Pressure and volume in the first law of black hole thermodynamics

arXiv:1106.6260v3 [gr-qc] 11 Nov 2011 Pressure and volume in the first law of black hole thermodynamics Brian P. Dolan Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland and Dublin Institute for Advanced Studies, 10 Burlington Rd., Dublin, Ireland e-mail: [email protected] November 14, 2011 Abstract The mass of a black hole is interpreted, in terms of thermodynamic potentials, as being the enthalpy, with the pressure given by the cosmological constant. The volume is then defined as being the Legendre transform of the pressure and the resulting relation between volume and pressure is explored in the case of positive pressure. A virial expansion is developed and a van der Waals like critical point determined. The first law of black hole thermodynamics includes a PdV term which modifies the maximal efficiency of a Penrose process. It is shown that, in four dimensional space-time with a negative cosmological constant, an extremal charged rotating black hole can have an efficiency of up to 75%, while for an electrically neutral rotating back hole this figure is reduced to 52%, compared to the corresponding values of 50% and 29% respectively when the cosmological constant is zero. PACS nos: 04.60.-m; 04.70.Dy 1 Introduction The thermodynamics of black holes is a rich and fascinating area of research which continues to yield surprises. The first law of black hole thermodynamics is usually written as dM =TdS + ΩdJ + ΦdQ (1) where T=κ 2πis the Hawking temperature of the black hole (with κthe surface gravity), S=A 4the entropy (with Athe area in Planck units), Ω the angular velocity, Jthe angular momentum, Φ the electrostatic potential difference between infinity and the horizon, Qthe electric charge and M the mass. The mass is usually interpreted as the internal energy, in the thermodynamic sense, of the black hole, but it was suggested in [1] that it is more correctly interpreted as the enthalpy. In this context it is notable that there is no PdV term in (1), corresponding to a change in volume at ambient pressure P. When a cosmological constant, Λ, is included there is a natural candidate for a pressure, P=−Λ 8π, and it was proposed in [2] that the volume of the black hole be defined as the thermodynamic variable conjugate to P. Interpreting the mass as the enthalpy, equation (1) should then be modified to dM =TdS +V dP + ΩdJ + ΦdQ, (2) where the thermodynamic volume is defined to be V=∂M ∂P S,J,Q, [1]. The idea that Λ should be thought of as a thermodynamic variable that can be varied is not new and has been considered by a number of authors, [3]-[9]. Equation (2), with Q= 0, was studied in [8], in the context of varying Λ, without any particular physical interpretation being given to the thermodynamic conjugate of Λ, often denoted Θ. One may question whether it is appropriate to identify Λ with a thermodynamic pressure. While a cosmological constant gives a pressure term in Einstein’s equations, a fluid dynamical pressure is not necessarily the same as a thermodynamic pressure. In equilibrium situations however it is presumably correct to identify the fluid dynamical pressure with the thermodynamic pressure and we shall do so here. In general the enthalpy, H, is the heat energy beloved of chemists, it is not the internal energy, U, of the first law of thermodynamics. That distinction goes to the Legendre transform of the enthalpy, U=H−PV, (3) 1 where His a function of S,P,Jand Qwhile U(S, V, J, Q) is a function of purely extensive variables. Then we get the usual form of the first law, dU =TdS −PdV + ΩdJ + ΦdQ. (4) This equation was written down in [9], using Θ −Λ notation, but its consequences were not pursued. When Λ is non-zero we should expect the P dV term to contribute to the mechanical energy that can be extracted from a black hole, by a Penrose process for example. For a negative Λ (positive pressure) the PdV term gives a positive contribution to dU if the black hole shrinks, and the P dV term reduces the amount of energy available for extraction as mechanical work W, with dW =−dU, hence reducing the efficiency. However we shall show that the maximal efficiency actually increases, relative to the Λ = 0 case, when Λ<0, because the maximal angular momentum of a black hole in AdS is greater than that of one with Λ = 0 and this can outweigh the reduction in efficiency: the engine may not be as efficient at a given Jbut it can be pushed to higher J. Conversely one would expect that, for a positive Λ more energy becomes available at a given J, relative to Λ = 0, as the black hole shrinks. Of course there are no pistons pushing against a gas for a black hole, but a negative cosmological constant contributes a negative energy density to space-time so a shrinking black-hole exposes negative energy, thus increasing the black hole’s internal energy and decreasing the amount of energy available for mechanical work. Conversely a positive cosmological constant would presumably release extra energy as the black hole shrinks, that can be used to do work, hence increasing the efficiency at a given J. The most efficient way to extract energy from a black hole is in an isentropic process, with dS = 0 and the area of the event horizon constant. So dU ≥dUmin =−PdV + ΩdJ + ΦdQ. (5) We shall see that, for a rotating black hole, it is possible to reduce Vwhile keeping Sconstant. The maximum amount of mechanical work that can be extracted in passing from an initial state ito a final state fis Wmax =−Zf i dUmin.(6) 2 The efficiency is defined to be the ratio of the mechanical energy extracted to the initial heat energy (enthalpy), η=Wmax Mi ,(7) where the initial enthalpy is identified with the initial mass, Mi. It will be shown that, for Λ <0, this can be as high as 52% for a rotating neutral black hole and 75% for a charged black hole. The volume of a black hole has only recently been considered as a thermodynamic variable, [2, 13]. At the simplest level, there is a natural tendency to assume that the area and the volume are related geometrically and are not independent. For a Schwarzschild black hole with radius rh, for example, the area is of course 4πr2 hand indeed the thermodynamic volume works out to be 4π 3r3 h, but this seems co-incidental and no particular significance should be attached to it. At a deeper level it is not even clear how to define a volume geometrically, as the metric is not static in the interior of a black hole. For a Schwarzschild black hole which is not rotating we shall see that the thermodynamic volume and the area are not independent: fixing Sfixes Vso dV = 0 in an isentropic process, and the P dV term does not contribute to the first law. But for a rotating black hole the area of the event horizon does not determine the thermodynamic volume uniquely and it is possible to vary the volume keeping the entropy constant, by changing the angular momentum and/or the charge. The properties of the thermodynamic volume and its contribution to the first law of black hole thermodynamics are explored in detail in this paper for a rotating charged black hole in four dimensional space-time with a negative Λ. In section 2 thermodynamic potentials and the equation of state are discussed and the Legendre transform from the enthalpy to the internal energy is given explicitly. In section 3 the efficiency of a Penrose type process is analysed and section 4 contains a discussion and outlook. Two appendices are dedicated to the technicalities of deriving some results used in the text 2 The internal energy Including a pressure term in the first law gives dU =TdS −PdV + ΩdJ + ΦdQ 3 where the internal energy, U(S, V, J, Q), is a function of extensive variables. The thermodynamic volume Vis the conjugate variable to the pressure and is obtained from the mass, which is identified in [1] with the enthalpy, M= H(S, P, J, Q), by V=∂H ∂P S,J,Q .(8) It was proposed in [2] that (8) be defined to be the thermodynamic volume of the black hole. The line element for a charged rotating black hole in 4-dimensional anti-de Sitter space is [10] ds2=−∆ ρ2dt −asin2θ Ξdφ2 +ρ2 ∆dr2+ρ2 ∆θ dθ2+∆θsin2θ ρ2adt −r2+a2 Ξdφ2 , (9) where ∆ = (r2+a2)(L2+r2) L2−2mr +q2,∆θ= 1 −a2 L2cos2θ, ρ2=r2+a2cos2θ, Ξ = 1 −a2 L2,(10) and the cosmological constant is Λ = −3 L2, which is related to the pressure by 1 L2=8πP 3. The physical properties of this space-time are well known [11], and the first law, applied to this metric, was discussed in [12], but without a PdV term. The metric parameters mand qare related to the mass and charge by M=m Ξ2, Q =q Ξ.(11) The event horizon, r+, lies at the largest root of ∆(r) = 0, so M=(r2 ++a2)(L2+r2 +) + q2L2 2r+L2Ξ2,(12) and the area of the event horizon is A= 4π(r2 ++a2) Ξ.(13) 4 The temperature is T=(L2+ 3r2 +)r2 +−a2(L2−r2 +)−q2L2 4πL2r+(r2 ++a2).(14) The angular momentum, J=aM and the relevant thermodynamic angular velocity is Ω = a(L2+r2 +) L2(r2 ++a2).(15) The electrostatic potential is Φ = qr+ r2 ++a2.(16) One can scale Lout from all the above expressions by defining dimensionless variables M=M/L, ¯a=a/L, ¯r+=r+/L, etc., but we prefer to keep Lexplicit to expose more clearly the rˆole of the pressure, and make the comparison with the L→ ∞ limit clear. Under the assumptions made here the thermodynamic volume for the Kerr-Newman-AdS black hole works out to be V=2π 3(r2 ++a2)(2r2 +L2+a2L2−r2 +a2) + L2q2a2 L2Ξ2r+,(17) which is a simple generalisation of the Kerr-AdS volume derived in [13]. A direct derivation of (17) from (12) is most easily achieved by first writing the mass as a function of (S, P, J, Q), differentiating with respect to P, and then transforming back to (r+, a, q, L). When a= 0 the area (13) and volume (17) are not independent and the area determines the volume uniquely, but when ais non-zero the area and the volume become independent. For asymptotically flat space, with L→ ∞, one has V=2π 3 (r2 ++a2)(2r2 ++a2) + q2a2 r+ .(18) While this reduces to the na¨ıve result, 4πr3 + 3, for the Schwarzschild black hole, a geometrical interpretation when ais non-zero is not so clear. A correct description of the thermodynamics of the black hole, in terms of thermodynamic potentials, requires replacing the geometric variables (r+, L, a, q) 5 with thermodynamic variables (S, P, J, Q). The relevant expression for the mass, and hence the enthalpy, was derived in [9], H(S, P, J, Q) := 1 2sS+πQ2+8P S2 32+ 4π21 + 8P S 3J2 πS .(19) This generalises the Christodoulou-Ruffini formula [14] for the mass of a rotating black hole in terms of its irreducible mass, Mirr. The irreducible mass for a black hole with entropy Sis the mass of a Schwarzschild black hole with the same entropy, M2 irr =S 4π. In terms of thermodynamic variables the temperature 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 the thermodynamic volume is V=∂H ∂P S,J,Q =2 3πH SS+πQ2+8PS2 3+ 2π2J2,(21) which is manifestly positive. The Legendre transform U=H−PV gives the thermal energy, a function of purely extensive variables. The transform is evaluated in an appendix to 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 that Uis only a function of three independent variables, since U→λU when S→λ2S,V→λ3V,J→λ2Jand Q→λQ. One must be careful taking the J→0 limit of these potentials. In this limit the enthalpy H(S, P, 0, Q) = 1 2rS π1 + πQ2 S+8SP 3(23) 6 is linear in Pand the Legendre transform is singular: V=∂H ∂P S,Q =4π 3S π3/2 (24) is independent of Pand so cannot be inverted to obtain P(V). Conversely, when Jis zero, the Legendre transform of (23) is U=1 2rS π1 + πQ2 S,(25) which is independent of the volume and is not equal to the J→0 limit of (22), unless a constraint, (3V 4π)2=S π3, is imposed. Indeed (25) gives the wrong J= 0 temperature (unless Q=P= 0, in which case U=Hand T=∂U ∂S =1 4πr+is the correct Hawking temperature for a Schwarzschild black hole). To get the correct temperature from (22) in the J→0 limit we must take the partial derivative with respect to Sbefore setting Jto zero, and take note of the fact that (3V 4π)2=S π3when J= 0. This constraint can be derived from (22) directly by observing that Uis not differentiable with respect to Jat J= 0 unless (3V 4π)2−(S π)3J2+SQ2 πvanishes there. It can also be seen directly when a= 0 in (13) and (17). To derive the relation between the pressure and the volume in general we first define v:= 3V 4πand s:= S π. Then (22) becomes U(s, v, J, Q) = 1 s3nv 2(s2+s Q2+ 2J2)−|J|p(v2−s3) (J2+s Q2)o(26) with temperature T=1 π ∂U ∂s =|J|{3J2(2v2−s3) + s Q2(5v2−2s3)} 2πs4p(v2−s3)(J2+s Q2)− vs2+ 2s Q2+ 6J2 2πs4 (27) and pressure P=−3 4π ∂U ∂v =3v|J| 4πs3rJ2+s Q2 v2−s3−3 8πs3s2+s Q2+ 2J2.(28) In the J→0 limit |J|and √v2−s3must vanish together for finite Tand P. The equation of state, in the form of the relation between the pressure, 7 This allows us to re-express Has a function of V, H=1 2sβ2−4αγ V2−γ.(47) We can immediately conclude that V2>4π 32S π3 ,(48) in agreement with the observation in [13]. It is now straightforward to determine U=H−PV =H−HV 2 γ−βV 2γ=βV 2γ−p(V2−γ)(β2−4αγ) 2γ,(49) which immediately gives (22) in the text. Appendix 2 In this appendix the virial expansion is developed. For simplicity we set Q= 0, but the same techniques can be applied to the case of non-zero Q. To develop the expansion we use dimensionless variables y:= v J3/2, x := s J, p := 8πP J 3, t := 2πT J1/2,(50) in terms of which equations (27) and (28) can be written t=3(2y2−x3) x4py2−x3−y(x2+ 6) x4(51) p=2y x3py2−x3−x2+ 2 x3.(52) When yand xare large let y2≈x3, with y2−x3=z2, then tis finite provided z≈3 y2/3tin which case p≈2t 3y1/3. Replacing xwith zin equations (51) and (52) gives 3y2+z2=nt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 powers of u=1 y1/3. Let z=3u2 tζfor some ζ(t, u), in terms of which (51) becomes t1 + 9u10ζ2 t2=(t1−9u10ζ2 t24/3 +u1−9u10ζ2 t22/3 + 6u5)ζ, (55) from which we can immediately conclude that ζ=t t+u+ 6u5+ω(t, u),(56) where an expansion of ωin ustarts at order 10. Any desired order can be obtained by further expanding ω ω=u10 ∞ X n=0 an(t)un,(57) with the co-efficients an(t) to be determined. Putting (57) and (56) into (55), and equating co-efficients of powers of u, gives an iterative procedure for evaluating the co-efficients anwhich can then be used to show that: ω= 7u10 t2−16u11 t3+ 27u12 t4−40u13 t5+ 55u14 t6−36 2 + 3 t4 t7u15 +Ou16.(58) Finally using this expansion in (54) gives the required virial expansion p=2tu 3−1 3u2+ 2 u6−u10 8u t−9u2 t2+ 10u3 t3−11u4 t4+ 12u5 t5(59) −13 1 + 6 t4 t6u6+Ou17, the first three terms of which are used in the text. References [1] D. Kastor, S. Ray and J. Traschen, Class. Quantum Grav. 26 (2009) 195011, [arXiv:0904.2765 [hep-th]]. [2] B.P. Dolan, Class. Quantum Grav. 28 (2011) 125020, [arXiv:1008.5023[gr-qc]]. 15 [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] S. Wang, S-Q. Wu, F. Xie and L. Dan, Chin. Phys. Lett.,23 (2006) 1096, [arXiv:hep-th/0601147]. [6] Y. Sekiwa, Phys. Rev. D73 (2006) 084009, [arXiv:hep-th/0602269]. [7] E.A. Larra˜naga Rubio, Stringy Generalization of the First Law of Thermodynamics for Rotating BTZ Black Hole with a Cosmological Constant as State Parameter, [arXiv:0711.0012 [gr-qc]]. [8] 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: variable cosmological constant, [arXiv:gr-qc/0606109]. [9] M.M. Caldarelli, G. Cognola and D. Klemm, Class. Quantum Grav. 17 (2000) 399, [arXiv:hep-th/9908022]. [10] B. Carter, Comm. Math. Phys. 10 (1968) 280. [11] M. 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The critical point for this value of Jis Tc≈0.0413, Pc≈0.00280 and vc≈3.08. J > 0 causes the rapid rise in Pat low values of v. 17 Figure 2: CVas a function of S/L2and J/L2, with fixed volume set to L3. 18 Figure 3: CPas a function of S/L2and J/L2, with fixed pressure set to 3 8π,i.e L= 1. CPdiverges along curve IV in figure 4 and vanishes along curve 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 Figure 4: phase diagram for q= 0, plotted in terms of S/L2and J/L2. The region above I is forbidden, because T < 0; in the region above curve II the 3-d Einstein universe at infinity rotates faster than the speed of light; curve III bounds the region of local stability (an analysis of the Gibbs free energy shows that the black hole is locally unstable above curve III); CPdiverges on curve IV; in the region above curve V the black hole is unstable due to the HawkingPage phase transition; in region below curve V the black hole is stable. 20