DOI: 10.1016/S0375-9601(97)00257-0We provide a physical basis for the local gravitational superenergy tensor. Furthermore, our gravitoelectromagnetic deduction of the Bel-Debever-Robinson superenergy tensor permits the identification of the gravitational stress-energy tensor. This {\it local} gravitational analog of the Maxwell stress-energy tensor is illustrated for a plane gravitational wave.Two of the authors (B. M. and J. C. M.) gratefully acknowledge the hospitality extended
to them at ICN-UNAM, Mexico City, where part of this work was done

Mashhoon, Bahram

McClune, James C. (James Cephas), 1961-

Quevedo, Hernando

University of Missouri: MOspace

arXiv:gr-qc/9609018v1  6 Sep 1996GRAVITATIONAL SUPERENERGY TENSORBahram Mashhoon1, James C. McClune1 and Hernando Quevedo21Department of Physics and Astronomy, University of Missouri-ColumbiaColumbia, MO 65211, USA2 Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de Me´xicoAp. Postal 70-543, 04510 Me´xico D. F., Me´xicoAbstractWe provide a physical basis for the local gravitational superenergy tensor. Further-more, our gravitoelectromagnetic deduction of the Bel-Debever-Robinson superenergy ten-sor permits the identification of the gravitational stress-energy tensor. This local gravi-tational analog of the Maxwell stress-energy tensor is illustrated for a plane gravitationalwave.PACS numbers: 04.20.Cv, 04.20.Me, 04.30.+xKeywords: Bel-Debever-Robinson tensor, gravitoelectromagnetic stress-energy tensor1The concept of energy and the law of conservation of energy play crucial roles in allphysical theories. The notion of particle energy is already present in Newtonian mechan-ics. In more general physical theories, the notion of energy becomes more sophisticated,while still retaining its fundamental significance. In general relativity, the energy associ-ated with matter is represented by a stress-energy tensor TαβM that allows us to define, ina consistent way, the local energy density of matter as measured by a given observer. Thecondition TαβM ;β = 0 represents the law of local conservation of energy of matter, but itdoes not guarantee, in general, that the total energy is conserved. This is because TαβMcontains only the contribution of matter as well as all nongravitational fields, while thegravitational energy, which is also expected to contribute to the total energy, has not beentaken into account. However, in general relativity there is no physically meaningful notionof energy of the gravitational field. In fact, one would expect that in general relativity thegravitational energy should be given by an expression quadratic in the first derivatives ofthe metric, such as the Landau-Lifshitz pseudotensor. However, it follows from Einstein’sprinciple of equivalence that it is not possible to construct locally a covariant expressionfor energy only in terms of the spacetime metric and its first derivatives. In this paper,we present a physical derivation of an expression that is, in a certain average sense, thelocal stress-energy tensor of the gravitational field, an expression which is analogous to thestress-energy tensor of the electromagnetic field. In fact, the validity of this novel gravita-tional stress-energy tensor rests upon the approximate correspondence – in a quasi-inertialneighborhood surrounding the worldline of a geodesic observer – between gravitation andelectromagnetism. Our gravitoelectromagnetic stress-energy tensor turns out to be directlyproportional to the Bel-Debever-Robinson tensor, which is thereby furnished with a properphysical interpretation.The electromagnetic field is endowed with a symmetric traceless stress-energy tensorΘαβ that is given in terms of the Faraday tensor Fαβ byΘαβ =18π(FαγF βγ +∗Fαγ ∗F βγ), (1)where ∗Fαβ is the dual of Fαβ . In 1958, Bel suggested the notion of a gravitational2superenergy tensor by constructing an analogous fourth-rank tensor from the Riemanntensor Rµνρσ , namely,Tαβγδ =12( RαµγνRβ δµ ν +∗Rαµγν ∗Rβ δµ ν ), (2)which is unambiguously defined only for spacetimes with Rµν = Λgµν [1]. In this paper,we assume that the cosmological constant vanishes (Λ = 0), and the signature of the Ricciflat spacetime is +2; moreover, units are chosen such that Newton’s constant of gravitationand the speed of light in vacuum are set equal to unity.The gravitational superenergy tensor (2) has been used to define the local energydensity and Poynting vector for source-free gravitational fields [1]. Contributions to thisapproach have also been made by Debever [2] and Robinson [3]; a detailed treatment ofthis topic as well as further references to the original literature is contained in Zakharov’smonograph [4].The Bel-Debever-Robinson (BDR) superenergy tensor Tαβγδ is totally symmetric andtraceless. Moreover, it satisfies a conservation law Tαβγδ;δ = 0 in analogy with Θαβ;β = 0 forsource-free electromagnetic fields. The BDR tensor has been particularly useful in numeri-cal relativity and has therefore been the subject of studies by a number of investigators [5].However, a basic derivation as well as physical interpretation of the superenergy tensor hasbeen lacking thus far. It is the purpose of the present work to ameliorate this situation.To provide a physical basis for the analogy with electrodynamics, essential use willbe made here of the notion of a Fermi system. Consider an observer freely falling in agravitational field; the observer carries a tetrad frame λµ(α), where λµ(0) = dxµ/dτ is theobserver’s velocity and τ is the proper time along its path. The Fermi system – which isthe simplest generalization of a local inertial frame along the path of the observer – is ageodesic coordinate system based on a nonrotating frame along the observer’s worldline[6]. Let Xα = (τ,X) be the Fermi coordinates along the observer’s worldline; the spacetimemetric in these coordinates is given by F gαβ = ηαβ +Fhαβ , where ηαβ is the Minkowskimetric and Fhαβ(τ,X) may be expressed as a power series in X away from the observerwith coefficients that depend on the Riemannian curvature of spacetime. Only the lowest3order terms away from the path will be taken into account throughout this paper. It isuseful to define the gravitoelectric and gravitomagnetic potentials Φg and Ag, respectively,via Fh00 = 2Φg andFh0i = −2(Ag)i in analogy with electrodynamics.It follows from the construction of the Fermi system thatΦg = −12FR0i0j(τ)XiXj , (3)(Ag)i =13FR0jik(τ)XjXk, (4)where FRαβγδ(τ) = R(α)(β)(γ)(δ) = Rµνρσλµ(α)λν(β)λρ(γ)λσ(δ) is the curvature as measured bythe fiducial observer. A discussion of the general properties of this curvature is beyond thescope of this paper [7,8]. We now define the gravitoelectric (E) and gravitomagnetic (B)fields in complete analogy with electrodynamics; hence,Ei(τ,X) =FR0i0j(τ)Xj, (5)Bi(τ,X) = −12ǫijkFRjk0l(τ)Xl, (6)which agree with the identification of “electric” and “magnetic” components of the Rie-mann tensor. The curvature tensor generally consists of “electric”, “magnetic,” and “spa-tial” components; in a Ricci flat region, however, the “spatial” components are given by the“electric” components. Therefore, Eqs. (5) and (6) contain the full information regardingthe gravitational field along the path. Moreover, it is possible to express the gravitationalfield equations in the Fermi system in a form that is analogous to the Maxwell equationsusing Eqs. (5) and (6). This partial agreement between the field equations of gravitationand electrodynamics is well known and will not be further elaborated here [9].The Lorentz force law is needed in addition to Maxwell’s equations to produce acomplete picture of classical electrodynamics. In the Fermi system, this equation to firstorder in velocity is4md2Xdτ2= qEE + qBdXdτ× B, (7)where m is the inertial mass of a test particle, and qEand qBare the gravitoelectricand gravitomagnetic charges of the particle, respectively. It follows from the gravitationalLarmor theorem [10] that qE= −m and qB= −2m; the negative signs of the gravito-electromagnetic charges reflect the attractive nature of gravitation, while qB/qE= 2 sincegravitation is a spin-2 field. Equation (7) turns out to be the generalized Jacobi equa-tion [11], which is the equation of motion of a free test particle relative to the fiducialobserver, valid to first order in X and dX/dτ . This agreement between the deviationequation and the Lorentz force law is a remarkable result, since it makes it possible, inprinciple, to introduce the stresss-energy tensor for gravitoelectromagnetism in completeanalogy with the standard deduction of the electromagnetic stress-energy tensor. However,the electromagnetic stress-energy tensor is defined globally while the gravitoelectromag-netic stress-energy tensor would have to be defined in a narrow tube along the timelikepath of the observer in spacetime. The invariance under translations in time and spaceis ultimately responsible for the existence of conservation laws of energy and momentum,respectively, as well as the stress-energy tensor in Minkowski spacetime. The existence ofthe gravitational pseudotensor can be similarly justified in an asymptotically Minkowskianspacetime manifold. On the other hand, the gravitoelectromagnetic stress-energy tensorwould owe its existence to an inertial Fermi region (called the “tidal frame” in [12]) in theform of a thin Minkowskian cylinder in spacetime along the timelike path of the observer.It is possible to define a gravitational Faraday tensor in the Fermi system byFαβ = −FRαβ0lXl, (8)which contains Eqs. (5) and (6). Let us define the stress-energy tensor associated withFαβ in the standard manner byTαβ =14π(FαγFβγ−14ηαβFγδFγδ), (9)5which is symmetric and traceless by construction. The substitution of Eq. (8) in Eq. (9)reveals that T αβ is zero along the path of the fiducial observer (X= 0); this is ultimately aconsequence of Einstein’s principle of equivalence. On the other hand, T αβ is nonzero in aneighborhood of this path as the gravitoelectromagnetic field initially varies linearly withdistance away from the fiducial observer. Imagine a second observer with a worldline onlyinfinitesimally distant from the fiducial observer; if a second Fermi system is establishedalong this path and our construction of the gravitoelectromagnetic stress-energy tensor isrepeated, then this tensor would vanish identically along the second path but would givea finite result along the original reference path. It follows from these considerations thatthe proper stress-energy tensor along the path of an observer must be defined through anaveraging procedure. At any given proper time τ , let us average the tensor given by Eq.(9) over a limiting tube along the worldline. More precisely, consider a sphere of properradius ǫL, 0 < ǫ≪ 1, centered around X= 0 at a given τ such thatT (α)(β)(τ) = limǫ→0ǫ−2 < T αβ >, (10)where the angular brackets denote the operation of averaging over the sphere. Here L isa constant invariant length scale that is otherwise unspecified; in practice, it could be acharacteristic length scale for the problem under consideration (e.g., in the exterior field ofa black hole, L could be the mass of the black hole), or, in the absence of a natural scalein spacetime as in a pure gravitational radiation field, one could set L equal to the Plancklength. Using the fact that < X iXj >= (ǫ2L2/3)δij, we can express Eq. (10) in arbitrarycoordinates asTµν =L212πTµνρσλρ(0)λσ(0), (11)where Tµνρσ is symmetric and traceless in the first pair of indices and symmetric in thesecond pair of indices by construction, and is given byTµνρσ =12(RµξρςRξ ςν σ +RµξσςRξ ςν ρ )−14gµνRαβργRαβ γσ . (12)6This tensor is identical with that first defined by Bel [1] in 1958 for Rµν = Λgµν and coin-cides with Eq. (2). It is important to remark, however, that in our gravitoelectromagneticdeduction, i.e. Eqs. (3) - (12), no restriction has been placed on the Ricci tensor; therefore,Eqs. (11) and (12) define the gravitoelectromagnetic part of the local stress-energy tensorof a general gravitational field. Furthermore, the linear treatment of the fields in Eqs.(5) and (6) is sufficient to obtain the general results in Eqs. (11) and (12) since terms ofhigher order would simply drop out of Eq. (10) in the limit ǫ→ 0.To define the local gravitational stress-energy tensor at an event in spacetime, ageodesic observer with ideal gyroscope axes is needed at the event; then, T(α)(β) =L2T(α)(β)(0)(0)/12π for the observer, and the stress-energy tensor for any other observer atthat event can be obtained from T(α)(β) by an appropriate Lorentz transformation. More-over, the local physical quantities defined in this way (i.e. energy density, energy flux,momentum density, and the gravitational stresses) have now their proper dimensionalityas a consequence of the introduction of the constant length L. The freedom in the choiceof L implies, however, that these quantities are fixed up to a constant scale factor.To illustrate these results, let us consider a plane gravitational wave given by−ds2 = −dt2 + dx2 + U2(e2hdy2 + e−2hdz2), (13)which propagates along the x–axis and is linearly polarized. Here U and h are functionsof u = t − x and are related by U ′′ + h′ 2 U = 0, where a prime indicates differentiationwith respect to u. This gravitational field is of Petrov type N and is a member of a classof plane wave spacetimes that has been extensively studied [13].In the coordinate system under consideration here, observers located at fixed spatialcoordinates follow geodesics. Furthermore, the natural tetrad system λµ(α) associated withthese observers is nonrotating. The nonzero elements of λµ(α) are given by λ0(0) = λ1(1) =1, λ2(2) = U−1 exp(−h), and λ3(3) = U−1 exp(h). The local gravitational stress-energytensor according to these observers is given by7(T (α)(β)) =L26π1 1 0 01 1 0 00 0 0 00 0 0 0K2(u), (14)K(u) = h′′ + 2U ′Uh′. (15)The spacetime is flat if K(u) = 0. All the nonzero components of T(α)(β)(γ)(δ) can beobtained from Eqs. (14) and (15). As expected, the only nonzero component of thegravitational stresses is the local pressure of the radiation along the x–axis.Let ρg be the local energy density of the gravitational radiation field according to thefiducial static observers; then, ρg = L2K2(u)/6π. What is the energy density ρˆg measuredby observers boosted along the x–axis with speed β? The boost can be expressed interms of coordinates as u = ∆uˆ, ∆v = vˆ, y = yˆ, and z = zˆ, where v = t + x and∆ = (1− β)1/2/(1 + β)1/2 is the Doppler factor. By transforming the tetrads at a givenevent, one can show explicitly that ρˆg = ∆4ρg. Alternatively, this result may be obtainedby noting that under the boost the metric (13) and hence the magnitudes of U and hremain invariant; then, the form of Eq. (15) leads directly to the formula for ρˆg.It follows that the local gravitational radiation energy density transforms in this caselike the square of the energy density in standard field theory in Minkowski spacetime. Thiscould possibly have observable consequences for the local inertia of gravitational radiation;however, such radiation can not be physically confined as a consequence of the universalityof gravitational interaction.The application of these concepts to rotating gravitational waves [14] will be publishedelsewhere.Two of the authors (B. M. and J. C. M.) gratefully acknowledge the hospitality ex-tended to them at ICN-UNAM, Mexico City, where part of this work was done. We wishto thank R. Sussman for helpful discussions.8References[1] L. Bel, C. R. Acad. Sci. Paris 247 (1958) 1094; 248 (1959) 1297; Cah. Phys. 16(1962) 59.[2] R. Debever, Bull. Soc. Math. Belg. 10 (1958) 112.[3] See the paper of L. Bel, La Radiation Gravitationnelle, in: Colloques Internationauxdu CNRS, Les The´ories Relativistes de la Gravitation (CNRS, Paris, 1962) pp. 119-126.[4] V. D. Zakharov, Gravitational Waves in Einstein’s Theory, translated by R. N. Senfrom the 1972 Russian edition (Halsted, New York, 1973).[5] J. A. Wheeler, Phys. Rev. D 16 (1977) 3384; A. Peres, Phys. Rev. D 18 (1978) 608;J. Iba´n˜ez and E. Verdaguer, Phys. Rev. D 31 (1985) 251; J. Isenberg, M. Jackson andV. Moncrief, J. Math. Phys. 31 (1990) 517; N. Bre´ton, A. Feinstein and J. Iba´n˜ez,Gen. Rel. Grav. 25 (1993) 267.[6] For recent interesting discussions of the Fermi system see K.-P. Marzlin, Phys. Rev.D 50 (1994) 888; Gen. Rel. Grav. 26 (1994) 619.[7] J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry, secondedition (Marcel Dekker, New York, 1996) ch. 2.[8] B. Mashhoon, Phys. Lett. A 163 (1992) 7; B. Mashhoon and J. C. McClune, Mon.Not. Roy. Astron. Soc. 262 (1993) 881.[9] A. Matte, Canadian J. Math. 5 (1953) 1; B. Mashhoon, Found. Phys. 15 (1985) 497;R. T. Jantzen, P. Carini and D. Bini, Ann. Phys. (NY) 215 (1992) 1; W. B. Bonnor,Class. Quantum Grav. 12 (1995) 1483.[10] B. Mashhoon, Phys. Lett. A 173 (1993) 347.[11] B. Mashhoon, Astrophys. J. 197 (1975) 705, N.B. Eq.(A7) of the Appendix; D. E.Hodgkinson, Gen. Rel. Grav. 3 (1972) 351.[12] B. Mashhoon, Astrophys. J. 216 (1977) 591.[13] H. Bondi, F.A.E. Pirani and I. Robinson, Proc. Roy. Soc. London A 251 (1959) 519.[14] B. Mashhoon and H. Quevedo, Phys. Lett. A 151 (1990) 464; H. Quevedo and B.Mashhoon, in: Relativity and Gravitation: Classical and Quantum, Proc. SILARG9VII, eds. J. C. D’Olivo et al. (World Scientific, Singapore, 1991) pp. 287-293.10

Gravitational Superenergy Tensor

Abstract

Similar works

Full text

Available Versions

University of Missouri: MOspace