We investigate a possible way of establishing a parallel between the third law of black hole mechanics, and the strong version of the third law of thermodynamics. We calculate the surface gravity and area for a naked singular null surface in the Gamma-metric and explain in what sense this behaviour violates thermodynamics

Lochan, K.

Malafarina, D.

Singh, T.P.

Nazarbayev University Repository

arXiv:1009.4585v1  [gr-qc]  23 Sep 2010Thermodynamics and the naked singularity in the Gamma-metricK. Lochan,1, ∗ D. Malafarina,1, † and T. P. Singh1, ‡1Tata Institute of Fundamental Research, Homi Bhabha road, Colaba, Mumbai 400005, IndiaWe investigate a possible way of establishing a parallel between the third law of black hole mechanics,and the strong version of the third law of thermodynamics. We calculate the surface gravity andarea for a naked singular null surface in the Gamma-metric and explain in what sense this behaviourviolates thermodynamics.PACS numbers: 04.20.Dw, 04.70.DyKeywords: Black holes, naked singularities, thermodynamicsI. BLACK HOLE THERMODYNAMICS ANDCOSMIC CENSORSHIPWe begin by making a few observations about the secondand third law of black hole mechanics, and the third lawof thermodynamics.The proof of the second law of black hole mechanics[i.e. the area theorem] requires the validity of the cosmiccensorship hypothesis. By now, various examples of theformation of naked singularities are known, in modelsof gravitational collapse studied in the general theory ofrelativity. It is also known that general relativity admitsextremal stationary Kerr-Newman black hole solutions,which posses in fact naked singularities. What is theimpact of these examples on the validity of the afore-mentioned proof of the second law? The general outlookin the community is that the Kerr-Newmann naked sin-gularities are non-generic and hence they have no impacton the proof [1].It is well-known that the suggestive analogy betweenarea and entropy becomes precise only when quantum ef-fects [Hawking radiation] are taken into account. Couldit then not be the case that the area theorem itselfstrictly holds [in the generalized form where entropy ofthe radiated matter is also taken into account] only whenquantum effects are taken into account, the role of quan-tum effects now being that of preventing the formationof naked singularities?The third law of black hole mechanics was stated byBardeen, Carter and Hawking [BCH] as: ‘It is impossibleby any procedure, no matter how idealized, to reduce thesurface gravity of a black hole to zero by a finite sequenceof operations’ [2]. A precise formulation and proof ofthis law was given by Israel [3]. The proof requires cos-mic censorship to hold. The thermodynamic analog ofthis third law is the weak form of the third law of ther-modynamics : ‘The temperature of a system cannot bereduced to zero in a finite number of operations’.∗Electronic address: kinjalk@tifr.res.in†Electronic address: daniele.malafarina@polimi.it‡Electronic address: tpsingh@tifr.res.inThe strong form of the third law of thermodynamicsstates that: ‘As the temperature of a system goes to zero,its entropy goes to an absolute constant, which may betaken as zero’. The weak form of the third law followsas a consequence of the strong form. There is no analogof the strong form of the third law of thermodynamicsin black hole mechanics, because there are examples ofextremal black holes for which surface gravity goes tozero, but the area of the outer horizon remains non-zeroand does not go to a universal constant (recall that thisis an example of a Kerr-Newmann naked singularity). Itis also known that there are theoretical counterexamplesto the strong form of the third law in thermodynam-ics. One such example was constructed by Wald [4] byconsidering a rotating bosonic gas at low temperatureswith a specific density of states. However, although someof these counterexamples can be regarded as ‘physicallyreasonable’, they have not been experimentally realized.What then should the analog between the third law ofblack hole mechanics and third law of thermodynamicsbe? There are various possible options to consider:(i) The strong form of the third law of thermodynam-ics is not a law, because there are counterexamples to it.The analog should be between the weak form and theBCH statement of the third law of black hole mechanics,for which Israel has given a proof.(ii) The counterexamples to the strong form of thethird law of thermodynamics have not been realized ex-perimentally, hence this form of the law should be ac-cepted only as empirically valid, notwithstanding thetheoretical counterexamples. Furthermore, there possi-bly are reasons as to why extremal black-holes are ‘exoticobjects’ which are not continual limits of non-extremalblack holes, and hence should not be regarded as viola-tions of a possible strong version of the third law of blackhole mechanics [viz. as surface gravity goes to zero areagoes to an absolute constant]. This point of view hasbeen nicely expressed by Racz [5].(iii) The strong form of the third law of thermody-namics is a correct law, and there are (as yet unknown)reasons why the theoretical counterexamples are not re-alized experimentally. Extremal black holes violate thestrong version of the third law of black hole mechanics,2and they violate cosmic censorship. We propose thatthe strong version of the third law of black hole mechan-ics holds when quantum effects are taken into account,the role of quantum effects being to avoid occurrence ofnaked singularities.In summary, the observations that: (i) the analogybetween area and entropy becomes precise only afterquantum effects are taken into account, (ii) classical gen-eral relativity does admit some examples of formation ofnaked singularities in gravitational collapse, (iii) forma-tion of naked singularities is possibly avoided in quantumtheory (quantum effects become important in the finalstages of collapse) and (iv) the theoretical counterexam-ples to the third law of thermodynamics have not beenrealized empirically, lead us to suggest that:The analogy between the strong version of the thirdlaw of thermodynamics and the strong version of thethird law of black hole mechanics does not hold at thetheoretical level (because there are counterexamples) butit holds at the empirical level [counterexamples are notrealized in the real world].The idea that General Relativity can be thought asthe theory of gravity that emerges at large scales fromquantum statistical effects justifies the claim that theconnection between thermodynamics and gravity mustbe valid in a more generic sense and not only for theevent horizons. In this view the laws of thermodynamicsfirst established for event horizons of stationary space-times may be generalized to ‘Isolated Horizons’ [6] andnaked singularities might be investigated from a thermo-dynamical point of view.On the other hand one could also say that naked singu-larities, which belong to another class of configurations,different from Black Holes (be it because of initial datafor gravitational collapse, or because of the values of theparameters in Kerr-Newman type spacetime), preventthe precise connection between gravity and thermody-namics from being realized in a context more generalthan just black holes. In fact the laws of thermodynam-ics when extended to Killing/Isolated horizon of nakedsingularities seem to be violated or at least seem to con-tradict our experience. However recent studies permita thermodynamical interpretation of gravity [7] and theassociation of an entropy with null surfaces [8].In this note we present one such instance of a violation,by observing the properties of the Gamma-metric, whichis a static axially symmetric vacuum spacetime belongingto the Zipoy-Voorhees class [9]. We point out that themetric has a singular null surface on which the surfacegravity diverges, and which is also naked. Furthermore,the null surface has zero area. Thus this surface in theGamma-metric has zero entropy and a divergent surfacegravity [equivalently temperature] - behaviour that is notin accordance with what we expect from the laws of ther-modynamics. One may hence conclude that the validityof thermodynamics, once assumed or demonstrated ona general level should prevent the possible occurrence ofsuch naked singularities.II. THE GAMMA-METRICThe most general static, axially symmetric line elementin cylindrical coordinates takes the formds2 = −e2λdt2 + e2µ−2λ[dρ2 + dz2] + r2e−2λdφ2 , (2.1)where λ(ρ, z) and µ(ρ, z) are solutions of∇2λ = 0 , (2.2)µ,ρ = ρ(λ2,ρ + λ2,z) , (2.3)µ,z = 2ρλ,ρλ,z . (2.4)Here ∇ is the flat Laplace operator in two dimensionsimplying that there is a one to one correspondencebetween solutions of the Laplace equation in flat spaceand static axial symmetry; therefore in principle allstatic axially symmetric space-times are known. In fact,once we find λ as a solution of equation (2.2), then µcan be obtained by quadrature.The Gamma-metric is the static axially symmetricspace-time that corresponds to the solution of Laplaceequation for a linear distribution of matter of constantdensity γ2 and length 2M distributed symmetricallyalong the z axis. In Erez-Rosen coordinates (or prolatespheroidal coordinates), given by ρ2 = (r2−2Mr) sin2 θ,z = (r −M) cos θ this solution takes the form:λ(r, θ) =γ2ln(1− 2Mr), (2.5)to which correspondsµ(r, θ) =γ22ln(1− 2Mr1− 2Mr+ M2r2sin2 θ). (2.6)The line element for the Gamma-metric can be writtenas:ds2 = −∆γdt2 + ∆γ2−γ−1Σγ2−1dr2 + (2.7)+r2∆1−γ[(∆Σ)γ2−1dθ2 + sin2 θdφ2],where∆ =(1− 2Mr), Σ =(1− 2Mr+M2r2sin2 θ).(2.8)The Gamma-metric reduces to the well knownSchwarzschild space-time in spherical coordinates for3γ = 1 and represents the external field of an oblate (pro-late) object for γ > 1 (γ < 1 respectively).Furthermore the Gamma-metric exhibits a naked cur-vature singularity on the surface r = 2M when γ 6= 1[10]. Investigation of the Kretschmann scalar shows thatthe singularity is a strong curvature singularity that cannot be removed by a coordinate transformation. Fur-thermore the investigation of null geodesics terminatingat the singular surface proves that the singularity has adirectional behavior for γ > 1. In fact the singularity inthis case is not visible along the polar axis θ = 0, mean-ing that photons emitted by the singularity along thisaxis need an infinite time to reach far away observers.A. Isolated horizonAn isolated horizon Ξ is a closed, convex null sur-face which is in equilibrium i.e., across which there isno gravitational radiation or flux of matter fields. Itis a 3-dimensional hypersurface in space-time with thefollowing properties(i) Ξ is null, topologically S2 ×R, and equipped witha preferred foliation by 2-surfaces SΞ transverse toits null normal.(ii) All equations of motion are satisfied on Ξ.(iii) There is no flux of radiation either through or alongΞ.R˚ablb = R˚abnb = 0,where R˚ab is pull back of Rab on S∆ and la , na aretwo null normals to SΞ.(iv) Ξ is non rotating, non-expanding future boundaryof space-time under consideration.In the case of the Gamma-metric the surface r = 2M (asa limit) satisfies these criteria. So, it is worth calculatingthermodynamic quantities attached with this surface.B. Killing horizon in the Gamma-metricBeing static and axially symmetric the Gamma-metricadmits the two Killing vector fields ∂t and ∂φ.The surfacef = r − 2M = 0 (2.9)turns out to be a Killing horizon for 1+ γ− γ2 > 0 withthe condition γ > 0, as shown below.The fact that f(r, θ) = 0 is a null surface follows fromgµν ▽µ f ▽ν f = 0 , (2.10)which, once written for the Gamma-metric, becomesr2∆γ−γ2Σγ2−1(∂θf)2 −∆γ−γ2+1Σγ2−1(∂rf)2 = 0 ,(2.11)showing that f = r − 2M is in fact a null surface.Furthermore it’s easy to check that the Killing field ∂tis normal to f . In fact if we write ∂t as ξa = ξµeaµ, thenξµ = δµt and therefore on r = 2M we getξaξa|S = gµνδµt δνt = 0 . (2.12)We next proceed to compute the surface gravity forthis Killing Horizon in the usual manner. The surfacegravity is defined asκ =√−12(▽µξν)(▽µξν) . (2.13)For the Killing field ∂t, the index lowered componentsare ξµ = gµνξν = gµt. Since the Gamma-metric is diag-onal, only ξt survives and hence the only non vanishingcomponent of ξν,µ is ξt,r.It is easy to check that▽t ξr = −Γtrtξt = −▽t ξr , (2.14)and▽r ξt = 12gttgrr∂gtt∂r. (2.15)Hence,κ =√−14gttgrr(∂gtt∂r)2. (2.16)So that finally the surface gravity for the Gamma-metrichas the formκ =2Mγr2∆−(γ−1)22 Σγ2−12 . (2.17)Now the exponent of first term is negative for all val-ues of γ, including those also for which r = 2M is aKilling Horizon. Therefore, on this surface κ divergeseverywhere on the surface, except for θ = 0, pi possibly.III. ENTROPY OF KILLING HORIZONFor quite some time there have been attempts tomodel gravity as a thermodynamic theory [7], [11]. Ithas been suggested that any null surface acts as a hori-zon for a specialized set of observers, which traces outthe degrees of freedom associated with the inaccessibleregion of spacetime.Therefore, there is a notion of entropy associatedwith the null surfaces and if gravity is an emergentthermodynamic property, it is demanded that laws of4thermodynamics should be applicable to any surfacewhere Einstein’s equations hold, specifically on all localRindler causal horizons through each spacetime point.Thus all Rindler horizons can be associated with arelation of the form δQ = TδS where surface gravityand the area play the role of temperature and entropyrespectively [7], [8]. In this sense Einstein’s equationscan be regarded as equations of state. Although thisline of thinking calls into the attempts of quantizationof Einstein’s equations it also suggests that, if true,some of the ‘configurations’ of spacetime are thermo-dynamically not favourable. In accordance with thisline of thinking we can approach the r = 2M surface ina limiting fashion to test its ‘thermodynamic’ properties.We now explain how the area of the null surface un-der consideration has the interpretation of entropy. Ithas been shown [8] that by maximizing the entropy func-tionalS =∫VdDx√−g(4P cdab ▽c ξa ▽d ξb − Tabξaξb) (3.1)w.r.t. all the null vector fields ξa Einstein field equationsare obtained as an ‘equilibrium configuration’, where P cdabis a fourth rank tensor built out of the metric and othergeometric quantities which can be expanded in powersof number of derivatives of the metricPabcd(gij , Rijkl) = c1P(1)abcd(gij) + c2P(2)abcd(gij , Rijkl) + ...(3.2)with the m-th order P(m)ijka related to m-th order La-grangian density of Lanczos-Lovelock gravityP (m)ijka =∂L(D)m∂Raijk, (3.3)withL(D)m =116pi2−mδa1a2..a2mb1b2..b2m Rb1b2a1a2...Rb2m−1b2ma2m−1a2m . (3.4)Choosing cm = 0, ∀m 6= 1, and c1 = 1 Einstein fieldequations can be recovered. The tensor P cdab is analogousto the elastic constant in case of elastic solids and hasthe same symmetry properties as the Riemann tensor.Also the following equation is true at each order:▽d P cdab = 0 . (3.5)The entropy of the Killing horizon is calculated in theRindler limit. S|On Shell is calculated for a particular(non-null) surface in the Rindler frame and then this sur-face is made to approach the Killing horizon; thus theentropy of the horizon is obtained in a limiting sense.By S|On Shell we mean that equilibrium configuration isassumed, which for this case is the field equations (with-out correction terms from higher order P(m)cdab ). For thispurpose ξa is chosen normal to the surface andS|Ξ = 4pi∫Ξd2x⊥√g(2)L(2)0 . (3.6)In the case of the Gamma-metric we have a naked cur-vature singularity at r = 2M . Hence a Rindler framecannot be established at the surface.Therefore, one should calculate the entropy in the lim-iting case only. Now L(2)0 = 116pi , hence the entropy ofnull surfaces as we approach the Killing horizon isS|Ξ −→ 14∫Ξd2x⊥√g(2) . (3.7)which is proportional to (in fact 14 th of) the area of ther = 2M surface in the Gamma-metric.A. Area of revolution surfaces in theGamma-metricThe area of a spheroid of a fixed radius r > 2M in theGamma space-time isAγ =∫ 2pi0∫ pi0r2 sin θ∆(1−γ)22 Σ1−γ22 dθdφ . (3.8)For γ = 1 it reduces to the area of the SchwarzschildspheresAS = 4pir2 . (3.9)When γ 6= 1 we see that the reduced metric on the 2-surface r =const. and t = const. becomes degenerate onthe singular surface r = 2M and thereforeAγ | −−−−−→r−→2M0 . (3.10)Since the area goes to zero, the entropy tends to zerotoo. Since the surface gravity diverges on this surface,we may conclude that the combination of ‘infinite tem-perature and zero entropy’ is not consistent with usualthermodynamics, and hence the occurrence of such a sin-gularity might be forbidden by thermodynamics.IV. CONCLUSIONSIn this brief note we have suggested that the analogybetween the strong form of the third law of thermo-dynamics, and the third law of black-hole mechanics,might be taken seriously since counterexamples to themdo not seem to be realized in the real world. We havealso suggested that the applicability of the third law ofthermodynamics may extend to gravitational configura-tions other than black holes as well, as seen above in5the example of the naked singularity in the Gamma-metric. Here, the surface gravity [analog of the tem-perature] diverges on the Killing horizon, whereas thearea [analog of the entropy] goes to zero. Consistencywith the third law would require surface gravity to goto zero when the area goes to zero. Since that does nothappen here, this example of a naked singularity couldbe regarded as possessing an anti-thermodynamic fea-ture, and hence being unrealistic. Similarly it is reason-able to suppose that this behaviour is due to the choiceof the class of space-times to which the Gamma-metricbelongs (namely vacuum, static and axially symmetric)and therefore it will be shared by other examples of man-ifolds belonging to the same class. If that is the case thenthe anti-thermodynamic behaviour, i.e. the fact that thiskind of naked singularities may not emerge as a thermo-dynamic limit from the quantum theory of gravity, couldwell be regarded as a generic feature of space-times withnaked singularities. We note though that strictly speak-ing what we have here is a self-consistency argument,and not a deduction, since a thermodynamic interpreta-tion of gravity itself indeed requires cosmic censorship tohold [we recall the importance of Rindler horizons in thedevelopment of this interpretation].It will be interesting to examine how naked singulari-ties that arise in dynamical gravitational collapse [suchas the shell-focusing naked singularity in the Lemaitre-Tolman-Bondi spherical dust collapse] relate to the thirdlaw of thermodynamics. Can one construct analogs ofsurface gravity and area in this context, and examinewhether the strong form of the third law is being obeyedor not? This issue is at present under investigation.[1] R. Wald, Ann. Phys. 82, 548 (1974).[2] J. M. Bardeen. B. Carter and S. W. Hawking, Commun.Math. Phys. 31, 161 (1973).[3] W. Israel, Phys. Rev. Lett. 57, 397 (1986).[4] R. M. Wald, Phys. Rev. D 56, 6467 (1997).[5] I. Racz, Class. Quantum Grav. 17, 4353 (2000).[6] A. Ashtekar, C. Beetle and S. Fairhurst, Class. QuantumGrav. 16, L1-L7 (1999).[7] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).[8] T. Padmanabhan and A. Paranjape, Phys. Rev. D 75,064004 (2007).[9] D. M.Zipoy, J. Math. Phys. 7, 1137 (1966); B. H.Voorhees, Phys. Rev. D 2, 2119 (1970).[10] L. Herrera, G. Magli and D. Malafarina, Gen. Rel. Grav.37, 1371 (2005).[11] T. Padmanabhan, Gen. Rel. Grav. 40, 2031 (2008).

Thermodynamics and the naked singularity in the Gamma-metric

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Thermodynamics and the naked singularity in the Gamma-metric

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