/0601147];S. Wang, Thermodynamics of high dimensional Schwarzschild de Sitter spacetimes: variablecosmological constant, [arXiv:gr-qc/0606109].[8] M.M. Caldarelli, G. Cognola and D. Klemm, Class. Quantum Grav. 17 (2000) 399,[arXiv:hep-th/9908022].[9] D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes,[arXiv:0904.2765 [hep-th]].8[10] B.P. Dolan, Class. Quantum Grav. 28 (2011) 125020, [arXiv:1008.5023[gr-qc]].[11] M. Cvetic, G.W. Gibbons, D. Kubiznak and C.N. Pope, Black Hole Enthalpy and an EntropyInequality for the Thermodynamic Volume, [arXiv:1012.2888[gr-qc]].[12] B.P. Dolan, Class. Quantum Grav. 28 (2011) 235017, [arXiv:1106.6260].[13] B. Luka´cs and K. Martina´s, Acta Phys. Pol. B21 (1990) 177.[14] B.J. Carr and S.W. Hawking, Mon. Not. R. Astr. Soc., 168 (1974) 399.

Compressibility of rotating black holes

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