Energy-momentum complexes in f(R) theories of gravity
Tuomas Multamaki, Antti Putaja, Elias C. Vagenas, Iiro Vilja
aa r X i v : . [ g r- q c ] F e b Energy-momentum complexes in f ( R ) theories of gravity Tuomas Multam¨aki, ∗ Antti Putaja, † and Iiro Vilja ‡ Department of Physics, University of Turku, FIN-20014 Turku, FINLAND
Elias C. Vagenas § Research Center for Astronomy & Applied MathematicsAcademy of AthensSoranou Efessiou 4GR-11527, Athens, GREECE (Dated: October 30, 2018)Despite the fact that modified theories of gravity, in particular the f ( R ) gravity models haveattracted much attention in the last years, the problem of the energy localization in the frameworkof these models has not been addressed. In the present work the concept of energy-momentumcomplexes is presented in this context. We generalize the Landau-Lifshitz prescription of calculatingthe energy-momentum complex to the framework of f ( R ) gravity. As an important special case,we explicitly calculate the energy-momentum complex for the Schwarzschild-de Sitter metric for ageneral f ( R ) theory as well as for a number of specific, popular choices of f ( R ). I. INTRODUCTION
It has been almost a century since the birth of General Relativity and there are still problems that remain unsolved.The energy-momentum localization is one of them which till today is treated as a vexed problem. Much attention hasbeen devoted for this problematic issue. Einstein was the first who tried to solve it by introducing the methodologyof energy momentum pseudotensors. He presented the first such prescription [1] and after that a plethora of differentenergy-momentum prescriptions were proposed [2, 3, 4, 5, 6, 7]. All these prescriptions were restricted to computethe energy as well as the momenta distributions in quasi-Cartesian coordinates. Møller was the first to present anenergy-momentum prescription which could be utilized in any coordinate system [8].The idea of energy-momentum pseudotensors was gravely criticized for several reasons [9, 10, 11, 12] (actually one ofthe drawbacks was the aforesaid use of quasi-Cartesian coordinates which was solved by Møller’s prescription). Firstly,although a symmetric and locally conserved object, its nature is nontensorial and thus its physical interpretationseemed obscure [13]. Secondly, different energy-momentum complexes could yield different energy distributions forthe same gravitational background [14, 15]. Thirdly, energy-momentum complexes were local objects while therewas commonly believed that the proper energy-momentum of the gravitational field was only total, i.e. it cannot belocalized [16]. For a long period of time the idea of energy-momentum pseudotensors was relinquished.The approach of energy-momentum pseudotensors for the thorny problem of energy-momentum localization wasrejuvenated in 1990 by Virbhadra and collaborators [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Since then, numerous workshave been performed on computing the energy and momenta distributions of different gravitational backgrounds usingseveral energy-momentum prescriptions (for a recent list of references see [27]). In 1996 Aguirregabiria, Chamorroand Virbhadra [28] showed that five different energy-momentum complexes yield the same energy distribution for anyKerr-Schild class metric. Additionally, their results were identical with the results of Penrose [29] and Tod [30] usingthe notion of quasi-local mass. Many attempts since then have been performed to give new definitions of quasilocalenergy in General Relativity [32, 33, 34, 35, 36]. Considerable efforts have also been performed in constructingsuperenergy tensors [37]. Motivated by the works of Bel [38, 39, 40] and independently of Robinson [41], manyinvestigations have been carried out in this field [42, 43, 44, 45, 46].In 1999 Chang, Nester and Chen [47] proved that every energy-momentum complex is associated with a Hamiltonianboundary term. Therefore, the energy-momentum complexes can be considered as quasi-local, boundary conditiondependent conserved quantities. Finally, it should be pointed out that though a long way has been trodden thesolution to the problem of energy-momentum localization in the framework of General Relativity is way ahead. ∗ Electronic address: tuomul@utu.fi † Electronic address: apjput@utu.fi ‡ Electronic address: vilja@utu.fi § Electronic address: [email protected]
Another challenge to the development of physical theory of gravitation is that, the plethora of observational datacollected recently indicates that our universe is undergoing an accelerated expansion. Motivated by this observationalevidence, we have been in a long hunt for the explanation for this speed-up. Till today, three possible reasons havebeen presented. Two of them, namely the cosmological constant and the quintessence field, are developed in theframework of General Relativity. The third one is developed in the framework of alternative theories of gravity.In particular, the simplest among the aforesaid models are that in which Einstein-Hilbert action is modified by anadditional term.Modified theories of gravity, especially the f ( R ) gravity models that replace the Einstein-Hilbert action of GeneralRelativity (henceforth abbreviated to GR) with an arbitrary function of the curvature scalar, have been extensivelystudied in recent years (see e.g. [48, 49, 50, 51, 52, 53, 54, 55, 56] and references therein). The challenges inconstructing viable models in the light of cosmological constraints (see e.g. [57, 58, 59] and references therein),instabilities [60, 61, 62], solar system constraints (see e.g. [63, 64, 65, 66] and references therein) and evolution largescale perturbations [67, 68, 69] are now known. The Solar System constraints are a major obstacle to most theories[70, 71, 72, 73] but they can be completely removed by certain types of models [58, 66, 74, 75, 76].It is widely known that when a new theory is introduced, it is expected this new theory to successfully answerall already-solved (in the framework of the old theory) problems. Moreover, it is anticipated that this new theorywill be able to address, alleviate, and finally solve problems that the existing old theory cannot. Following this lineof thought, we address here, to our knowledge, for the very first time the problematic issue of energy-momentumlocalization in the context of f ( R ) gravity models which as was mentioned above intend to replace GR. We take firststeps in this direction and consider energy-momentum complexes within f ( R ) gravity models.The remainder of the paper is as follows. In Section II, we present the basic equations and formalism of f ( R ) gravity.In Sections III and IV, the Landau-Lifshitz energy-momentum complex and the Schwarzschild-de Sitter (henceforthabbreviated as SdS) metric, or the SdS black hole background, are reviewed. In Section V, we extend the concept ofthe Landau-Lifshitz energy-momentum complex into the framework of f ( R ) theories. As a special case, we computethe energy-momentum complex of the SdS metric for a number of commonly considered f ( R ) theories. In the finalsection we summarize the results and present our conclusions. II. f ( R ) GRAVITY FORMALISM
The action for f ( R ) gravity is S = Z d x √− g (cid:16) πG f ( R ) + L m (cid:17) , (1)where the standard Einstein-Hilbert action is replaced by a general function of scalar curvature f ( R ). The corre-sponding field equations (in the metric approach) are found by varying with respect to the metric g µν and readas F ( R ) R µν − f ( R ) g µν − ∇ µ ∇ ν F ( R ) + g µν (cid:3) F ( R ) = 8 πGT mµν (2)where T mµν is the standard minimally coupled stress-energy tensor and F ( R ) ≡ df /dR . In contrast to the standardEinstein’s equations from the Einstein-Hilbert action, the field equations are now of higher order in derivatives.Contracting the field equations gives F ( R ) R − f ( R ) + 3 (cid:3) F ( R ) = 8 πG ( ρ − p ) (3)where we have assumed that we can describe the stress-energy tensor with a perfect fluid. From the contractedequation it is clear that in vacuum, any constant scalar curvature metric with R = R is a solution of the contractedequation as long as F ( R ) R = 2 f ( R ). In general, the whole set of field equations is solved exactly by the SdSmetric [77] (for a more recent work on spherically symmetric solutions of modified field equations in f ( R ) gravity seealso [78]) ds = Bdt − B − dr − r dθ − r sin θdφ (4)with B ( r ) = 1 − Mr − Λ r . (5)The scalar curvature for this metric is R = − f ( R ) theory, including the standard General Relativity,satisfying the constant curvature condition F ( R ) R = 2 f ( R ) has the SdS (black hole) metric as an exact solution.We will return to this important special case in a later section. III. THE LANDAU-LIFSHITZ ENERGY-MOMENTUM COMPLEX
In general, the energy-momentum complex τ µν (henceforth abbreviated as EMC) carries coordinate dependentinformation on the energy content of the gravitational and matter fields. It sums up the energies of the matter fieldsthrough the energy-momentum tensor T µν (henceforth abbreviated as EM), and that of the gravitational field throughthe energy-momentum pseudotensor t µν (henceforth abbreviated as EMPT), which depends on the coordinate systemused to describe the system. The EMPT cannot be defined uniquely and a number of suggestions, with differentmathematical properties, exist. All of them lead to conserved quantities of the gravitational theory.The most straightforward conserved quantity is the integrated EMC over the three-dimensional space integral E EMC = Z B (0 ,r ) d x τ (6)which represents both the energy of the gravitational field and that of matter inside the coordinate volume B (0 , r ).In the case of a black hole, it consists of two parts: the black hole mass M and the energy stored in the gravitationalfield t , therefore E EMC = M + Z B (0 ,r ) d x √− g t . (7)In the construction of Landau and Lifshitz [2], one looks for an object, η µνα , that is antisymmetric in its indicessince then ∂ ν ∂ α η µνα = 0 due to the covariant continuity equation, which in a locally Minkowskian coordinate system,simplifies to ∂ ν T µν = 0 . (8)Hence T µν = ∂ α η µνα . (9)Einstein’s equations give T µν = 1 κ ( R µν − Rg µν ) (10)where κ = 8 πG (we have set c = 1), and in the locally Minkowskian coordinates Ricci tensor and Ricci scalar readsas R µν = 12 g µα g νβ g γδ ( ∂ α ∂ δ g γβ + ∂ γ ∂ β g αδ − ∂ α ∂ β g γδ − ∂ γ ∂ δ g αβ ) (11)and R = ∂ α ∂ β g αβ − g αβ (cid:3) g αβ = ( g αµ g βν − g αβ g µν ) ∂ µ ∂ ν g αβ . (12)Using these expressions in Eq. (10), one can rewrite the EM tensor as [2] T µν = ∂ α η µνα (13)where η µνα = 12 κ − g ) ∂∂x β (cid:2) ( − g )( g µν g αβ − g µα g νβ ) (cid:3) . (14)In a locally Minkowskian coordinate system ∂ α g µν = 0 and hence one can define( − g ) T µν ≡ ∂ α h µνα ≡ ∂ α ∂ β H µναβ (15)where two new tensors, so-called superpotentials, h µνα = ( − g ) η µνα = 12 κ ∂∂x β (cid:2) ( − g )( g µν g αβ − g µα g νβ ) (cid:3) (16) H µναβ = 12 κ [( − g )( g µν g αβ − g µα g νβ )] (17)have been defined.In a general coordinate system, Eq. (15) is no longer valid and one defines a new object t µν such that( − g )( T µν + t µν ) ≡ ∂h µνα ∂x α . (18)The new object, namely the EMPT t µν , is straightforwardly computed in a general coordinate system employing Eq.(18) since T µν can be expressed in terms of the geometric quantities by using the Einstein’s equation, i.e. Eq. (10),and h µνα is given in Eq. (16).Carrying out this somewhat lengthy but routine exercise, one obtains [2] t µνLL = κ n (2Γ ǫαβ Γ ωǫω − Γ ǫαω Γ ωβǫ − Γ ǫαǫ Γ ωβω )( g µα g νβ − g µν g αβ )+ g µα g βǫ (Γ ναω Γ ωβǫ + Γ νβǫ Γ ωαω − Γ νǫω Γ ωαβ − Γ ναβ Γ ωǫω )+ g να g βǫ (Γ µαω Γ ωβǫ + Γ µβǫ Γ ωαω − Γ µǫω Γ ωαβ − Γ µαβ Γ ωǫω )+ g αβ g ǫω (Γ µαǫ Γ νβω − Γ µαβ Γ νǫω ) o . (19)The Landau-Lifshitz EMC, i.e. τ µνLL , can now be evaluated either as a sum of the EM and the EMPT, namely τ µνLL = ( − g ) ( t µνLL + T µν ) (20)or directly using Eq. (18) which is now written as τ µνLL ≡ ∂h µνα ∂x α . (21) IV. THE SCHWARZSCHILD-DE SITTER METRIC
In GR, the empty space solution outside a static spherically symmetric mass distribution in a universe with acosmological constant is the SdS metric, or the SdS black hole metric. In spherically symmetric coordinates, it readsoutside the mass distribution as (in units where G = 1) given in eqs. (4) and (5). M is the total mass and Λ isthe cosmological constant. Due to the fact that some EMPTs are calculated in cartesian coordinates, we need tore-express the SdS black hole metric in cartesian terms. The metric (4) then reads as ds = Bdt − B − x + y + z x + y + z dx − x + B − y + z x + y + z dy − x + y + B − z x + y + z dz − ( B − − yzx + y + z dydz − ( B − − xyx + y + z dxdy − ( B − − xzx + y + z dxdz. (22)For this metric, but not for the metric in spherically symmetric coordinates, the determinant reads as g = −
1. Thisfeature is used regularly in the following as outer factors like √− g equal unity.The EMPTs corresponding to the SdS solution are straightforwardly calculable (in a cartesian coordinate system).Working in a space where r > − g ) t = − κ M + 12 M Λ r + Λ r (Λ r − r (6 M + r (Λ r − = − κ − r + (6 M + Λ r ) r (Λ r − r + 6 M ) . (23)For comparison, one can easily repeat this exercise for the other well-know EMPTs: we find that the WeinbergEMPT is equal to that of Landau and Lifshitz, the Einstein and Tolman EMPT is t = − Λ κ , (24)and the Møller EMPT is t = − κ . (25)It is evident that the EMPT of the SdS metric strongly depends on the chosen construction, even though EMPTsderived in different prescriptions sometimes can be identical. Furthermore, their behavior far from the mass sourceis wildly different. Therefore, by looking only the functional form of a single EMC, it is obscure what is the precisephysical interpretation of it, even though it represents a conserved quantity. However, we emphasize, that the differentEMCs differ by a divergence term related to the boundary conditions of the physical situation [47]. V. EMPT IN f ( R ) THEORIES OF GRAVITY
Among the different EMPTs studied in the literature, Landau-Lifshitz’s and Weinberg’s prescriptions appear tobe most straightforwardly suitable for extending into f ( R ) gravity theories. Here we consider extending the Landau-Lifshitz’s prescription and leave others for future work.In f ( R ) theories, one can write the field equations as T µν = 1 κ (cid:26) f ′ ( R ) R µν − g µν f ( R ) − D µ D ν f ′ ( R ) + g µν (cid:3) f ′ ( R ) (cid:27) . (26)Like in GR, the covariant continuity equation holds, i.e. D µ T µν = 0 [82], suggesting that one should write the RHSof Eq. (26) as a divergence of an object antisymmetric in its indices, i.e. in a form ∂ α h µνα .Following Landau’s and Lifshitz’s prescription and considering a locally Minkowskian coordinate system at a givenpoint we obtain κ T µν = f ′ ( R ) R µν − g µν f ( R ) + ( g µν g αβ − g µα g νβ ) ∂ α ∂ β f ′ ( R )= f ′ ( R ) G µν + 12 g µν ( f ′ ( R ) R − f ( R )) + ∂ α (cid:2) ( g µν g αβ − g µα g νβ ) ∂ β f ′ ( R ) (cid:3) = ∂ α (cid:2) f ′ ( R ) κ η µνα + ( g µν g αβ − g µα g νβ ) f ′′ ( R ) ∂ β R (cid:3) − ∂ α f ′ ( R ) κ η µνα + 12 g µν ( f ′ ( R ) R − f ( R ))= ∂ α (cid:2) f ′ ( R ) κ η µνα + ( g µν g αβ − g µα g νβ ) f ′′ ( R ) ∂ β R (cid:3) + 12 g µν ( f ′ ( R ) R − f ( R )) (27)where η µνα is that defined in (14). It is noteworthy that the term ∂ α f ′ ( R ) κ η µνα vanishes in locally Minkowskiancoordinate system, because η µνα is linear in the first derivatives of the metrics. We are partially able to write the RHSof the field equations, namely Eq. (27), as a divergence. The remaining term, absent in GR, remains problematicin a general case without a clear method which would enable us to write it as a four divergence. We can, however,proceed in an important special case where the scalar curvature is a constant, R = R . The SdS black hole metricbelongs to such a class of metrics. A. The Landau-Lifshitz -energy momentum complex for a metric with constant scalar curvature
For a metric with a constant scalar curvature, R = R , Eq. (27) simplifies to T µν = ∂ α [ f ′ ( R ) η µνα ] + 12 κ g µν ( f ′ ( R ) R − f ( R ))= ∂ α [ f ′ ( R ) η µνα ] + 16 κ ∂ α ( g µν x α − g µα x ν )( f ′ ( R ) R − f ( R ))= ∂ α [ f ′ ( R ) η µνα + 16 κ ( g µν x α − g µα x ν )( f ′ ( R ) R − f ( R ))] (28)so that the generalized Landau-Lifshitz superpotential takes the form˜ h µνα = f ′ ( R ) η µνα + 16 κ ( g µν x α − g µα x ν ) [ f ′ ( R ) R − f ( R )] . (29)The EMPT t µν in a general coordinate system defined in the Landau-Lifshitz prescription can now be read out fromthe expression for the EMC (remember, that g = − τ µν ≡ T µν + t µν ≡ ∂ α ˜ h µνα . (30)Hence the generalized Landau-Lifshitz EMC reads as τ µν = f ′ ( R ) τ µνLL + 16 κ [ f ′ ( R ) R − f ( R )] ∂ α ( g µν x α − g µα x ν ) , (31)where τ µνLL is the Landau-Lifshitz EMC evaluated in the framework of GR (see Eq. (21)). The 00-component reads as τ = f ′ ( R ) τ LL + 16 κ [ f ′ ( R ) R − f ( R )] ∂ α (cid:0) g x α − g α x (cid:1) = f ′ ( R ) τ LL + 16 κ ( f ′ ( R ) R − f ( R )) (cid:0) ∂ i g x i + 3 g (cid:1) . (32)Eq. (31) is a general formula valid for any f ( R ) theory when the studied metric has constant scalar curvature. Thestandard GR result is recovered when f ( R ) = R . B. Energy-momentum complex of the SdS metric of some f ( R ) models Using Eq. (31) we can compute the EMC of the SdS metric in a general f ( R ) theory: τ = f ′ ( R ) τ LL + 16 κ ( f ′ ( R ) R − f ( R )) ( rB ′ ( r ) + 3 B ( r ))= f ′ ( R ) τ LL + 16 κ ( f ′ ( R ) R − f ( R )) (cid:18) − Mr − r (cid:19) = − κ − r + (6 M + Λ r ) r (Λ r − r + 6 M ) f ′ ( R ) + 16 κ ( f ′ ( R ) R − f ( R )) (cid:18) − Mr − r (cid:19) . (33)This result is valid for any f ( R ) theory that has the SdS metric as a vacuum solution i.e. any theory which satisfiesthe vacuum equation f ′ ( R ) R − f ( R ) = 0. Again note that when f ( R ) = R we recover the standard, i.e. GR,form of Landau-Lifshitz EMPT, Eq. (23), as expected.An important special case encompassing popular choices of f ( R ) is a generic action function f ( R ) = R − ( − n − aR n + ( − m − b R m , (34)where n and m are positive integers and a, b any real numbers. This form of function f ( R ) is widely used incosmological context. In this case the generalized Landau-Lifshitz EMC takes the form τ = 2 − (1+2 n ) Λ − n r κ (cid:26) r (cid:2) (12 M − r + 5Λ r )( a (1 + n ) + b ( m − m + n ) (cid:3) − (cid:2) − r + (6 M + Λ r ) (cid:3) [ a n + (4Λ) n (4Λ + b m (4Λ) m )]Λ(6 M − r + Λ r ) (cid:27) . (35)For the form of f ( R ) considered above, i.e. Eq. (34), and recalling that for the SdS metric we have R = − n +1 = a ( n + 2) + b ( m − m + n . (36)In the special case where m = 2 or b = 0 we get a = (4Λ) n +1 n + 2 . (37)Note, that not all type (34) models are cosmologically viable. It is known, that those vacuum solutions with R suchthat f ′′ ( R ) > f ( R ) theory of gravity is f ( R ) = R − µ R − ǫR , (38)which has a stable vacuum whenever ǫ > / (3 √ µ ). The 00-component of the corresponding generalized Landau-Lifshitz EMC for this model is written as τ = 118 r κ R (cid:26) r ( ǫR − µ ) (cid:2) M − r + 5 r Λ (cid:3) − (cid:2) − r + (6 M + Λ r ) (cid:3) (cid:2) µ + R − ǫR (cid:3) R (6 M − r + Λ r ) (cid:27) , (39)which for the special case of the SdS black hole metric with the cosmological vacuum R = −√ µ , reduces to τ = 2 + 3 √ ǫµ √ r κ (cid:26) √ µ r + 3 µ r (4 M − r ) − (cid:2) √ M + 48 µ M r − µ r + √ µ r (cid:3) (24 M − r + √ µ r ) (cid:27) . (40)Another cosmologically interesting f ( R ) gravity model includes also logarithmic dependence on curvature. Thus itreads f ( R ) = R + ( − m − c R m − d ln (cid:18) | R | k (cid:19) , (41)where its parameters are related to the cosmological constant by the constant curvature condition written now as d + (4Λ) m c m = 2 (cid:20) (4Λ) m c + 2Λ + d ln (cid:18) k (cid:19)(cid:21) . (42)For this model the corresponding 00-component of the Landau-Lifshitz EMC is of the form τ = 118 κ r n − d + c m (4Λ) m + 4Λ) (cid:0) − r + (6 M + r Λ) (cid:1) Λ (6 M − r + Λ r ) + r (cid:0) M − r + 5 r Λ (cid:1) (cid:18) d + c ( m − m − d ln (cid:18) k (cid:19)(cid:19) o . (43)In any case the generalized Landau-Lifshitz EMPT of a f ( R ) model differs crucially from the GR Landau-LifshitzEMPT for the SdS metric. Taking into account the constant curvature condition we can write τ = f ′ ( R ) τ LL + 16 κ f ( R ) (cid:18) − Mr − r (cid:19) , (44)which coincides with the GR Landau-Lifshitz EMPT only if f ( R ) = 0 and f ′ ( R ) = 1 (implying, due to the constantcurvature condition, that there is no cosmological constant). This special case, while possible, is not a general propertyof physically meaningful f ( R ) models indicating that in a general f ( R ) model the Landau-Lifshitz EMPT will benon-trivially related to the corresponding EMPT in GR. For example, it is clear that at large r the two EMPTs havedifferent asymptotic limits with τ LL ∼ r − in GR and τ LL ∼ r in a general f ( R ) model. VI. CONCLUSIONS AND DISCUSSION
The problem of energy localization has been one of the first problems that was treated after the onset of GR.Although a number of scientists endeavored to solve it, the energy localization remains a vexed and unsolved problemtill to date. In this work, motivated by the recent interest in constructing extended models of gravity and in particular f ( R ) gravity models that replace the standard Einstein-Hilbert action of GR, we have introduced for the very firsttime, to our knowledge, the energy localization problem in the framework of f ( R ) theories of gravity. In particular,we have extended the concept of energy-momentum complex in the prescription of Landau-Lifshitz. Although we areunable to formulate a completely general expression for the EMC valid for all theories and metrics, we can proceedin an important special case where the scalar curvature of the considered metric is constant. In this case, we havepresented a general formula for the Landau-Lifshitz energy-momentum complex for a general f ( R ) theory. We findthat the general relativity result is generalized to encompass an additional term.Metrics satisfying the requirement of constant scalar curvature include the Schwarzschild-de Sitter metric, which e.g. describes the space-time around spherically symmetric objects in a universe with a cosmological constant. Wehave computed the generalized Landau-Lifshitz EMC for a general f ( R ) theory that accepts the SdS metric as asolution as wells as for a number of f ( R ) commonly considered in the literature. We find that the GR result isgeneralized by the presence of additional term. The new term is non-trivial as it has a different dependence on thecoordinate r than the term arising from the GR part.It is more than obvious that further study is needed, e.g. other EMC’s and their interpretation, i.e. correspondingphysical boundary conditions, in f ( R ) models need to be considered. A particularly interesting and a potentiallyfruitful direction to follow in the future is to consider the problem of energy localization in Weinberg’s formulation.For a more general EMC covering also the non-constant curvature case, a construction of a new type of EMC maybe a more direct way to proceed as generalization of the Landau-Lifshitz EMC is challenging. The calculations of theintegrated constants of motion in different models and systems is another example of a relevant open question. Wehope to address these issues in future work. Acknowledgments
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