The Quasi-localized Einstein and Møller Energy Complex as Thermodynamic Potentials
aa r X i v : . [ g r- q c ] O c t The Quasi-localized Einstein and Møller EnergyComplex as Thermodynamic Potentials
I-Ching Yang Systematic and Theoretical Science Research Groupand Department of Applied Science,National Taitung University, Taitung 95002, Taiwan (R. O. C.)
ABSTRACT
To begin with, in this article, I obtain the Einstein and Møller energy complexin PG coordinates. According to the difference of energy within region M between Einstein and Møller prescription, I could present the difference ofenergy of RN black hole like the fomula of Legendre transformation andpropose that the Møller and Einstein energy complex play the role of internalenergy and Helmholtz energy in thermodynamics.PACS No.: 04.70.Dy; 04.70.Bw; 04.20.CvKeywords: Einstein and Møller energy-momnetum pseudotensor, Hawkingtemperature, Bekenstein-Hawking entropy E-mail:[email protected] Introduction
In the theory of general relativity (GR), one of the most important issueswhich is still unsolved is the localization of energy. According to Noether’stheorem, one would define a conserved and localized energy as a consequenceof energy-momentum tensor T µν satisfying the differential conservation law ∂ ν T µν = 0 . (1)However, in a curved space-time where the gravitational field is presented,the differential conservation law becomes ∇ ν T µν = 1 √− g ∂∂x ν (cid:16) √− gT µν (cid:17) − g νρ ∂g νρ ∂x λ T µλ = 0 , (2)and generally does not lead to any conserved quantity. In GR, we shall lookfor a new quantity Θ µν = √− g ( T µν + t µν ) instead of T µν , which satisfies thedifferential conservation equation ∂ ν Θ µν = 0 , (3)if we want to maintain the localization characteristics of energy. Here, Θ µν is an energy-momentum complex of matter plus gravitational fields and t µν is regarded as the contribution of energy-momentum from the gravitationalfield. It should be noted that Θ µν can be expressed as the divergence of the“superpotential” U µ [ νρ ] that is antisymmetric in ν and ρ asΘ µν = U µ [ νρ ] ,ρ . (4)Mathematically, it is freedom on the choice of superpotential, because onecan add some terms ψ µνρ , whose divergence or double divergence is zero, to U µνρ . A large number of definitions for the gravitational energy in GR havebeen given by many different authors, for example Einstein [1], Møller [2],Landau and Lifshitz [3], Bergmann and Thomson [4], Tolman [5], Wein-berg [6], Papapetrou [7], Komar [8], Penrose [9] and Qadir and Sharif [10].On the other hand, Chang, Nester and Chen [11] showed that every energy-momentum complex is associated with a legitimate Hamiltonian boundaryterm and actually quasilocal. 2ne of those problems for using several kinds of energy-momentum com-plexes is that they may give different results for the same space-time. Es-pecially Virbhadra and his colleagues [12] showed that Einstein, Landau-Lifshitz, Papapetrou, and Weinberg prescriptions (ELLPW) lead to the sameresults in Kerr-Schild Cartesian coordinates for a specific class of spacetime,i.e. the general nonstatic spherically symmetric space-time of the Kerr-Schildclass ds = B ( u, r ) du − dudr − r d Ω (5)and the most general nonstatic spherically symmetric space-time ds = B ( t, r ) dt − A ( t, r ) dr − F ( t, r ) dtdr − D ( t, r ) r d Ω , (6)but not in Schwarzschild Cartesian coordinates. Afterward Xulu [13] pre-sented Bergmann-Thomson complex also “coincides” with ELLPW com-plexes for a more general than the Kerr-Schild class metric. Mirshekari andAbbassi [14] find a unique form for a special general spherically symmetricmetric in which the energy of Einstein and Møller prescriptions lead to thesame result. In particular, whatever coordinates do not exist the same en-ergy complexes associated with using definitions of Einstein and Møller insome space-time solutions, i.e. Reissner-Nordstr¨om (RN) black hole. On theother hand, Yang and Radinschi [15] attemptd to investigate the differencebetween the energy of Einstein prescription E Einstein and Møller prescription E Møller , and observed the difference ∆ E = E Einstein − E Møller can be relatedto the energy density of the matter fields T as∆ E ∼ r × T . (7)Matyjasek [16] also presented two analogous relations which are∆ E = 4 πr T (8)for the simplified stress-energy tensor of the matter field and∆ E = 4 πr h T rr i ( s ) ren (9)for the approximate renormalized stress-energy tensor of the quantized mas-sive scalar ( s = 0), spinor ( s = 1 /
2) and vector ( s = 1) field. Later, Vage-nas [17] hypothesized that α (Einstein) n and α (Møller) n are the expansion coefficients3f E Einstein and E Møller in the inverse powers of r , and found out an interestingrelation between these two coefficients α (Einstein) n = 1 n + 1 α (Møller) n . (10)Finally, Matyjasek [16] and Yang et. al. [18] pointed out the following formularespectively E Møller = E Einstein − r dE Einstein dr . (11)It should be noted that these relations in Eq. (7)-(11) offer us the mathemat-ical formula between E Einstein and E Møller only. The remainder of the article isorganized as follows. In section 2, I will calculate the energy distribution forgeneralized Painlev´e-Gullstrand (PG) coordinates [19] by using the Einsteinand Møller complex. In section 3, the physical explanation of the difference∆ E will be given. I will summarize and conclude finally in section 4. In thisarticle, I use geometrized units in which c = G = ¯ h = 1 and the metric hassignature (+ − −− ). The continuation of black holes across the horizon is a well understoodproblem discussed on GR. The difficulties of the Schwarzschild coordinates( t, r, θ, φ ) at the horizons of a nonrotating black hole provide a vivid illus-tration of the fact that the meaning of the coordinates is not independentof the metric tensor g µν in GR. Several coordinate systems produce a metricthat is manifestly regular at horizons, i.e. the Kruskal-Szekeres, Eddington-Finkelstein, and PG coordinates. However, PG coordinates have often beenemployed to study the physics of black holes. They have been applied toanalyse quantum dynamical black holes [20], and used extensively in deriva-tions of Hawking radiation as tunneling following the work of Parikh andWilczek [21]. In this section, while using PG coordinates, I will find out theenergy of static spherically symmetric black hole solutions in Einstein andMøller prescriptions. In four-dimensional theory of gravity, I can write thestatic spherically symmetric metrics in the form ds = f dt − f − dr − r d Ω , (12)4here f is a function of r , i.e. f = f ( r ). Let me transform to generalizedPG coordinates [19] and introduce the PG time dt p = dt + βdr , thus 4-metriccan be written as ds = f dt p − s − fA dt p dr − A dr − r d Ω , (13)where A ≡ q f / (1 − f β ).At the outset, the energy component in the Einstein prescription [1] isgiven by E Einstein = 116 π Z ∂H l ∂x l d x, (14)where H l is the corresponding von Freud superpotential H l = g n √− g ∂∂x m h ( − g )( g n g lm − g ln g m ) i , (15)and the Latin indices take values from 1 to 3. For performing the calculationsconcering the energy component of the Einstein energy-momentum complex,I have to transform the spatial parts of above metric (13) into the quasi-Cartesian coordinates ( x, y, z ) ds = A dt p − s − fA dt p ( xr dx + yr dy + zr dz ) − ( 1 A − xr dx + yr dy + zr dz ) − ( dx + dy + dz ) . (16)Then, the required nonvanishing components of the Einstein energy-momentumcomplex H l are H = 2 Cxr ,H = 2
Cyr ,H = 2
Czr , (17)and these are easily shown in spherical coordinates to be a vector H r = 2 Cr ˆ r, (18)where C = 1 − f and ˆ r is the outward normal. Applying the Gauss theroremI obtain E Einstein = 116 π I H r · ˆ rr d Ω , (19)5nd the integral being taken over a sphere of radius r and the differentialsolid angle d Ω. The Einstein energy complex within radius r reads E Einstein = r − f ) . (20)Next, the energy component of the Møller energy-momentum complex [2]is described as E Møller = 18 π Z ∂χ l ∂x l d x, (21)where χ l is the Møller superpotential χ l = √− g ∂g α ∂x β − ∂g β ∂x α ! g β g lα , (22)and the Greek indices run from 0 to 3. However, the only nonvanishingcomponent of Møller’s superpotential is χ = dfdr r sin θ. (23)Applying the Gauss theorem, I evaluate the integral over the surface of asphere within radius r , and find the energy distribution is E Møller = r dfdr . (24)Here, I consider the results of calculation for two cases of the simplestblack hole solutions, i.e. Schwarzschild and RN solution. In the first case Ihave f = 1 − M/r , therefore the energy complex of Einstein is E Einstein = M, (25)and of Møller is also E Møller = M. (26)For the next case it is defined that f = 1 − M/r + Q /r , so the energycomplex in Einstein prescription is E Einstein = M − Q r , (27)and in Møller prescription is E Einstein = M − Q r . (28)6t should be noted that the above results of Einstein energy complex in PGCartesian coordinates are equivalent to in Schwarzschild Cartesian and Kerr-Schild Cartesian ones [12], but the time coordinate of these three coordinatesis different to each other. Using the Schwarzschild black hole as an example,the time coordinate of Schwarzschild Cartesian coordinates t , of Kerr-SchildCartesian coordinates v = t + r + 2 M ln (cid:12)(cid:12)(cid:12)(cid:12) r M − (cid:12)(cid:12)(cid:12)(cid:12) , (29)and of PG Cartesian coordinates t p = t + 4 M r r M + 12 ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q r/ M − q r/ M + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (30)are not the same. In other words, independent of the choices of these threekinds of time coordinate, the energy complex of Einstein within radius r is E Einstein = M . It is indefinite that the energy complex of Einstein is universalfor any kinds of time coordinate. Some quasi-local energy expressions [22] andthe Einstein energy-momentum pseudotensors are coordinate-independentin spherically symmetric space-time. It remains to investigate whether thecoordinate-independent is the property of the spherically symmetric space-time. To understand the physical meaning of difference ∆ E , let me to begin withexamining the RN black hole, which is a static spherically symmetric solutionwith two horizons, as an example. The line element of RN black hole can bewritten as ds = f ( r ) dt − f − ( r ) dr − r d Ω , (31)where f ( r ) = (cid:18) − r + r (cid:19) (cid:18) − r − r (cid:19) , (32) r + = M + √ M − Q is the event horizon and r − = M − √ M − Q isthe inner Cauchy horizon. According to Eq.(19) and Eq.(23), the Einsteinenergy complex with radius r is E Einstein = r + + r − − r + r − r , (33)7nd the Møller energy complex is E Møller = r + + r − − r + r − r . (34)Therefore, the difference of energies with radius r between the Einstein andMøller prescription can be obtained as∆ E = r + r − r . (35)In the article of Nester et. al. [11], they had stated that “Consequently, thereare various of energy, each corresponding to a different choice of boundarycondition; this situation can be compared with thermodynamics with its var-ious energies: internal, enthalpy, Gibbs, and Helmholtz.” Hence, I insert theidea of black hole thermodynamics to compare energy-momentum complexwith thermodynamic potential.Afterward, I would introduce two thermodynamic qualities of black hole,the Hawking temperature [19] T H = 14 π ∂f∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r h (36)and the Bekenstein-Hawking entropy [23] S BH = A (cid:12)(cid:12)(cid:12)(cid:12) r h = πr h . (37)Because those two qualities are only defined on event horizon, at r = r + , thetemperature is given as T + = r + − r − πr = √ M − Q π ( M + √ M − Q ) (38)and the entropy is also given as S + = πr = π ( M + q M − Q ) , (39)Supposing that we consider the region between those two horizons, shown as M = B ( r + ) − B ( r − ), the difference of energies will be obtained in the form∆ E | r = r + r = r − = − r + − r − − q M − Q = − T + S + . (40)8ere, B ( r ) is a 3-sphere within a radius r . The term T + S + can be consideredthat the heat flow streams out the region M by passing the boundary of B ( r + ). Therefore, Eq. (39) would be rewitten as E Møller | r + r − − E Einstein | r + r − = 2 T + S + . (41)It is meaning that the difference of energies between Einstein and Møllerprescription equal to the double of the heat flow stearms out by passing thebounday of B ( r + ) in the region M of RN black hole.On the other hand, to base on Zhao’s study [24], the entropy of blackhole, which has two horizons, is defined as ˜ S = S + + S − , where the entropyof the inner Cauchy horizon can be shown as S − = πr − = π (cid:18) M − q M − Q (cid:19) , (42)and the temperature of the inner Cauchy horizon is given as T − = κ − π , (43)where the surface gravity of the inner Cauchy horizon is [25] κ − = lim r → r − − r − r − ) s − g g = r + − r − r − . (44)So the difference of energy between the Einstein prescription and Møllerprescription within the region M can be written as E Møller | r + r − − E Einstein | r + r − = T + S + + T − S − , (45)and the heat flow will be with respect to both two boundaries of M . Torewrite Eq. (44) as E Einstein | M = E Møller | M − X ∂ M T S, (46)these heat flows are exhibited on every boundary of M . Comparing Eq. (45)with the Legendre transformation, E Møller and E Einstein in the region M playthe role of internal energy U and Helmholtz energy F in thermodynamics,even so there is a puzzle where E Møller or E Einstein are not a function of T or S . We could obtain not only a physical meaning of the difference of energiesin Eq. (34), although the statement can only be used to RN black hole, butalso such a result agreed with the entropy redefined in Zhao’s article.9 Conclusion and Discussion
I have attempted to answer two questions in this article. One is whetherthe calculation of Einstein energy-momentum complex is acceptable in PGcoordinate, and the other is whether those energy-momentum complex canbe described as a thermodynamic potential. Here, the expression for energyof the static spherically symmetric space-time with the PG Cartesian coor-dinates, Eq.(19), is obtained E Einstein = (1 − f ) r/
2. This is a reasonable andsatisfactory result, because Virbhadra [12], using the Kerr-Schild Cartesiancoordinates, and Yang et al. [18], using the Schwarzschild Cartesian coordi-nates, also got the same expression. It is interesting to investigate whetherthere is any coincidence between the energy expressions with those three timecoordinate.In addition, I have showed that the relational formula about E Einstein and E Møller is similar to the Legendre transformation. The E Møller and E Einstein are regarded as the equivalent of internal energy U and Helmholtz energy F in the region M . Although, the transformation takes us from a functionof one pair variables to the other. It means that S BH and T H must be thevariable of E Møller and E Einstein , but I do not verify that yet. On the otherhand, when I set S ⊥ = πr to be a variable, the second term in the right-handside of Eq.(11) can be replaced as E Møller = E Einstein − S ⊥ dE Einstein d S ⊥ . (47)To compare with F = U − T S , we could obtain T H = dE Einstein d S ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r h . (48)Here the formula of Eq.(44) presents that E Einstein and E Møller play the roleof U and F , and is opposite to the view of above.In summary, I have obtained the Einstein and Møller energy complexesof static spherically symmetric black hole with generalized PG coordinates inwhich has been used in derivations of Hawking radiation as tunneling. Baseon the calculation of energy expression in generalized PG coordinates, inEq.(45) I have combined the difference ∆ E with the temperature and entropyof black hole, but Eq.(4g) do not fit in with the Legendre transformation.10evertheless, it is an example to show that the energy-momentum complexesof RN black hole will compare with thermodynamic potential, and futureresearch should be considered on more kinds of space-time. ACKNOWLEDGMENTS
I would like to thank Prof. Chopin Soo and Prof. Su-Long Nyeo for usefulsuggestions and discussions. This work is partially supported by NationalCenter for Theoretical Sciences, Taiwan.
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