Energy-Momentum Localization for a Space-Time Geometry Exterior to a Black Hole in the Brane World
aa r X i v : . [ g r- q c ] M a y Energy-Momentum Localization for a Space-TimeGeometry Exterior to a Black Hole in the BraneWorld
Irina Radinschi *1 , Theophanes Grammenos **2 and Andromahi Spanou ***31 Department of Physics“Gh. Asachi” Technical University,Iasi, 700050, Romania Department of Civil Engineering,University of Thessaly, 383 34 Volos, Greece School of Applied Mathematics and Physical Sciences,National Technical University of Athens, 157 80, Athens, Greece * [email protected], ** [email protected], *** [email protected] Abstract
In general relativity one of the most fundamental issues consists in defininga generally acceptable definition for the energy-momentum density. As a con-sequence, many coordinate-dependent definitions have been presented, wherebysome of them utilize appropriate energy-momentum complexes. We investigatethe energy-momentum distribution for a metric exterior to a spherically symmet-ric black hole in the brane world by applying the Landau-Lifshitz and Weinbergprescriptions. In both the aforesaid prescriptions, the energy thus obtained de-pends on the radial coordinate, the mass of the black hole and a parameter λ ,while all the momenta are found to be zero. It is shown that for a special valueof the parameter λ , the Schwarzschild space-time geometry is recovered. Someparticular and limiting cases are also discussed. Keywords : Energy-momentum complexes; Brane world black holes.
PACS Numbers : 04.20.-q, 04.20.Cv, 04.70.-s, 11.25.-w
In general relativity there has not been given so far a generally accepted expression forthe energy density in gravitational fields, while none of the various approaches used has1rovided a strong indication for its candidacy as the best for the energy-momentumlocalization. Over the past years, numerous attempts have been made in order to calcu-late the energy-momentum distribution of the gravitational field, utilizing tools such assuperenergy tensors [1], quasi-local expressions [2], energy-momentum complexes [3]-[10]and even the tele-parallel (tetrad) theory of gravity [11].A few remarks concerning the aforementioned different approaches are deemed neces-sary. The approach using the superenergy tensors has improved the energy-momentumexpressions a great deal in recent years, while the definitions of the quasi-local massexhibit the advantage of being applicable to any coordinate system. On the otherhand, pseudotensorial definitions which make use of the energy-momentum complexesof Einstein [3], Landau-Lifshitz [4], Papapetrou [5], Bergmann-Thomson [6], Møller [7],Goldberg [8], Weinberg [9] and Qadir-Sharif [10] have been applied to many spacetimegeometries, yielding also significant results. Here it is worth mentioning that the Ein-stein, Landau-Lifshitz, Papapetrou and Weinberg (henceforth ELLPW) prescriptionsuse quasi-Cartesian coordinates, whereas the Møller energy-momentum complex can beapplied to any coordinate system. Finally, as far as the 3 + 1, 2 + 1 and 1 + 1 dimensionalspace-times are concerned, we may point out how useful the pseudotensorial definitionshave been proven for the evaluation of the gravitational energy-momentum. We maynotice that for many gravitational backgrounds different prescriptions have given thesame expression for the energy-momentum (e.g., [12] and references therein, mainly onthe LL and W prescriptions).In the tele-parallel theory of gravity a regularized expression for the gravitationalenergy-momentum is derived. It has been shown that the theory of general relativitycan be reformulated in the context of the tele-parallel (Weitzenb¨ock) geometry. Re-cently, there has been an increasing interest in calculations employing this theory andmany significant results for various space-times ([11]) have been obtained. At this point,the similarity of some results generated by pseudotensorial prescriptions and their tele-parallel versions (see, e.g., [13]) should also be stressed.However, even if the aforementioned approaches have not led to a generally accepted,well-defined expression for the gravitational energy density, they have contributed toestablishing a basis for the performance of valid calculations.The remainder of this paper is organized as follows: in Section 2 a short presentationof the energy-momentum complexes is given along with some details on the Landau-Lifshitz and Weinberg energy-momentum complexes utilized for the calculations in thepresent paper. Section 3 is devoted to the calculations of the energy-momentum for anew black hole solution in the brane world. Finally, the Discussion contains a summaryof the results and a presentation of some particular limiting cases. Throughout thepaper we have used geometrized units ( c = 1; G = 1) and the signature (+ , − , − , − ) forthe Landau-Lifshitz and Weinberg prescriptions in Schwarzschild-Cartesian coordinates,while Greek indices range from 0 to 3 while Latin indices from 1 to 3.2 Energy-Momentum Complexes - A Short Presen-tation
The energy-momentum complexes of Einstein [3], Landau-Lifshitz [4], Papapetrou [5],Bergmann-Thomson [6], Møller [7], Goldberg [8], Weinberg [9] and Qadir-Sharif [10]have been employed for numerous gravitational backgrounds, leading to acceptable re-sults for the energy-momentum localization. An energy-momentum complex is madeup of three components describing the gravitational field, the matter and the possiblenon-gravitational fields, respectively. The energy-momentum complexes conserve thedifferential conservation law.Among the characteristic features, or better said, weaknesses of these complexes, weshould mention the fact that they are coordinate dependent. This, together with the factthat the calculations can be performed only in quasi-Cartesian coordinates, except forthe Møller prescription, has led to their rather scarce utilization for energy-momentumlocalization.Chang, Nester and Chen [14] attempted to rehabilitate the energy-momentum com-plexes, by showing that these are quasi-local and by emphasizing their importance. Oneshould point out that different quasi-local definitions correspond to different boundaryconditions. Also, So, Nester and Chen [15], in an attempt to improve the pseudotensorialmechanism, showed that “for small vacuum regions the Einstein, Landau-Lifshitz, Papa-petrou, Weinberg, Bergmann-Thomson and Goldberg pseudotensors have the zero ordermaterial limit required by the equivalence principle”. The Møller prescription thoughdoes not comply to this.Furthermore, the aforementioned pseudotensors are not proportional to the Bel-Robinson tensor. Thus, some attempts have been made in order to find a profitablecombination of different pseudotensors. This way, they managed to make improvementsand to elaborate an independent combination of Bergmann-Thomson, Papapetrou andWeinberg energy-momentum complexes. They also formulated a one parameter set oflinear combinations of classical pseudotensors with the required Bel-Robinson connec-tion.Part of the significance of the energy-momentum complexes relies on the proof thatseveral energy-momentum complexes “coincide” for any metric of the Kerr-Schild class[16]-[17]. Furthermore, we may notice that the definitions provided by Einstein, Landau-Lifshitz, Papapetrou, Bergmann-Thomson, Weinberg and Møller agree with the quasi-local mass definition introduced by Penrose [18] and developed by Tod [19], at least forsome space-times and some energy-momentum complexes. Moreover, it has been shownthat different prescriptions yield the same result for a given gravitational backgroundunder the condition that the calculations are performed in Schwarzschild Cartesian andKerr-Schild coordinates, while satisfying results have also been obtained for 2 and 3dimensional space-times [12], [16]. However, there do exist some cases where differentenergy-momentum complexes yield different results when the calculations are performedin Schwarzschild Cartesian and Kerr-Schild coordinates [17].Last but not least, we should mention two more viewpoints supporting the importance3f the energy-momentum complexes. Lessner [20] has argued that “the Møller definitionis a powerful concept of energy and momentum in general relativity”, while Cooperstock[21] has emphasized, in his hypothesis of utmost importance, the fact that “the energyand momentum are confined to the regions of non-vanishing energy-momentum tensorfor the matter and all non-gravitational fields”.Now, we shall present the energy-momentum complexes used in the present work.The Landau-Lifshitz energy-momentum complex [4] reads L µν = 116 π S µρνσ, ρσ (1)and the corresponding superpotentials are given by: S µνρσ = − g ( g µν g ρσ − g µρ g νσ ) . (2) L and L i represent the energy and the momentum density components, respectively.The Landau-Lifshitz energy-momentum complex satisfies the local conservation law L µν, ν = 0 . (3)The integration of L µν over the 3-space gives the expression for the energy-momentumfour-vector: P µ = Z Z Z L µ dx dx dx . (4)By using Gauss’ theorem one obtains P µ = 116 π Z Z S µ iν,ν n i dS = 116 π Z Z U µ i n i dS. (5)The Weinberg energy-momentum complex [9] is given by W µν = 116 π D λµν, λ , (6)where the superpotentials are D λµν = ∂h κκ ∂x λ η µν − ∂h κκ ∂x µ η λν − ∂h κλ ∂x κ η µν + ∂h κµ ∂x κ η λν + ∂h λν ∂x µ − ∂h µν ∂x λ , (7)with h µν = g µν − η µν and W , W i represent the energy and the momentum density components, respectively.The Weinberg energy-momentum complex satisfies the local conservation law W µν, ν = 0 . (8)The integration of W µν over the 3-space yields the expression for the energy-momentumfour-vector: P µ = Z Z Z W µ dx dx dx . (9)By applying Gauss’ theorem we obtain P µ = 116 π Z Z D i µ n i dS. (10)4 Energy-Momentum for a New Black Hole Solutionin the Brane World
Since the introduction of the brane world notion and the role of gravity in it (for a reviewsee, e.g., [22] and references therein), an increasing interest in black hole solutions in thiscontext has led to an accordingly increasing number of relevant works, such as [23] just toselect but a few publications that review the subject. On the other hand, there is ratherlittle work concerning the calculation of energy-momentum in brane world models, see,e.g., [24] and in particular for black hole solutions in such models [25].Searching for black hole solutions on the brane, Casadio, Fabbri and Mazzacurati[26] found a new static, spherically symmetric and asymptotically flat solution that hasthe line element: ds = (1 − Mr ) dt − (1 − M r )(1 − Mr )(1 − λ r ) dr − r ( dθ + sin θdϕ ) , (11)where λ ∈ R + and M is the ADM mass. In fact, it is shown [27] that the line element(11) can be also used to describe space-time geometry exterior to a homogenous staron the brane, while for λ > M it gives a wormhole geometry [28]. Furthermore, themetric described by (11) represents a black hole solution in the brane world also in thecontext of the Teleparallel Equivalent of General Relativity (TEGR) [25].For the specific values λ = 0, M = 0 the space-time geometry given by (11) is flat,while for λ = M , (11) becomes the Schwarzschild line element.The line element (11) can be written in Schwarzschild-Cartesian coordinates, whichwe need for the utilization of the Landau-Lifshitz and Weinberg prescriptions, as follows: ds = B ( r ) dt − ( dx + dy + dz ) − A ( r ) − r ( xdx + ydy + zdz ) , (12)with A ( r ) and B ( r ) given by A ( r ) = (1 − M r )(1 − Mr )(1 − λ r ) , B ( r ) = (1 − Mr ) . (13)The calculation for the superpotentials yields the following non-vanishing componentsfor the • Landau-Lifshitz prescription U ttx = 2 xr [(1 − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) , (14) U tty = 2 yr [(1 − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) , (15) U ttz = 2 zr [(1 − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) . (16)where the U ’s are defined by Eq.(5). 5 Weinberg prescription D xtt = 2 xr [(1 − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) , (17) D ytt = 2 yr [(1 − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) , (18) D ztt = 2 zr [(1 − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) . (19)Utilizing Eq.(5) along with the metric (12) and the quantities (14)-(16) we obtain forthe energy distribution inside a 2-sphere of radius rE LL = r − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) , (20)while the momenta are found to be zero.In a similar way, using Eq.(10) and (17)-(19) we get, for the Weinberg prescription,an expression for the energy inside a 2-sphere of radius r that is equal to the one obtainedin the Landau-Lifshitz case E W = E LL = r − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) , (21)while, again, the momenta are found to be zero.Inserting the specific value λ = M in (21), we obtain E LL = E W = M (1 − Mr ) − , (22)which is identical to the expression obtained by Virbhadra for the Schwarzschild metric[17].It is seen from (21) that the energy in both the Landau-Lifshitz and the Weinbergprescriptions depends on the mass M of the black hole, the parameter λ and the radialcoordinate r . In our paper we have studied the energy-momentum for a new black hole solution in thebrane world described by the line element ds = (1 − Mr ) dt − (1 − M r )(1 − Mr )(1 − λ r ) dr − r ( dθ + sin θdϕ ) .
6n particular, we have used the definitions of Landau-Lifshitz and Weinberg in order tocalculate the expressions for the energy and the momentum. It is found that, in thegravitational background considered, all the momenta vanish while the energy dependson the mass M , the radial coordinate r and the positive parameter λ , for both theLandau-Lifshitz and Weinberg prescriptions. In fact, the energy obtained is found to bethe same in both prescriptions and it is given by E W = E LL = r − M r ) − (1 − λ r )(1 − Mr )](1 − Mr )(1 − λ r ) . Specifically, for the value λ = M the expression for the energy reads E LL = E W = M (1 − Mr ) − , being the result obtained by Virbhadra for the Schwarzschild metric [17].As it is evident, for r → ∞ the last expression gives E LL = E W = M which representsthe ADM mass, while for r → E LL = E W = 0.In the following table we summarize the limiting behavior of the energy in bothprescriptions for two particular cases:Limit Energy LL Energy W r → r → M ∞ ∞ r → ∞ ( M + λ ) ( M + λ )At r = 2 M lies the event horizon of the black hole as in the Schwarzschild case.The result E LL = E W = ( M + λ ) for r → ∞ was also obtained by Gamal G.L.Nashed [25] in the context of TEGR. If we consider this limiting case for λ = M weget again E = M which is consistent with the energy obtained for the Schwarzschildsolution, whereby M is the ADM mass. The agreement between our result and theresult obtained in the context of TEGR is worth to be pointed out if one considers thatgeneral relativity is a geometrical theory (the metric tensor being its fundamental field),while TEGR is actually a gauge theory (with a gauge potential as its fundamental field)[29]. Thus, we have come up with the same result not only by using a different theorybut also by applying a different methodology.Even if our present work does not settle the problem of the energy-momentum lo-calization of the gravitational field by using energy-momentum complexes, the resultsobtained can be considered as a contribution to the ongoing debate concerning this issue. Acknowledgements
The authors would like to thank Dr. G. O. Papadopoulos for his valuable suggestions.7 eferences [1] L. Bel, C. R. Acad. Sci. Paris , 3105 (1958); L. Bel, C. R. Acad. Sci., Paris, , 1094 (1958); L. Bel, C. R. Acad. Sci. Paris , 1297 (1959); I. Robinson,unpublished lectures, King’s College, London (1958); I. Robinson, Class. QuantumGrav. , 4331 (1997); J. M. M. Senovilla, “Remarks on superenergy tensors” in Gravitation and Relativity in General , eds. A. Molina, J. Martin, E. Ruiz and F.Atrio (Singapore: World Scientific, 1999) pp.175–182; J. M. M. Senovilla, Class.Quantum Grav. , 2799 (2000).[2] J. D. Brown and J. W. York, “Quasilocal energy in general relativity” in Mathe-matical Aspects of Classical Field Theory (Seattle, WA, 1991), pp. 129-142, Con-temporary Mathematics 132, Amer. Math. Soc., Providence, RI, 1992; J. D. Brownand J. W. York, Phys. Rev. D , 1407 (1993); S. W. Hawking and G. T. Horowitz,Class. Quantum Grav. , 1487 (1996); Sean A. Hayward, Phys. Rev. D , 831(1994); G. Bergqvist, Class. Quantum Grav. , 1917 (1992); R. Penrose, Proc. R.Soc. London A , 53 (1982); K. P.Tod, Proc. R. Soc. London A , 457 (1983);A. Komar, Phys. Rev. , 934 (1959); C. M. Chen, J. M. Nester, Class. QuantumGrav. , 1279 (1999).[3] A. Einstein, Preuss. Akad. Wiss. Berlin , 778 (1915); Addendum-ibid. , 799(1915); A. Trautman, in Gravitation: An Introduction to Current Research , ed. L.Witten (Wiley, New York, 1962, p. 169).[4] L. D. Landau and E.M. Lifshitz,