Invariant conserved currents in gravity theories with local Lorentz and diffeomorphism symmetry Yuri N. Obukhov* Instituto de Fı́sica Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, Brazil and Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia Guillermo F. Rubilar† Instituto de Fı́sica Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, Brazil and Departamento de Fı́sica, Universidad de Concepción, Casilla 160-C, Concepción, Chile (Received 16 July 2006; published 1 September 2006) We discuss conservation laws for gravity theories invariant under general coordinate and local Lorentz transformations. We demonstrate the possibility to formulate these conservation laws in many covariant and noncovariant(ly looking) ways. An interesting mathematical fact underlies such a diversity: there is a certain ambiguity in a definition of the (Lorentz-) covariant generalization of the usual Lie derivative. Using this freedom, we develop a general approach to the construction of invariant conserved currents generated by an arbitrary vector field on the spacetime. This is done in any dimension, for any Lagrangian of the gravitational field and of a (minimally or nonminimally) coupled matter field. A development of the ‘‘regularization via relocalization’’ scheme is used to obtain finite conserved quantities for asymptotically nonflat solutions. We illustrate how our formalism works by some explicit examples. DOI: 10.1103/PhysRevD.74.064002 PACS numbers: 04.20.Cv, 04.20.Fy, 04.50.+h I. INTRODUCTION As is well known, in the Lagrangian approach every continuous symmetry of the action gives rise to a conser- vation law. This is the substance of the Noether theorem which provides a construction of the corresponding con- served currents that satisfy certain algebraic and differen- tial identities. In particular, the energy and momentum of a physical system are described by the currents that are generated by the symmetry with respect to time and space translations. In a similar way, the angular momentum is described by a current related to the spacetime rotations. In gravity theory, the definition of energy, momentum, and angular momentum is a nontrivial problem that has a long and rich history (see the reviews [1–3], for example). The difficulty is rooted deeply in the geometric nature of the gravitational theory and is related to the equivalence principle which identifies locally gravity and inertia. As a result, the conservation laws that arise from the general coordinate (diffeomorphism) invariance normally have the form of covariant, but not invariant, equations. The energy- momentum currents in general transform nontrivially under a change of the local coordinate system or a local frame. In a recent paper [4] we have discussed the covari- ance properties of conserved quantities in the framework of the tetrad approach to general relativity theory. Along with the covariant conservation laws that are formulated in terms of the gravitational field variables independently of the additional geometric structures on a spacetime manifold, there exist a class of invariant conser- vation laws. The corresponding conserved currents are usually associated with a vector field that acts on spacetime as a generator of a certain symmetry of the gravitational configuration. However, no systematic and general method for the construction of such invariant conserved quantities was ever developed in the literature, at least not to our knowledge. For example, quite a long time ago Komar [5,6] proposed a very nice formula that gives reasonable values of the total energy (mass) and angular momentum for asymptotically flat configurations. However, although these formulas have proved their apparent feasibility, and despite certain achievements [7–21] in explaining their relevance to the concepts of the gravitational energy and angular momentum, many important issues remained un- clear. In particular, the freedom in the choice of the gravi- tational Lagrangian, of the gravitational field variables, of the coupling to matter, of the matter Lagrangian, of the background geometry (is it needed at all?), of the class of relevant vector fields (should they be necessarily Killing vector fields? why?); all these questions did not have a satisfactory answer (in our opinion) in the existing litera- ture. One of the motivations for the current study can be formulated as ‘‘understanding Komar.’’ Another motivation is related to the interesting work of Aros et al. [22,23] who have proposed a new conserved quantity for asymptotically anti-de Sitter (AdS) space- times. Actually, their result appeared both surprising and confusing to us. The reason is that their conservation law was derived from an invariant Lagrangian with the help of (apparently) covariant computations, however, the result- ing conserved current (and the corresponding total charge) was not a scalar under local Lorentz transformations. How could this happen? Two immediate guesses naturally can *Electronic address: yo@ift.unesp.br †Electronic address: grubilar@udec.cl PHYSICAL REVIEW D 74, 064002 (2006) 1550-7998=2006=74(6)=064002(19) 064002-1 © 2006 The American Physical Society http://dx.doi.org/10.1103/PhysRevD.74.064002 be formulated. Either the construction [22,23] is funda- mentally inconsistent, then it is necessary to find the source of the inconsistency. Or, if this result is nevertheless con- sistent, then perhaps one can improve it in such a way that from an invariant Lagrangian, by using covariant manipu- lations, one would be able to derive invariant conserved quantities. At the beginning, we considered the first guess to be most probably true. However, a careful analysis has demonstrated that the results of [22,23] are consistent, but can be improved. In this paper, we present a detailed exposition of the corresponding analysis. At the same time, we show that indeed the results of Aros et al. can be improved along the lines mentioned above. Namely, one of the aims of our paper is to systematically investigate the derivation of covariant and invariant conservation laws for the local Lorentz and the diffeomorphism symmetry in gravity theo- ries. As a result, we derive new explicitly invariant con- servation laws for the currents that are true scalars under general coordinate and local Lorentz transformations. The Lagrangian approach seems to be most appropriate when one discusses generally covariant models. We will thus use the Lagrangian framework in this study. Nevertheless, it is worthwhile to recall that the Hamiltonian approach reveals many other important aspects, for example, the role of the conserved charges in defining the generators of the corre- sponding symmetry transformations [24]. These aspects are discussed in a review [3] (see also the references therein). For a covariant Hamiltonian formulation see, in particular, [25–28]. Since both the diffeomorphism and the local Lorentz symmetry are in the center of our attention, we naturally turn to the so-called Poincaré gauge approach to gravity [24,29–33] in which both symmetries are naturally real- ized. Einstein’s general relativity arises as a particular (degenerate) case in this framework. In the gauge theory of gravity based on the Poincaré group (the semidirect product of the Lorentz group and the spacetime transla- tions) mass (energy-momentum) and spin are treated on an equal footing as the sources of the gravitational field. The gravitational gauge potentials are the local coframe 1-form #� and the 1-form �� � of the metric-compatible connec- tion. The absence of nonmetricity and use of orthonormal frames yield the skew symmetry of the connection, ��� � ����. The spacetime manifold carries a Riemann-Cartan geometric structure with nontrivial curvature and torsion that arise as the corresponding gauge field strengths: R�� � d�� � � �� � ^ �� � and T� � d#� � �� � ^ #�. The structure of the paper is as follows. In Sec. II we consider the consequences of the invariance of a general Lagrangian of the gravitational field under local Lorentz transformations of the frames and under spacetime diffeo- morphisms. The Noether identities corresponding to the Lorentz symmetry are derived. We then demonstrate that these identities underlie the possibility to recast the Lie equation for the diffeomorphism symmetry in an explicitly covariant form. However, such a ‘‘covariantization’’ is essentially nonunique, and it is determined by what we call a generalized Lie derivative. The latter can be intro- duced on a Riemann-Cartan manifold with the help of an arbitrary (1, 1) tensorial field that has a ‘‘connection’’-like transformation law under the action of the Lorentz group. These observations are subsequently used in Sec. III A to derive several equivalent forms of the Noether identities for the diffeomorphism symmetry of the gravitational ac- tion. In Sec. IV we extend the Lagrange-Noether machi- nery to the Lagrangian of a matter field with an arbitrary coupling to gravity (i.e., we allow also for nonminimal coupling). Section V presents the main results of our paper: here we give the explicit construction of the invariant currents associated with an arbitrary vector field. The next Sec. VI provides a natural extension of the construc- tion to the case of an arbitrary generalized Lie derivative. This, in particular, demonstrates the consistency of the original construction of Aros et al. By applying our general construction to the case of the Einstein-Cartan theory in Sec. VII we show that the Komar formula arises as a special case of our invariant current. Section VIII describes the relocalization of the currents that is induced by the change of the Lagrangian by a boundary term. In order to demonstrate how the general formalism works, in Sec. IX we apply our derivations to the computation of the con- served charges (total mass and angular momentum) for solutions without and with torsion. In the asymptotically nonflat cases, the conserved charges turn out to be diver- gent and we need to regularize them. We show that the ‘‘regularization via relocalization’’ method (which we re- cently used in [4]) successfully works also here. Finally, Sec. X contains a discussion of the results obtained and gives an outlook of further possible developments and of open problems. Our general notations are as in [32]. In particular, we use the Latin indices i; j; . . . for local holonomic spacetime coordinates and the Greek indices �;�; . . . label (co)frame components. Particular frame components are denoted by hats, 0̂, 1̂, etc. As usual, the exterior product is denoted by ^, while the interior product of a vector � and a p-form � is denoted by �c�. The vector basis dual to the frame 1- forms #� is denoted by e� and they satisfy e�c#� � ���. Using local coordinates xi, we have #� � h�i dx i and e� � hi�@i. We define the volume n-form by � :� # 0̂ ^ � � � ^ #n̂. Furthermore, with the help of the interior product we define �� :� e�c�, ��� :� e�c��, ���� :� e�c���, etc., which are bases for (n� 1)-, (n� 2)-, and (n� 3)-forms, etc., respectively. Finally, ��1����n � e�n c��1����n�1 is the Levi-Civita tensor density. The �-forms satisfy the useful identities: #� ^ �� � ����; (1.1) YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-2 #� ^ ��� � ����� � � � ���; (1.2) #� ^ ���� � ������ � � � ���� � � � ����; (1.3) #� ^ ����� � ������� � � � ����� � � � ����� � �������; (1.4) etc. The line element ds2 � g��#� � #� is defined by the spacetime metric g�� of signature (�;�; � � � ;�). II. GENERAL LAGRANGE-NOETHER MACHINERY We work on a n-dimensional spacetime manifold. The Poincaré gauge potentials are the coframe #� and the Lorentz connection �� �. We assume that the gravitational Lagrangian n-form V has the form V � V�#�; T�; R���; (2.1) and that it is invariant under local Lorentz transformations. As usual, let us introduce, according to the canonical prescription, the following translational and rotational gauge field momenta (n� 2)-forms [34]: H� :� � @V @T� ; H� � :� � @V @R� � : (2.2) Moreover, we define the canonical energy-momentum and spin (n� 1)-forms E� :� @V @#� ; E�� :� �#�� ^H� ; (2.3) for the gravitational gauge field. A general variation of the gravitational Lagrangian then reads �V � �#� ^ E� � ��� � ^ C�� � d��#� ^H� � ��� � ^H� ��; (2.4) where we have defined the variational derivatives with respect to (w.r.t.) the gravitational potentials: E � :� �V �#� � �DH� � E�; (2.5) C � � :� �V ��� � � �DH � � � E � �: (2.6) For an infinitesimal Lorentz transformation, �� � � ��� � " � �, with "�� � �"��, we have �#� � "��# �; ��� � � �D"��: (2.7) Substituting this into (2.4), we find �V � "���# � ^ E� �DC � ��: (2.8) Thus, the local Lorentz invariance of the gravitational Lagrangian, �V � 0, yields the Noether identity DC�� � #�� ^ E� 0: (2.9) Now, let us derive the consequences of the diffeomor- phism invariance of V. Let f be an arbitrary local diffeo- morphism on the spacetime manifold. It acts with the pullback map f� on all the geometrical quantities, and the invariance of the theory means that [35] V�f�#�; f�T�; f�R� �� � f��V�#�; T�; R� ���: (2.10) Consider an arbitrary vector field � and the corresponding local 1-parameter group of diffeomorphisms ft generated along this vector field. Then, using ft in the above formula and differentiating w.r.t. the parameter t, we find the iden- tity �‘�#�� ^ E� � �‘�T�� ^H� � �‘�R��� ^H� � � ‘�V: (2.11) Here the Lie derivative is given on exterior forms by ‘� � d�c � �cd: (2.12) At first sight, the left-hand side (l.h.s.) of (2.11) does not look to be invariant under local Lorentz transformations because of the usual (not covariant) derivatives in (2.12). However, it is invariant. In order to see this, let us recall the property of the Lie derivative ‘��’!� � ’‘�!� ��cd’�! which is valid for any exterior form ! and any function (0-form) ’. Then, taking into account the local Lorentz transformations #0� � �� �# �, E0� � ���1���E�, T0� � �� �T �, H0� � ���1���H�, R0� � �� ��� �1���R� �, and H0�� � �� ��� �1���H � �, we straightforwardly find that under the action of the local Lorentz transformation, the l.h.s. of (2.11) will be shifted by the term ���1�����cd�� ���DC � � � #� ^ E��. This is zero in view of the Noether identity (2.9). Let us now notice that the 1-form ���1���d�� � is a flat Lorentz connection. Then the above observation can be used as follows. We can take an arbitrary Lorentz-valued 0-form B����� � �B����� and add a zero term, B� ��DC�� � #� ^ E��; (2.13) to the l.h.s. of (2.11). As is easily verified, this addition is equivalent to the replacement of the usual Lie derivative (2.12) with a generalized Lie derivative L� :� ‘� � B� � �� when applied to coframe, torsion, and curvature. This generalized derivative will be covariant provided B�� transforms according to (A2). Here �� denote the corre- sponding Lorentz generators for the object on which the derivative acts. There are many options for the choice of the generalized covariant Lie derivative. One family of generalized Lie derivatives corresponds to the case in which B� � � �cA� �, where A� � is an arbitrary Lorentz connection. For this INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-3 family, one can verify that ‘� � �cA� � �� � �cD A �D A �c, where D A :� d� A� � �� denotes the covariant derivative defined by the connection A. We will consider the follow- ing two possibilities: 8� :� ‘� � �c�� � �� � �cD�D�c; (2.14) 8 fg � :� ‘� � �c� fg � � �� � �cD fg �D fg �c: (2.15) Here �� � is the dynamical Lorentz connection of the theory, whereas � fg � � is the Riemannian connection (i.e., the anholonomic form of the Christoffel symbols). We refer the reader to Appendix A for further details. Yet another possibility satisfying the condition (A2), that is not directly related to a connection, is given by L � :� ‘� ��� � ��; (2.16) i.e., B�� :� ��� �, where ��� :� e��c‘�#� : (2.17) This, our third choice of generalized covariant Lie de- rivative, is very different from the other two, as its exis- tence is specific for the models with a coframe (tetrad) field (and a metric). We will call (2.16) a Yano derivative since its counterpart for the linear group was first introduced in [36]. See also [37] for similar, but different definitions. III. GRAVITATIONAL NOETHER IDENTITIES FOR DIFFEOMORPHISM SYMMETRY All the above three choices of the covariant Lie deriva- tive are useful. The most common is the first option (2.14) which directly yields covariant Noether identities [32]. (For a general discussion of the Noether theorems in models with local symmetries see [38]). A. Covariant Noether identities Using the covariant Lie derivative 8� we recast (2.11) into �8�# �� ^ E� � �8�T �� ^H� � �8�R� �� ^H� � � 8�V � A� dB � 0; (3.1) from where (with �� :� �c#� denoting the components of the vector field �), we find A � ����DE� � e�cT� ^ E� � e�cR�� ^ C���; (3.2) B � ���E� � e�cT � ^H� � e�cR� � ^H� � � e�cV�: (3.3) Since the diffeomorphism invariance holds for arbitrary vector fields �, A and B must necessarily vanish and thus we find the familiar Noether identities: DE� e�cT� ^ E� � e�cR�� ^ C��; (3.4) E� e�cV � e�cT � ^H� � e�cR� � ^H� �: (3.5) B. Covariant Noether identities: another face The second choice (2.15) brings (2.11) into a similar form: �8 fg �#�� ^ E� � �8 fg �T�� ^H� � �8 fg �R��� ^H� � � 8 fg �V � A0 � dB � 0: (3.6) For B we still find (3.3), whereas A0 is different, acquiring an additional piece proportional to the rotational Noether identity, namely A0 � ����DE� � e�cT � ^ E� � e�cR� � ^ C�� � �e�cK����#� ^ E� �DC��� : (3.7) In accordance with the above general analysis, the differ- ence of (3.1) and (3.6) is proportional to the difference of the connections K� � :� � fg � � � �� �: (3.8) This quantity is known as contortion 1-form. In particular, the torsion is recovered from it as T� � K� � ^ # �. The corresponding curvature 2-forms are related via R� � � R fg � � �D fg K� � � K� � ^ K� �: (3.9) Combining together the terms with E�, with the help of (3.8) and (3.9) we can recast (3.7) into A0 � ����D fg �E� � C��e�cK� �� � C�� ^ �e�cR fg � � � 8 fg �K��� : (3.10) Again, since the diffeomorphism invariance holds for an arbitrary vector field �, we recover the corresponding Noether identity in the alternative form D fg �E� � C��e�cK� �� �e�cR fg � � � 8 fg �K��� ^ C��: (3.11) The identities (3.4) and (3.11) are equivalent, but in con- trast to the usual (3.4), the alternative form (3.11) is less known. It was derived previously in [33] using a different method. C. ‘‘Noncovariant’’ Noether identities Besides the above choices, it is still possible to work with the noncovariant ordinary Lie derivative (2.12). Then YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-4 we start directly with the identity (2.11) which we recast into �‘�#�� ^ E� � �‘�T�� ^H� � �‘�R��� ^H� � � ‘�V �A� dB � 0: (3.12) Again B is given by (3.3), but A changes into A � ����DE� � e�cT � ^ E� � e�cR� � ^ C�� � �e�c�����#� ^ E� �DC��� � ����dE� � �e�cd#�� ^ E� � �e�cd�� �� ^ C�� � �e�c�� ��dC�� : (3.13) As a result of the diffeomorphism invariance for an arbi- trary � we find yet another form of the Noether identity (3.4), namely d�E� � C��e�c�� �� �‘e�# �� ^ E� � �‘e��� �� ^ C��: (3.14) One may call this form of the Noether identity ‘‘noncovar- iant’’ since it explicitly involves the noncovariant gravita- tional field potentials �� � (and not the corresponding field strengths) and the ordinary (noncovariant) Lie derivative. However, just like the noncovariantly looking expression (2.11), the identity (3.14) is in fact covariant. Moreover, all the three forms (3.4), (3.11), and (3.14) are completely equivalent. IV. MATTER: DYNAMICS AND NOETHER IDENTITIES Matter fields can be represented by scalar-, tensor-, or spinor-valued forms of some rank, and we will denote all of them collectively as �A, where the superscript A indicates the appropriate index (tensor and/or spinor) structure. We will assume that the matter fields �A belong to the space of some (reducible, in general) representation of the Lorentz group. For an infinitesimal Lorentz transformation (2.7) the matter fields transform as �0A � �A � ��A; ��A � "��� � �� A B�B: (4.1) Here � ��� A B denote the corresponding matrices of gen- erators of the Lorentz group. For definiteness, we consider the case when the matter field �A is a 0-form on the spacetime manifold. This includes scalar and spinor fields of any rank [39]. A. Lagrangian and field equations We assume that the matter Lagrangian n-form L de- pends most generally on �A, d�A and the gravitational potentials #�, �� �. According to the minimal coupling prescription, derivatives of the gravitational potentials are not permitted. We usually adhere to this principle. How- ever, Pauli-type terms and Jordan-Brans-Dicke-type terms may occur in phenomenological models or in the context of a symmetry breaking mechanism. Also the Gordon decom- position of the matter currents and the discussion of the gravitational moments necessarily requires the inclusion of Pauli-type terms, see [40,41]. Therefore, we develop our Lagrangian formalism in sufficient generality in order to cope with such models by including in the Lagrangian also the derivatives d#�, and d�� � of the gravitational poten- tials. In view of the assumed invariance of L under local Lorentz transformations, the derivatives can enter only in a covariant form, namely L � L��A;D�A; #�; T�; R� ��: (4.2) That is, the derivatives of the matter fields appear only in a covariant combination, D�A � d�A � �� � ^ � ��� A B�B, whereas the derivatives of the coframe and connection can only appear via the torsion and the curva- ture 2-forms. For the total variation of the matter Lagrangian with respect to the material and the gravitational fields, we find �L � �#� ^ @L @#� � �T� ^ @L @T� � �R� � ^ @L @R� � � ��A ^ @L @�A � ��D�A� ^ @L @D�A (4.3) � �#� ^�� � ��� � ^ �� � ��A ^ �L ��A � d � �#� ^ @L @T� � ��� � ^ @L @R�� � ��A ^ @L @D�A � : (4.4) Here, as usual, we denote the covariant variational deriva- tive of L with respect to the matter field �A as �L ��A :� @L @�A �D @L @�D�A� ; (4.5) and the canonical currents of energy-momentum and spin are defined, respectively, by [42] �� :� �L �#� � @L @#� �D @L @T� ; (4.6) �� :� �L ��� � � � ��� A B�B ^ @L @�D�A� � #�� ^ @L @T� �D @L @R� � : (4.7) The principle of stationary action � R Vtot � 0, V tot :� V � L, for the coupled system of the gravitational and material fields, see (2.4) and (4.4), then yields the dynami- cal equations: INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-5 �L ��A � 0; (4.8) E � ��� � 0; (4.9) C � � � � � � 0: (4.10) B. Lorentz symmetry It is straightforward to derive the Noether identities following from the invariance of L under the local Lorentz group and under diffeomorphisms. Using the in- finitesimal transformations (2.7) and (4.1), we find for the Lorentz symmetry: �L � "�� � #� ^�� �D �� � � � �� A B�B ^ �L ��A � � d � "�� � � �� � � � �� A B�B ^ @L @D�A � #� ^ @L @T� �D @L @R�� �� � 0: (4.11) The term in the total derivative vanishes identically in view of the definition (4.7) of the spin current ��. Then, from the arbitrariness of "��, we find the corresponding Noether identity: D �� � #�� ^�� �� ���AB�B ^ �L ��A � 0: (4.12) With� we denote ‘‘weak identities,’’ i.e., an identity valid assuming that the matter field equations are satisfied. C. Diffeomorphism symmetry For a diffeomorphism generated by an arbitrary vector field �, we find directly from (4.3): ‘�L � �‘�# �� ^ @L @#� � �‘�T �� ^ @L @T� � �‘�R� �� ^ @L @R�� � �‘��A� ^ @L @�A � �‘�D�A� ^ @L @D�A : (4.13) In complete analogy to the purely gravitational case, we have a wide choice of covariant Lie derivatives which we can use in the above identity instead of the ordinary Lie derivative ‘�. However, we will not repeat once again the detailed analysis of all possibilities, taking into account the equivalence of the final results. Instead, here we give de- tails for the covariant option (2.14) only. We then find � 8�L� �8�#�� ^ @L @#� � �8�T�� ^ @L @T� � �8�R��� ^ @L @R�� � �8��A� ^ @L @�A � �8�D�A� ^ @L @D�A � A� dB � 0; (4.14) where, after some algebra, we find A � ���c#��D �L �#� � ��cT�� ^ �L �#� � ��cR� �� ^ �L ��� � � ��cD�A� ^ �L ��A ; (4.15) B � ��c#�� @L @#� � ��cT�� ^ @L @T� � ��cR��� ^ @L @R� � � ��cD�A� ^ @L @D�A � �cL: (4.16) Using the fact that � is a pointwise arbitrary vector field, we derive the following Noether identities from B � 0 and A � 0, respectively: �� e�cL� �e�cD�A� ^ @L @D�A �D @L @T� � �e�cT�� ^ @L @T� � �e�cR� �� ^ @L @R� � ; (4.17) and D�� �e�cT �� ^�� � �e�cR� �� ^ �� � w� � �e�cT�� ^�� � �e�cR� �� ^ ��; (4.18) where w� :� �e�cD�A� �L ��A � 0: (4.19) Equation (4.17) yields the explicit form of the canonical energy-momentum of matter, with the first line representing the result known in the context of the special-relativistic classical field theory. The second line in (4.17) accounts for the possible Pauli terms as well as for Lagrange multiplier terms in the variations with constraints and it is absent for the case of minimal coupling. The first line in (4.18) is given in the strong form, without using the field equations for matter (4.8). In complete analogy to Sec. III B and III C it is possible to derive the ‘‘Riemannian’’ and the ‘‘noncovariant’’ ver- sions of the Noether identity (4.18). By using instead (2.14) the covariant Lie derivative (2.15) we then obtain D fg ��� � � �e�cK� �� � �e�cR fg � � � 8 fg �K� �� ^ ��; (4.20) whereas replacement of 8� by the ordinary Lie derivative (2.12) yields d��� � ��e�c�� �� � �‘e�# �� ^�� � �‘e��� �� ^ ��: (4.21) YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-6 The three forms of the Noether identity (4.18), (4.20), and (4.21) are equivalent. In [43] the metric-affine counter- part of (4.20) was used to demonstrate that structureless test particles always move along Riemannian geodesics. V. CURRENTS ASSOCIATED WITH A VECTOR FIELD In the two previous sections we have analyzed the Noether identities which follow from the diffeomorphism symmetry generated by any vector field �. The correspond- ing covariant currents do not depend on the latter. However, there exists a class of invariant conserved cur- rents which are associated with a given (though arbitrary) vector field. A notable example is the well-known Komar construction [5,6]. As another example, take a symmetric energy-momentum tensor Tj i (which is covariantly con- served in diffeomorphism-invariant theories) and a Killing field � � �i@i (that generates an isometry of the space- time). Then ji :� �jTj i is a conserved current, and a conserved charge is defined as the integral R S j i@ic� over a (n� 1)-hypersurface S. Moreover, it is possible to construct a conserved current (n� 1)-form for any solu- tion of a diffeomorphism-invariant model even when � is not a Killing field [19]. Such a current and the correspond- ing charge are scalars under general coordinate transformations. In this section, we derive globally conserved currents associated with a vector field by making use of the third covariant Lie derivative, namely, the Yano derivative (2.16). Note that it is defined without any connection. A. Gravitational current We start from the identity (2.11) for the gravitational Lagrangian, and replace the ordinary Lie derivative ‘� with the covariant Yano derivative L�. Then using the proper- ties (A18) and (A19), we find �L�#�� ^ E� � �L�T�� ^H� � �L�R��� ^H� � �L�V � L�#� ^ E� �L��� � ^ C�� � dJ grav�� � 0; (5.1) where we introduced the scalar (n� 1)-form J grav�� :� �cV �L�# � ^H� �L��� � ^H� �: (5.2) By making use of the definitions of the Yano derivative (A12) and (A13), and taking into account the Noether identity (3.5), we recast this current into the equivalent form J grav�� � d���H� ��� �H� �� � � �E� ��� �C��: (5.3) B. Matter current In complete analogy with the previous subsection, start- ing now from the identity (4.13) for the material Lagrangian, and replacing the ordinary Lie derivative ‘� with the covariant Yano derivative L�, we obtain L �# � ^ @L @#� �L�T � ^ @L @T� �L�R� � ^ @L @R�� �L��A ^ @L @�A �L�D�A ^ @L @D�A �L�L � L�#� ^�� �L��� � ^ �� �L��A ^ �L ��A � dJ mat�� � 0: (5.4) Here the (n� 1)-form of the matter current is defined by J mat�� :� �cL�L�# � ^ @L @T� �L��� � ^ @L @R� � �L��A ^ @L @D�A : (5.5) Again using the definitions of the Yano derivative (A12) and (A13), together with the Noether identity (4.17), we find that J mat�� � �d � �� @L @T� ��� � @L @R�� � � ���� ��� � ��: (5.6) C. Total current Finally, for the coupled system of gravitational and matter fields described by the total Lagrangian Vtot � V � L, the diffeomorphism invariance of Vtot gives rise to the total current (n� 1)-form J �� :� J grav�� � J mat�� � �c�V � L� �L��A ^ @L @D�A �L�# � ^ � H� � @L @T� � �L��� � ^ � H� � � @L @R�� � : (5.7) By combining (5.1) and (5.4), we verify that the exterior derivative of this current is dJ �� � L�# � ^ �E� � ��� �L��� � ^ �C�� � � �� �L��A ^ �L ��A : (5.8) INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-7 Finally, from (5.3) and (5.6), we find J �� � d � ��c#�� � H� � @L @T� � ��� � � H� � � @L @R�� �� � ��c#���E� � ��� ��� ��C�� � � ��: (5.9) As follows from (5.8), for solutions of the coupled system of gravitational plus matter field equations (4.8), (4.9), and (4.10), the total current (n� 1)-form (5.7) is conserved: dJ �� � 0 for any �. Hence, it is possible to define a corresponding conserved charge Q�� by integrating J �� over a (n� 1)-dimensional spatial hypersurface S. Moreover, as we see from the Eq. (5.9), on the solutions of the field equations (4.8), (4.9), and (4.10), this current is expressed in terms of a superpotential (n� 2)-form. As a result, the corresponding charge can be computed as an integral over the spatial boundary @S: Q�� :� Z S J �� � Z @S � ��c#�� � H� � @L @T� � ��� � � H� � � @L @R�� �� : (5.10) For the usual case of minimally coupled matter @L=@T� � 0 and @L=@R�� � 0, and this expression then simplifies considerably. As a historic remark, let us mention that similar con- structions for the matter current were described in the literature in the framework of the Einstein-Cartan theory [7,44,45], and also in the context of metric-affine gravity [46]. However, the conservation of those currents was derived assuming a (Killing) symmetry of the gravitational configuration. We can recover the latter result as follows. If we assume minimal coupling of matter to gravity and choose � as a generalized Killing vector which satisfies L�#� � 0 (cf. Eq. (A21)) and L��� � � 0, then (5.6) yields the matter current J mat�� � ���� ��� � ��, which is conserved in view of (5.4) on the solutions of the matter field equations. For spinless matter, i.e. �� � 0, it is sufficient for � to satisfy the usual Killing equation, L�#� � 0. In contrast to that, it is worthwhile to stress that in our derivations we never made any assumptions concerning the vector field �. The conservation of the current J �� and the existence of the corresponding charge Q�� do not depend on whether � is a usual/generalized Killing vector or not. Note also that we did not specify either the dimension of spacetime or the form of the gravitational Lagrangian. VI. GENERALIZED LIE DERIVATIVES AND COVARIANT CURRENTS In the previous section, we have applied the Lie deriva- tive L� in the sense of Yano to obtain invariant conserved currents associated with a vector field. This is one of our main results. However, the existence of arbitrary general- ized Lie derivatives (see Appendix A) opens additional possibilities. Let us recall that in Sec. II, in particular, see Eq. (2.13), we discovered that all generalized Lie derivatives (i.e., for any (1, 1) field B��) are admissible for the analysis of the conservation laws related to the diffeomorphism symmetry. The specific examples for the corresponding Noether identities were considered in Sec. III. So, a natural question arises: Can we find other con- served currents for the generalized Lie derivatives (for an arbitrary B��)? The answer is positive. Without repeating the computations, we just formulate the result. Given a generalized Lie derivative L� defined by some B��, the diffeomorphism invariance of the total Lagrangian V tot of the coupled gravitational and matter fields associates to a vector field � (that generates a diffeo- morphism) a current (n� 1)-form JL�� :� �c�V � L� � L��A ^ @L @D�A � L�# � ^ � H� � @L @T� � � L��� � ^ � H� � � @L @R�� � : (6.1) The subscript L reflects the fact that this current is defined with the help of the generalized Lie derivative L�. Moreover, this current has two basic properties: (i) it sat- isfies d�JL�� � � L�# � ^ �E� � ��� � L��� � ^ �C�� � � �� � L��A ^ �L ��A : (6.2) (ii) it admits the representation JL�� � d � ��c#�� � H� � @L @T� � � ��c�� � � B� �� � H� � � @L @R�� �� � ��c#���E� � ��� � ��c��� � B����C�� � � ��: (6.3) Consequently, for solutions of the field equations (4.8), (4.9), and (4.10), the new currents are conserved, d�JL�� � � 0, and the corresponding charges can be com- puted as an integral over the spatial boundary: QL�� :� Z S JL�� � Z @S � ��c#�� � H� � @L @T� � � ��c��� � B��� � H� � � @L @R�� �� : (6.4) When B�� � ��� �, we recover (5.7), (5.8), (5.9), and (5.10) in view of (A15). Below we describe a number of additional particular cases. YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-8 A. Covariant current for natural covariant Lie derivative The covariant Lie derivative 8� � D�c � �cD is defined by the dynamical connection �� � of the gravitational theory (hence the name natural). It corresponds to the choice B�� � �c���. As a result, the conserved current J��� (which is given by (6.1) with L� replaced with 8�) gives rise to the charge Q��� :� Z S J��� � Z @S � ��c#�� � H� � @L @T� �� : (6.5) This quantity is nontrivial only when the gravitational Lagrangian V depends on the torsion, and/or for nonmini- mal coupling of matter. For all other cases this charge is identically zero. For example, for the usual Einstein-Cartan theory we have H� � 0, and for the minimal coupling the derivative @L=@T� � 0 also vanishes, hence the above charge is trivial. B. Covariant current for Riemannian covariant Lie derivative The Riemannian covariant Lie derivative 8 fg � � D fg �c � �cD fg is defined by the Christoffel connection � fg � � (hence the notation with the subscript fg ). It corresponds to the choice B�� � �c� fg � �. The corresponding current Jfg�� is obtained by substituting L� with 8 fg � in (6.1). It yields the charge Qfg�� :� Z S Jfg�� � Z @S � ��c#�� � H� � @L @T� � � �cK� � � H� � � @L @R� � �� : (6.6) Here K� � is the contortion, see (3.8). As a result, this charge reduces to (6.5) for all torsion-free solutions, as is the case, for example, in the standard general relativity theory. C. Covariant current for background covariant Lie derivative Yet another option arises when we introduce a nondy- namical background connection ��� � (that is different from both �� � and � fg � �) and define a generalized Lie derivative with the help of B�� � �c ��� �. Then the key combination is �c��� � B�� � �c��� �, and the corresponding con- served charge reduces to Q��� :� Z S J��� � Z @S � ��c#�� � H� � @L @T� � � �c��� � � H� � � @L @R� � �� : (6.7) The difference ��� � :� �� � � ��� � normally should be chosen so that it provides a finite value for integral over the spatial boundary. Similar constructions for the computa- tions of conserved quantities in the gauge theories of gravity were used in [4,25–28,47,48]. D. Noncovariant current for ordinary Lie derivative Finally, one can also consider the case when B�� � 0. Then the generalized Lie derivative reduces to the ordinary one, L� � ‘�. As a result, we find the following charge Q0�� :� Z S J0�� � Z @S � ��c#�� � H� � @L @T� � � �c��� � H� � � @L @R� � �� : (6.8) Because of the noncovariant character of ‘�, this quantity is not invariant under local Lorentz transformations, i.e. Q0�� depends in general on the choice of frame on the spatial boundary @S. The general expression (6.8) reduces to the result found by Aros et al. for the specific Lagrangian V considered in [22], see also [23,49]. VII. EINSTEIN(-CARTAN) GRAVITY Komar [5] gave a formula for the computation of the gravitational energy in Einstein’s general relativity theory which proved to give correct (i.e., physically reasonable) results for asymptotically flat configurations. Later [6] it was recognized that this formula also yields the angular momentum for rotating configurations. Yet, it remained unclear (for us, at least) how this nice formula fits into a general Noether scheme. In the few relevant studies, the explanations were based on an assumption that � is a Killing vector either of the physical spacetime geometry or of a background geometry [7–12]. One of the motiva- tions for the current work was to clarify the status of the Komar construction. Let us consider the Einstein-Cartan theory that is de- scribed by the Hilbert-Einstein Lagrangian plus, in general, a cosmological term: V � � 1 2� �R�� ^ ��� � 2���: (7.1) Here � is the gravitational coupling constant, and � is the cosmological constant (with a dimension of the inverse length square). Making use of (2.2), (2.3), and (3.5), we INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-9 find explicitly H� � 0; E� � � 1 2� �R�� ^ ���� � 2����; (7.2) H�� � 1 2� ���; E�� � 0: (7.3) We assume minimal coupling of matter, so that @L=@T� � 0 and @L=@R�� � 0. The covariant current (5.7) then reduces to J �� � �c�V � L� �L��A ^ @L @D�A � 1 2� L���� ^ ���: (7.4) This current is conserved, dJ �� � 0, on the solutions of the field equations (4.8), (4.9), and (4.10), which now read 1 2R �� ^ ���� � ��� � ���; (7.5) 1 2T � ^ ���� � � ��; (7.6) �L ��A � 0: (7.7) On the other hand, ‘‘on shell’’ from (5.9) we read off J �� � 1 2� df��dk� �c�#� ^ T�� g; (7.8) where we used (7.2), (7.3), and (A17) (recall that k :� ��#� is the 1-form dual to the vector �). For the solutions of (7.6) for �� � 0, i.e. for spinless matter or in vacuum, the torsion vanishes, T� � 0, and hence the total charge (5.10) finally reduces to Q �� � 1 2� Z @S �dk: (7.9) This invariant conserved quantity Q�� is precisely the (n-dimensional generalization of the) Komar formula. VIII. RELOCALIZATION OF THE CURRENTS All our constructions are invariant under coordinate and local Lorentz transformations. However, besides the local coordinate and the local Lorentz freedom, there is another ambiguity in the definition of the conserved quantities. Namely, the field equations always allow for a relocaliza- tion of the gravitational field momenta. As a result, the conserved currents and the values of the total charges can be changed by means of the relocalization of a translational and rotational momenta. More specifically, we consider here the case when a relocalization is produced by the change of the gravita- tional field Lagrangian by a total derivative: V 0 � V � d ; � �#�;���; T�; R���: (8.1) The term d changes only the boundary part of the action, leaving the field equations unchanged. We will assume a boundary (n� 1)-form whose general variation can be written as � � �#� ^ @ @#� � ��� � ^ @ @�� � � �T � ^ @ @T� � �R�� ^ @ @R�� : (8.2) Then, taking the exterior derivative of (8.2) and expressing the variations of d#�, dT�, d�� �, and dR�� in terms of the variations of #�, T�, �� �, and R��, we find @d @#� � �d @ @#� � �� � ^ @ @#� � R�� ^ @ @T� ; (8.3) @d @T� � d @ @T� � �� � ^ @ @T� � @ @#� ; (8.4) @d @R� � � d @ @R� � � �� � ^ @ @R� � � �� � ^ @ @R� � � @ @�� � � #�� ^ @ @T� : (8.5) We assume that, just like the original Lagrangian V, the boundary term d is invariant under Lorentz transforma- tions. Then (making use of the Lagrange-Noether machi- nery outlined above) one can verify that @d @�� � � �d @ @�� � � �� � ^ @ @�� � � �� � ^ @ @�� � � #�� ^ @ @#� � T�� ^ @ @T� � R� � ^ @ @R� � � R� � ^ @ @R�� 0: (8.6) Note that unlike the n-form d , the (n� 1)-form itself does not necessarily need to be a scalar under local Lorentz transformations. Using (8.3), (8.4), and (8.5) in the general definitions (2.2) and (2.3), we then find the relocalized momenta and gravitational currents: H0� � H� � d @ @T� � �� � ^ @ @T� � @ @#� ; (8.7) H0�� � H� � � d @ @R� � � �� � ^ @ @R� � � �� � ^ @ @R� � � @ @�� � � # �� ^ @ @T� ; (8.8) E0� � E� � d @ @#� � �� � ^ @ @#� � R� � ^ @ @T� ; (8.9) YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-10 E0�� � E�� � #�� ^ d @ @T� � #�� ^ �� � ^ @ @T� � #�� ^ @ @#� : (8.10) Accordingly, the relocalized translational and rotational momenta (8.7) and (8.8) determine the relocalized con- served currents and charges when they are substituted into the corresponding formulas (5.7), (5.8), (5.9), and (5.10). The choice of a boundary term is fairly arbitrary. For example, a ‘‘universal’’ relocalization (in the sense that it is available in all spacetime dimensions) is defined with the help of a background connection ��� �. The latter can be introduced as the limit of the dynamical connection �� � at spatial infinity @S, for instance, or fixed from other argu- ments. Given the background connection, one can always (for any n) add to a Lagrangian V the covariant boundary term �0d 0 with a constant �0 and the (n� 1)-form 0 :� ��� ^����; ���� :� ��� � ����: (8.11) Then the field momenta are relocalized as H0� � H� � �0���� ^����; H0�� � H�� � ��1�n�0���: (8.12) Alternatively, there is a possibility to make the arbitrari- ness in the choice of the boundary term more narrow by restricting one’s attention to the topological invariants which always can be added to the action. This option, however, depends on the spacetime dimension. For in- stance, in four dimensions (n � 4) we can consider a 3- parameter family of boundary forms :� �1 1 � �2 2 � �3 3; (8.13) 1 :� T� ^ #�; (8.14) 2 :� �� � ^ �R� � � 1 3�� � ^ �� ��; (8.15) 3 :� �������� ^ �R�� � 1 3� �� ^ �� ��: (8.16) These 3-forms correspond to the Nieh-Yan [50,51], the Pontryagin, and the Euler topological invariants, respec- tively. They represent the so-called gravitational (transla- tional and rotational) Chern-Simons 3-forms, see [52] for more details. Substituting this into (8.7), (8.8), (8.9), and (8.10) we then obtain a particular (‘‘topological’’) relocal- ization: H0� � H� � 2�1T�; (8.17) H0�� � H�� � �1#� ^ #� � 2�2R�� � 2�3�����R ��; (8.18) E0� � E� � 2�1DT�; (8.19) E0�� � E�� � 2�1#�� ^ T� : (8.20) We will use this specific relocalization in the subsequent discussion of the regularization of invariant conserved quantities. XI. EXAMPLES In order to illustrate how our general formalism works, in this section we specialize to the case of four-dimensional theories: n � 4, � � 8 G=c3. At first, we study the purely Riemannian (without torsion) solutions of the Einstein- Cartan theory which have asymptotic AdS behavior. After that, we consider similar configurations with torsion that arise as solutions in the quadratic Poincaré gauge gravity theory. A. Kerr-AdS solution in Einstein-Cartan theory When the cosmological constant � is nontrivial, the Einstein-Cartan field equations (7.5) and (7.6) admit in vacuum the generalized Kerr solution with AdS asymp- totics (� < 0). We use a spherical local coordinate system (t, r, �, ’), and choose the coframe as # 0̂ � ���� � � s �cdt� a sin2�d’ ; (9.1) # 1̂ � ���� � � s dr; (9.2) # 2̂ � ���� � f s d�; (9.3) # 3̂ � ���� f � s sin���acdt� �r2 � a2�d’ : (9.4) Here the functions and constants are defined by � :� �r2 � a2� � 1� � 3 r2 � � 2mr; (9.5) � :� r2 � a2cos2�; (9.6) f :� 1� � 3 a2cos2�; (9.7) m :� GM c2 ; :� 1 1� � 3 a 2 ; (9.8) and 0< t <1, 0< r <1, 0< �< , and 0<’< 2 . For this solution the curvature has two nonvanishing irre- ducible pieces: R�� � �1�R�� � �6�R�� (see Appendix B for definitions of the irreducible parts of the curvature). The curvature scalar is R � �4�, and the Weyl 2-form (B14) has a very special structure. Namely, it is expressed INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-11 only in terms of a certain scalar 2-form w � 1 2w��# � ^ #� as follows: W�� � ~w���w� w��w� 1 3 ��w ^ �w�#� ^ #� � 1 3 ��w ^ w����: (9.9) Here ~w�� :� 1 2� ����w��. The 2-form w is given by a simple formula: w � u# 0̂ ^ # 1̂ � v# 2̂ ^ # 3̂; (9.10) where its two nontrivial components u :� w0̂ 1̂ and v :� w2̂ 3̂ are functions of r and �. Their explicit form is not of interest since we will need only the two invariants ��w ^ �w� � 3mr�r2 � 3a2cos2�� �3 ; (9.11) ��w ^ w� � 3ma cos��3r2 � a2cos2� �3 : (9.12) One can straightforwardly find u and v from these two equations, but as we said, we do not need these functions explicitly. For a vector field � � �i@i with constant holonomic components, �i, in the coordinate system used in (9.1), (9.2), (9.3), and (9.4), the computation of conserved charges is fairly straightforward. Substituting (C1) into (7.9), and using (C2)–(C4), we find in the asymptotically AdS case (� < 0): Q �� � �0 � Mc2 2 � 4 c� 3� r1�r2 1 � a2� � � �3 2Mca: (9.13) Here r1 is the radius of the spatial boundary sphere @S. It is worthwhile to note that Q�@r � 0 and Q�@� � 0. When � � 0, we recover the usual Komar result with Q�@t � Mc2=2 and Q�@’ � �Mca. For a nontrivial negative cosmological constant, the conserved charge Q�@’ � � 2Mca is finite, but Q�@t diverges as r1 ! 1. Hence, a regularization is needed. B. Regularization via relocalization In a recent paper [4], we have demonstrated that total conserved quantities can be regularized by means of a relocalization of the gravitational field momenta. In Sec. VIII, we discussed a specific relocalization generated by a boundary term in the action. Here we will use the same method to remove the divergence of the conserved charge (9.13) for the Kerr-AdS configuration. A straightforward inspection shows that one can solve the regularization problem with the help of the relocaliza- tion (8.17), (8.18), (8.19), and (8.20) generated by the Chern-Simons boundary terms (8.14), (8.15), and (8.16). More exactly, we can verify that the translational and the rotational Chern-Simons forms (8.14) and (8.15) do not affect the total charge, whereas the Euler boundary term (8.16) does the job. Hence we put �1 � �2 � 0, and consider the relocalization H�� ! H0�� � H�� � 2�3�����R �� that is generated by the change of the Lagrangian V ! V 0 � V � �3d 3 by the boundary term (8.16). Using (7.3) and the irreducible decomposition of the curvature (B9), we find H0�� � � 1 2� � 4�3� 3 � ��� � 2�3����� �R��; (9.14) with �R�� :� R�� � � 3 # � ^ #�. The term ��� contributes to Q0�� with the usual Komar expression �dk which makes the conserved charge (9.13) infinite. Hence we choose �3 � 3=8�� and eliminate the corresponding term completely. As a result, we end with the regularized invariant conserved charge Q 0�� � Z @S ���H0�� � � 3 4�� Z @S ������e �cD��� �R��: (9.15) On the Kerr-AdS solution, the relocalized momentum (9.14) is constructed in terms of the Weyl 2-form H0�� � � 3 4�� �����W ��: (9.16) Making use of (A17) and (C1), and of the explicit form of the Weyl 2-form for the Kerr-AdS spacetime, c.f. Eq. (9.9), we derive ��������W�� � 2 3� ��w ^ �w���2!� �� � ��w ^ w��2!� �� : (9.17) Substituting now (9.11), (9.12), and (C1)–(C4), we finally calculate the regularized charge: Q 0�� � �0 Mc2 � �3 2Mca: (9.18) In other words, we found finite values of the covariant total charge Q for the Kerr-AdS solution, which read Q 0�@t � Mc2; Q0�@’ � � 2Mca; Q0�@r � 0; Q0�@� � 0: (9.19) Our results agree with those in [53,54]. It is worthwhile to note that the relocalized gravitational Lagrangian for �3 � 3=8�� can be written as V 0 � 3 8�� ����� �R�� ^ �R��: (9.20) This Lagrangian was studied extensively in the framework of various approaches to gravity on the basis of the de Sitter group, see [55–58], for example. The same action was also used in [22,23] for the derivation of the conserved current associated with a vector field. YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-12 Our results (9.19) agree with those in [22,23] for this particular case (note that the timelike Killing vector used in [22] corresponds to @’ in our notation). However, such an agreement appears to be a mere (although remarkable) coincidence since in [22] the authors consider the non- covariant charge (6.8), which depends in general on the choice of a frame at spatial infinity. In order to illustrate this difference, let us take the Kerr- AdS coframe (9.1), (9.2), (9.3), and (9.4) and evaluate the noninvariant charge Q0�@t for a tetrad #0� that is obtained from the original one by means of the local Lorentz trans- formation # 0̂0 � # 0̂ cosh��xi� � # 1̂ sinh��xi�; (9.21) # 0̂1 � # 0̂ sinh��xi� � # 1̂ cosh��xi�; (9.22) # 0̂2 � # 2̂; # 0̂3 � # 3̂: (9.23) Choosing the function as ��xi� � �0rt sin� (�0 is a con- stant) we find that Eq. (6.8) yields Q00�@t � Mc2 � �06 2 m=��. The value of the charge can thus be arbi- trary, depending on the constant �0. This charge can even be made divergent. For � � �0r 2t sin�, for example, we find Q00�@t � Mc2 � r1�06 2 m=�� which diverges when the spatial boundary is r1 ! 1. It is certainly true that the parameters of the Lorentz transformation above look quite exotic, so to say. However, this demonstrates, as a matter of principle, the fact thatQ0 depends on the choice of a frame. In contrast, our invariant formula (5.10) yields the same finite value for all coordinates and all frames, which is much more advantageous. C. Kerr-AdS solution with torsion In vacuum, the Einstein-Cartan theory coincides with Einstein’s general relativity theory, and all the solutions are characterized by a vanishing torsion. In order to test our approach for the configurations with torsion, we will con- sider a different model, namely, the quadratic Poincaré gauge theory with the Lagrangian [59–61] V � � 1 2� � T� ^ #� ^ ��T� ^ #�� � 3 4� R�� ^ �R�� � : (9.24) This model was extensively studied [53,59–62] and it was demonstrated that it is a natural generalization of the Einstein-Cartan theory. In particular, it was shown that this model has a correct Einsteinian limit. Note that we use a slightly different notation for the coupling constants in (9.24), as compared to [59–61]. Here � � 8 G=c3 and � has dimensions of �length��2. The translational and the rotational gauge field momenta (2.2) now read: H� � 1 � #� ^ ��T� ^ #��; (9.25) H�� � 3 4�� �R��: (9.26) Substituting these expressions into the vacuum field equa- tions E� � 0 and C�� � 0, one can verify that there is a generalized Kerr-AdS solution that is described as follows. The coframe is again given by the above formulas (9.1), (9.2), (9.3), and (9.4), whereas the components of the torsion 2-form read: T 0̂ � ���� � � s � �v1# 0̂ ^ # 1̂ � 2v4# 2̂ ^ # 3̂ � ���� � � s T ^ �v2# 2̂ � v3# 3̂� � ; (9.27) T 1̂ � T 0̂; (9.28) T 2̂ � ���� � � s T ^ �v5# 2̂ � v4# 3̂�; (9.29) T 3̂ � ���� � � s T ^ ��v4# 2̂ � v5# 3̂�: (9.30) Here we have denoted the functions v1 � m�r2 � a2cos2�� �2 ; v4 � � mra cos� �2 ; v5 � mr2 �2 ; (9.31) v2 � � ���� f � s mra2 sin� cos� �2 ; v3 � � ���� f � s mr2a sin� �2 ; (9.32) and introduced the 1-form T :� # 0̂ � # 1̂. One can verify that the axial torsion piece vanishes for this configuration, while the torsion trace is proportional to the above 1-form, that is: #� ^ T � � 0; T � e�cT � � � m�������� �� p T : (9.33) The Riemann-Cartan curvature 2-form of this solution consists of only two irreducible parts (see the definitions in Appendix B), R�� � �4�R�� � �6�R��; (9.34) which read explicitly as follows �4�R�� � �mr 3� �4�R��; �6�R�� � � 3 #� ^ #�: (9.35) INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-13 Here the nonvanishing components of the fourth irreduc- ible part are given by �4�R0̂ 2̂ � ��4�R2̂ 0̂ � �4�R1̂ 2̂ � ��4�R2̂ 1̂ � T ^ # 2̂, and �4�R0̂ 3̂ � ��4�R3̂ 0̂ � �4�R1̂ 3̂ � ��4�R3̂ 1̂ � T ^ # 3̂. Substituting all this into (9.25) and (9.26), we explicitly find the translational momentum H0̂ � 1 � ���� � � s � �2v4# 0̂ ^ # 1̂ � 2v5# 2̂ ^ # 3̂ � ���� � � s T ^ �v3# 2̂ � v2# 3̂� � ; (9.36) H1̂ � �H0̂; (9.37) H2̂ � 1 � ���� � � s T ^ ��v4# 2̂ � �v1 � v5�# 3̂ ; (9.38) H3̂ � 1 � ���� � � s T ^ ��v5 � v1�# 2̂ � v4# 3̂ ; (9.39) and the rotational field momentum H�� � 1 4� ��� � mr 4�� ��4�R��: (9.40) We consider again vector fields with constant compo- nents �i in the coordinate system used above. Then, from (9.1)–(9.4) and (9.36)–(9.39), for the translational contri- bution described by the first term in (5.10) we obtain: Z @S ��c#��H� � �0� Mc2� � �3 � � 2 3 Mac � : (9.41) On the other hand, taking into account that the axial torsion vanishes (9.33), the Eq. (A17) again yields ��� � 1 2 e�ce�c�dk�. Using then (9.40), after some algebra we find the second (rotational contribution) term in the charge (5.10): Z @S ���H�� � 1 4� Z @S ������ � m 4� Z @S r � �����4�R�� (9.42) � � �0 � 1 4 Mc2 � 2 �c 3� r1�r 2 1 � a 2� � � �3 � � 1 2 2Mac �� � � �3 � � 1 6 2Mac �� : (9.43) In (9.43) we have explicitly specified by the square brack- ets the contributions of each integral in (9.42). Combining (9.41) with (9.43), we obtain the invariant conserved charges Q �@t � 5 4 Mc2 � 2 �c 3� r1�r 2 1 � a 2�; Q�@’ � � 4 3 2Mac; (9.44) whereas again Q�@r �Q�@� � 0. As in the Riemannian case discussed in Sec. IX A, we thus find a finite value for the angular momentum Q�@’ , but a divergent energy Q�@t . The source of the divergence is easily detected: From (9.42) and (9.43) we can see that it is again the usual Komar term ������ � ��dk� that is responsible for all the infinite contributions to Q�@t . Therefore, we can try to regularize the conserved charges (9.44) by relocalizing the field mo- menta in the same way as it was done for the Riemannian case in (9.14). Namely, we add an Euler boundary term, see (8.16), to the gravitational action (9.24). In this case the translational momentum does not change, H0� � H�, see (8.17), but the rotational momentum does change, accord- ing to (8.18). Making use of (9.34), (9.35), and (9.40), we then find the relocalized Lorentz momentum: H0�� � � 1 4� � 4�3� 3 � ��� � mr � � 1 4� ��4�R�� � 2�3� 3 ����� �4�R�� � : (9.45) Clearly, the divergent contributions are canceled provided we fix the coefficient of the boundary term as�3 � 3 16�� . In addition, we recall the double duality property of the fourth irreducible piece of the curvature, namely, �4�R�� � 1 2����� �4�R��, see Eq. (164) of [33], for example. The resulting relocalized quantity then reads H0�� � mr 2�� ��4�R��: (9.46) Using the explicit expression (9.35) for �4�R��, together with (A17) and (9.33), we then find Z @S ���H0�� � �3 � � 1 3 2Mac � : (9.47) Combining (9.41) and (9.47), we finally obtain the regu- larized covariant conserved charge: Q 0�� � �0� Mc2� � �3�� 2Mac�: (9.48) It is satisfactory to see that the new current yields the ‘‘standard’’ values for the energy and the angular momen- tum, Q0�@t � Mc2, Q0�@’ � � 2Mac, as well as the trivial values Q0�@r � 0 and Q0�@� � 0. It seems worthwhile to mention that although in both examples (without and with torsion) the resulting values of the regularized charge are the same, there is an important difference in the way how these values are actually com- posed. As compared to the Riemannian case discussed in YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-14 Sec. IX A, where the rotational term ���H�� alone gave rise both to the total energy and to the total angular momentum, in the solution with torsion the whole energy comes only from the translational contribution ��H�, whereas the total angular momentum arises from the com- bined contributions of the translational and rotational terms. X. DISCUSSION AND CONCLUSION In this work, we have analyzed the problem of defining conserved currents in diffeomorphism- and local Lorentz- invariant theories. Einstein’s general relativity theory, Einstein-Cartan theory, and general Poincaré gauge theo- ries of gravity belong to this class of models. We have presented a systematic derivation of a general expression for conserved currents that are invariant under both coor- dinate and local Lorentz transformations. Such currents J �� are associated with a given, but completely arbitrary, vector field � on the spacetime manifold. The conservation law, dJ �� � 0, holds ‘‘on-shell,’’ i.e. on every solution of the coupled system of the gravitational and matter field equations. Our results are valid in any spacetime dimension for the field-theoretic models with arbitrary Lagrangians. Since the latter are always defined up to a total derivative, we discussed the relocalization of the gravitational mo- menta, induced by boundary terms in the action, in order to find how such a relocalization affects the conserved cur- rents and charges. As we stressed, the total invariant current J �� is con- served for any vector field � when the field equations are satisfied. In contrast, the separate gravitational J grav�� and matter J mat�� currents are not conserved in general, not even on-shell. However, if the vector field � generates a symmetry of the field configuration, i.e. if the generalized Killing equations are satisfied for the coframe and connec- tion, L�#� � 0 and L��� � � 0, then (5.1) and (5.4) yield two independent conservation laws, dJ grav�� � 0 and dJ mat�� � 0. Moreover, in the case of spinless matter ( �� � 0) or in vacuum, the separate conservation laws arise for the case when the vector field � is a usual isometry (L�gij � 0). In other words, separate invariant conserva- tion laws exist for the matter and/or gravitational currents under stronger (Killing symmetry) conditions on the field configurations, as compared to the conservation of the total (gravitational plus matter) current that only requires the field equations to be satisfied. As an immediate consequence of the general results, we have demonstrated that the Komar construction arises as a particular invariant current for the Hilbert-Einstein Lagrangian of the gravitational field. In this sense, the whole formalism can be viewed as a generalization of the Komar currents to arbitrary gravitational models with more complicated Lagrangians. As we verify, the usual Komar charges diverge for spacetimes which are not asymptoti- cally flat (in particular, for the asymptotically anti-de Sitter spacetimes). We have shown that the general scheme of ‘‘regularization via relocalization’’ can be used in this case to obtain finite total conserved charges. In addition, we considered asymptotically anti-de Sitter solutions of the quadratic Poincaré gauge theory in order to test how our general formalism works for models with nontrivial torsion degrees of freedom. We found divergent total charges, which can again be regularized with the help of a relocal- ization induced by a suitable boundary term added to the action. Rather curiously, we found the same total final finite charges for the Riemannian and for the quadratic Poincaré model. This result appears to be quite satisfactory and rather nontrivial, since in the quadratic Poincaré model the total regularized charges arise from different sectors of the dynamical field degrees of freedom (translational ver- sus rotational). Moreover, we have verified that Q�@r � Q�@� � 0 in both models. It seems worthwhile to mention some specific mathe- matical results obtained in this study. Namely, we have discovered that the local Lorentz invariance of the theory allows for a certain freedom in the description of the consequences of the diffeomorphism invariance. This free- dom is equivalent to a definition of a ‘‘generalized Lie derivative’’ that acts on the geometrical and matter fields. In other words, there is no unique or natural definition of the Lie derivative for the Lorentz-covariant fields (co- frame, torsion, and Lorentz-covariant matter fields). For example, it is always possible to define the Lie derivative of Lorentz-covariant fields as the usual Lie derivative ‘�. However, the result will not be a Lorentz-covariant field and the conserved currents derived from this choice will not be invariant under local Lorentz transformations. We have found a variety of consistent covariant Lie deriva- tives, all leading to the definition of invariant conserved currents. Technically, the latter is a consequence of the commuting property of the exterior and generalized Lie derivative. There is a number of interesting directions in which we can further develop the current formalism. In particular, the general approach can be naturally extended to include gravity interacting with gauge fields, thus allowing to obtain conserved quantities invariant also under gauge transformations in addition to the coordinate and local Lorentz invariance. Furthermore, it is possible to general- ize the current framework to metric-affine gravity, so that to include gravitational models with local invariance under the general linear group. These developments will be dis- cussed elsewhere. ACKNOWLEDGMENTS We would like to thank J. G. Pereira for useful discus- sions and for his hospitality at IFT-UNESP. This work was supported by FAPESP (for Y. N. O.) and by CNPq (for G. F. R.). We also thank F. W. Hehl for the careful reading of the paper and for his numerous helpful comments. INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-15 APPENDIX A: GENERALIZED LIE DERIVATIVES In this paper, generalized Lie derivatives are heavily used. Here we collect the corresponding mathematical definitions and describe properties of the generalized Lie derivatives. We call an operator L� in the algebra of tensor fields over spacetime a generalized Lie derivative if it is a deri- vation of the tensor algebra (as defined in [63], e.g.) which satisfies the following properties (for any Lorentz-valued p-form !A, q-form ’A, and constant �): (a) L��!A � �L�! A (linearity), (b) L��!A ^ ’B� � �L�!A� ^ ’B �!A ^ L�’B (Leibniz rule), (c) L�� � ‘�� � �cd�� d��c�� (usual Lie deriva- tive for any scalar-valued p-form �). In accordance with the general mathematical theory (see Proposition 3.3 of [63]), every derivation is constructed from the ordinary Lie derivative ‘� and a (1, 1) tensorial field. In this work, we explicitly define the generalized Lie derivative by the formula L�!A :� ‘�!A � B��� ��� A B ^! B (A1) when it acts on any (Lorentz-)covariant p-form!A. The (1, 1) field B�� depends linearly on � in order to provide the above property (a). If, under the local Lorentz transforma- tion #� ! # 0� � �� �# �, this field transforms as B0� � ���1� �B ��� � � �� �1�����cd�� ��; (A2) then the generalized Lie derivative (A1) of a covariant object is again a covariant object with the same trans- formation properties. In addition, we would like to know how L� acts on noncovariant objects, such as d!A, for example. In order to find this, it suffices to define the Lie derivative of the connection L��� �. Then we can compute L��d! A� by writing d!A � D!A � �� �� ��� A B ^! B, and by using the Leibniz rule and the definition (A1). We define the generalized Lie derivative of the connec- tion by L��� � :� ‘��� � � �dB�� � �� �B�� � �� �B� ��: (A3) If we recall the expression of the curvature 2-form in terms of the connection 1-form, we can recast the above defini- tion into an equivalent form: L��� � � D��c��� � B��� � �cR��: (A4) Without going into details, let us explain this important point. The ordinary Lie derivative commutes with the exterior differential, �‘�; d � 0. Using this fact together with the Leibniz rule, we can easily compute the Lie derivative of the curvature 2-form: ‘�R�� � ‘��d�� � � �� � ^ �� �� � d�‘��� �� � �‘��� �� ^ �� �� �� � ^ �‘��� �� � D�‘��� ��. Here we formally write the right-hand side as a covariant derivative, however in reality the resulting expression is not covariant. The generalized Lie derivative (A3) removes this deficiency. Indeed, the operation (A1) is well defined on the covariant curvature 2- form, and the result is the covariant 2-form L�R��. Now, by applying the covariant differential D to (A3), or, equiv- alently, to (A4), we straightforwardly verify that D�L��� �� � L�R��. In other words, in the replacement of the noncovariant ‘� with a covariant L�, the definition of the generalized Lie derivative of the connection (A3) pro- vides the correct replacement of the formal noncovariant relation ‘�R�� � D�‘��� �� with an appropriate covariant formula L�R�� � D�L��� ��. With these definitions, one can prove that the general- ized Lie derivative commutes with the exterior derivative, i.e. �L�; d � 0. For an arbitrary Lorentz-valued form !A, we have L��d!A� � L��D!A � �� �� ��� A B ^! B� � d‘�!A � �‘��� � � L��� ��� ��� A B ^! B � B��� ��� A Bd! B � B� ��� ��� ��� A C� � �� C B � � ��� A C� � �� C B ^! C: (A5) Using now (A3) and the commutation relation of the Lorentz generators, � ��; � A B � 1 2�g� �� � g�� � � g�� � � g� �� AB; (A6) we ultimately find L��d! A� � d‘�! A � dB� �� ��� A B ^! B � B��� ��� A Bd! B � d‘�! A � d�B� �� ��� A B! B� � d�L�! A�: (A7) Similarly, we can verify that L��d�� �� � d�L��� ��. The vanishing of the commutator �L�; d on all other geometric quantities follows then directly from these formulas and the Leibniz rule. In conclusion, let us give the following useful identities for the basic gravitational field variables: L�# � � D�� � �cT� � �B� � � �c�� �� ^ #�; (A8) L�T � � D�L�# �� � �L��� �� ^ #�; (A9) L�R� � � D�L��� ��; (A10) �L�;D ! A � �L��� ��� ��� A B ^! B: (A.11) YURI N. OBUKHOV AND GUILLERMO F. RUBILAR PHYSICAL REVIEW D 74, 064002 (2006) 064002-16 1. Yano’s derivative The covariant Lie derivative in the sense of Yano is defined by (2.16) and (2.17). Accordingly, this turns out to be a particular realization of the generalized Lie deriva- tive that corresponds to the choice B�� � ���� � �e��c‘�#� . This choice is special because in a certain sense it is ‘‘minimal.’’ Let us explain this property. The usual Lie derivative ‘� (in which case B� � � 0) of covariant geo- metrical and matter fields is not covariant under local Lorentz transformations. Specifically, let us consider the Lie derivative of the coframe. This is a 1-form and we can decompose it as follows: ‘�#� � �S� � � A� ��#�, where S� � is symmetric, S��� � 0, and A� � is antisymmetric, A���� � 0. Explicitly we find S�� � e��c‘�#�� hi�h j �‘�gij=2 and A�� � e��c‘�#� . We can verify that S� � is a tensor under local Lorentz transformations, but A� � is not. Therefore, the second piece is the source of the noncovariance of the usual Lie derivative ‘�# � of the coframe. There exists a unique B� � which introduces a covariant Lie derivative L�#� by just removing the second noncovariant term above, namely B�� � �A�� � �e��c‘�#� �: ����. This definition is minimal in the sense that it does not require the choice of any arbitrary constant. For instance, if t� � is a given tensor field, then the nonminimal choice B0� � ��� � � �t� � also leads to a covariant Lie derivative, but introduces an unknown scalar �. In [64–66] the choice B�� � ���� is referred to as ‘‘Kosmann lift’’ and it was derived with the help of fairly ad hoc assumptions. For this minimal choice we have, for vector-valued forms, connection, and for matter fields, respectively: L �!� � ‘�!� ��� �!�; (A12) L ��� � � D�� � � �cR� �; (A13) L ��A � ‘��A ��� �� ��� A B�B: (A14) Here, in accordance with the general relation (A4), we have �� � :� �c��� ��� �: (A15) For a given vector field �, we can define a corresponding 1- form by k :� ��c#��# � � ��# �. Then a straightforward computation shows that ��� e��cD�� � �ce��cT� (A16) 1 2e�ce�c�dk� �c�# � ^ T�� : (A17) It is worthwhile to calculate explicitly the Yano deriva- tive of the torsion, curvature, and of the coframe: L �T � � D�L�# �� � �L��� �� ^ #�; (A18) L �R� � � D�L��� ��; (A19) L �# � � D�� � �cT� ��� �#� (A20) � #�D fg �����: (A21) These formulas are quite useful for many practical compu- tations, but with a word of caution: The two last equations are actually somewhat misleading since one might have a wrong impression that a (either non-Riemannian or Riemannian) connection is involved. However, the Yano derivative L� is (like the ordinary Lie derivative ‘�) de- fined independently of any connection. This fact becomes more clear if we recall that the holonomic form of the second factor in (A21) reads D fg ����� � 1 2h i �h j ��� k@kgij � gik@j� k � gkj@i� k�: (A22) As a bonus, from this observation we learn that the Yano derivative of the coframe vanishes if and only if � is a Killing vector field. It is worthwhile to note that in the absence of torsion the Yano derivative of the usual spinor field reduces to the Lie derivative of Kosmann [67]. 2. Covariant Lie derivative 8� The covariant Lie derivative (2.14) corresponds to the choice B�� � �c���. Accordingly, the basic relations for this case read: 8�#� � D�� � �cT�; (A23) 8�! � � �cD!� �D��c!��; (A24) 8��A � �cD�A �D��c�A�; (A25) 8��� � � �cR� �: (A26) APPENDIX B: IRREDUCIBLE DECOMPOSITIONS In a Riemann-Cartan spacetime, the torsion and the curvature can be decomposed into three and six irreducible parts, respectively. Namely, the torsion 2-form is decomposed as T� � �1�T� � �2�T� � �3�T�, with �2�T� � 1 3# � ^ �e�cT ��; (B1) �3�T� � �1 3 ��#� ^ ��T� ^ #��� � 1 3e �c�T� ^ #��; (B2) �1�T� � T� � �2�T� � �3�T�: (B3) The curvature 2-form is decomposed as R�� �P6 I�1 �I�R��, with INVARIANT CONSERVED CURRENTS IN GRAVITY . . . PHYSICAL REVIEW D 74, 064002 (2006) 064002-17 �2�R�� � ���#�� ^�� �; (B4) �3�R�� � � 1 12 ��X#� ^ #��; (B5) �4�R�� � �#�� ^ � ; (B6) �5�R�� � �1 2# �� ^ e� c�#� ^ R��; (B7) �6�R�� � � 1 12R# � ^ #�; (B8) �1�R�� � R�� � X6 I�2 �I�R��: (B9) Here R� :� e�cR ��; R :� e�cR �; X� :� ��R�� ^ #��; X :� e�cX�; (B10) and �� :� X� � 1 4#�X� 1 2e�c�# � ^ X��; (B11) � :� R� � 1 4#�R� 1 2e�c�# � ^ R��: (B12) The components of the 2-form R�� � 1 2R��� �#� ^ #� are identified with the curvature tensor R��� �. Accordingly, the Ricci tensor is defined by the components of the 1-form R� � Ric��# �, where explicitly we have Ric�� � R��� �. This tensor is not symmetric, in general. The curvature scalar is, as usual, R � g��Ric��. It deter- mines the 6th irreducible part (B8) of the curvature. From (B12) we learn that the 4-th part of the curvature is given by the symmetric traceless Ricci tensor, � � �Ric���� � 1 4Rg���# �: (B13) The first irreducible part (B9) introduces the generalized Weyl tensor C���� which is defined by the components of the Weyl 2-form W� � :� �1�R� � � 1 2C��� �#� ^ #�: (B14) Accordingly, the 1st, 4th, and 6th curvature parts reproduce the well-known irreducible decomposition of the Riemannian curvature tensor into the Weyl, traceless Ricci, and curvature scalar parts. The 2nd, 3rd, and 5th curvature parts are purely non-Riemannian since they all arise from the nontrivial right-hand side of the first Bianchi identity R�� ^ #� � DT�, see (B10) and (B11). APPENDIX C: VECTOR FIELD IN KERR-ADS SPACETIME Let us take an arbitrary vector field � � �i@i, the com- ponents of which are four constant parameters �0, �1, �2, �3. 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