View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server ENERGY-MOMENTUM CONSERVATION LAWS IN AFFINE-METRIC GRAVITATION THEORY. Gennadi A Sardanashvily Department of Theoretical Physics, Physics Faculty, Moscow State University, 117234 Moscow, Russia E-mail: [email protected] Abstract The Lagrangian formulation of eld theory do es not provide any universal energy- momentum conservation law in order to analize that in gravitation theory. In La- grangian eld theory,we get di erent identities involving di erent stress energy- momentum tensors whichhowever are not conserved, otherwise in the covariant multimomentum Hamiltonian formalism. In the framework of this formalism, we have the fundamental identity whose restriction to a constraint space can b e treated the energy-momentum transformation law. This identity remains true also for grav- ity.Thus, the to ols are at hand to investigate the energy-momentum conservation laws in gravitation theory. The key p oint consists in the feature of a metric gravi- tational eld whose canonical momenta on the constraint space are equal to zero. 1 Intro duction In Hamiltonian mechanics, there is the conventional energy transformation law @ H dH = (1) dt @t on solutions of the Hamilton equations, otherwise in eld theory. The standard Hamiltonian formalism has b een applied to eld theory. In the straight- forward manner, it takes the form of the instantaneous Hamiltonian formalism when canonical variables are eld functions at a given instant of time. The corresp onding phase space is in nite-dimensional, so that the Hamilton equations in the bracket form are not the familiar di erential equations, adequate to the Euler-Lagrange eld equations. In Lagrangian eld theory,wehave no conventional energy-momentum transformation law. One gets di erent identities whichinvolve di erent stress energy-momentum tensors, in particular, di erent canonical energy-momentum tensors. Moreover, one can not say a priori what is really concerved. We follow the generally accepted geometric description of classical elds by sections of bred manifolds Y ! X: Their dynamics is phrased in terms of jet spaces [2, 7, 10 , 13 , 15]. GR-QC-9501009 k Given a bred manifold Y ! X , the k -order jet space J Y of Y comprises the equivalence k classes j s, x 2 X , of sections s of Y identi ed by the rst (k + 1) terms of their Taylor x 1 series at a p oint x. It is a nite-dimensional smo oth manifold. Recall that a k -order di erential op erator on sections of a bred manifold Y ,by de nition, is a morphism of k J Y toavector bundle over X . As a consequence, the dynamics of eld systems is played out on nite-dimensional con guration and phase spaces. In eld theory,we can restrict ourselves to the rst order Lagrangian formalism when 1 i 1 the con guration space is J Y . Given bred co ordinates (x ;y )ofY, the jet space J Y i i is endowed with the adapted co ordinates (x ;y ;y ): i i 0 0 @y @y @x i 0 j y =( : y + ) j 0 @y @x @x 1 A rst order Lagrangian density on the con guration space J Y is represented bya horizontal exterior density i i 1 n L = L(x ;y ;y )!; ! = dx ^ ::: ^ dx ; n = dim X: The corresp onding rst order Euler-Lagrange equations for sections s of the bred jet 1 manifold J Y ! X read i i @ s = s ; j j @ L(@ + s @ +@ s @ )@ L =0: (2) i j j i We consider the Lie derivatives of Lagrangian densities in order to obtain di erential conservation laws. Let i u = u @ + u @ i be a vector eld on a bred manifold Y and u its jet lift (15) onto the bred jet manifold 1 1 J Y ! X . Given a Lagrangian density L on J Y , let us computer the Lie derivative L L. On solutions s of the rst order Euler-Lagrange equations (2), wehave the equality u d i i s L L = [ (s)(u u s )+u L(s)]!; = @ L: (3) u i i i dx In particular, if u is a vertical vector eld such that L =0; L u the equality (3) takes the form of the current conservation law d i [u (s)] = 0: (4) i dx In gauge theory, this conservation law is exempli ed by the No ether identities. Let = @ 2 be a vector eld on X and i u = = (@ + @ ) i its horizontal lift onto the bred manifold Y by a connection on Y . In this case, the equality (3) takes the form d s L s)]! (5) L = [ T ( dx where i i T ( s)= (s ) L (6) i is the canonical energy-momentum tensor of a eld s with resp ect to the connection on Y . The tensor (6) is the particular case of the stress energy-momentum tensors [1, 3 , 6]. In particular, when the bration Y ! X is trivial, one can cho ose the trivial connection i = 0. In this case, the temsor (6) is precisely the standard canonical energy-momentum tensor, and if L L =0 for all vector elds on X (e.g., X is the Minkowski space), the conservation law (5) comes to the well-known conservation law d T (s)= 0 dx of the canonical energy-momentum tensor. In general, the Lie derivative L L fails to b e equal to zero as a rule, and the equality (5) is not the conservation law of a canonical energy-momentum tensor. For instance, in gauge theory of gauge p otentials and scalar matter elds in the presence of a background world metric g ,we get the covariant conservation law r t =0 (7) of the metric energy-momentum tensor. In Einstein's General Relativity, the covariant conservation law (7) issues directly from gravitational equations. But it is concerned only with zero-spin matter in the presence of the gravitational eld generated by this matter itself. The total energy-momentum conservation law for matter and gravityisintro duced by hand. It reads d N ) [(g ) (t + T ]= 0 (8) g dx where the energy-momentum pseudotensor T of a metric gravitational eld is de ned g to satisfy the relation 1 N N @ @ [(g ) (g g g g ) (g ) (t + T )= g 2 3 on solutions of the Einstein equations. The conservation law (8) is rather satisfactory only in cases of asymptotic- at gravitational elds and a background gravitational eld. The energy-momentum conservation law in the ane-metric gravitation theory and the gauge gravitation theory was not discussed widely [5]. Thus, the Lagrangian formulation of eld theory do es not provide us with any univer- sal pro cedure in order to analize the energy-momentum conservation lawingravitation theory, otherwise the covariantmultimomentum Hamiltonian formalism. In the framer- work of this formalism, we get the fundamental identity (32) whose restriction to the Lagrangian constraint space can b e treated the energy-momentum transformation lawin eld theory [10, 14]. Lagrangian densities of eld mo dels are almost always degenerate and the corresp ond- ing Euler-Lagrange equations are underdetermined. To describ e constraint eld systems, the multimomentum Hamiltonian formalism can b e utilized [9, 11 , 12 ]. In the framework of this formalism, the nite-dimensional phase space of elds is the Legendre bundle n = ^T X TX V Y (9) Y Y b over Y into which the Legendre morphism L asso ciated with a Lagrangian density L on 1 i J Y takes its values. This phase space is provided with the bred co ordinates (x ;y ;p ) i such that i i b (x ;y ;p ) L =(x ;y ; ): i i The Legendre bundle (9) carries the multisymplectic form i =dp ^ dy ^ ! @ : (10) i In case of X = R, this form recovers the standard symplectic form in analytical mechanics. Building on the multisymplectic form , one can develop the so-called multimomentum Hamiltonian formalism of eld theory where canonical momenta corresp ond to derivatives of elds with resp ect to all world co ordinates, not only the temp oral one. On the math- ematical level, this is the multisymplectic generalization of the standard Hamiltonian formalism in analytical mechanics to bred manifolds over an n-dimensional base X , not only R.Wesay that a connection on the bred Legendre manifold ! X is a Hamil- tonian connection if the form c is closed. Then, a Hamiltonian form H on is de ned to b e an exterior form such that dH = c (11) for some Hamiltonian connection .Every Hamiltonian form admits splitting i i i f H = p dy ^ ! p ! H ! = p dy ^ ! H!; ! = @ c!; (12) i i i where is a connection on Y ! X . Given the Hamiltonian form H (12), the equality (11) comes to the Hamilton equations i i @ y (x)= @ H; @ p (x)=@H (13) i i 4 for sections of the bred Legendre manifold ! X . The Hamilton equations (13) are the multimomentum generalization of the standard Hamilton equations in mechanics. The energy-momentum transformation law (32) which we suggest is accordingly the multimomentum generalization of the conventional energy transformation law (1). Its application to the Hamiltonian gauge theory in the presence of a background world metric recovers the familiar metric energy-momentum transformation law (7) [10 , 14]. The identity (32) remains true also in the gravitation theory. The to ols are nowat hand to examine the energy-momentum transformation for gravity. In this work, we restrict our consideration to the ane-metric gravitation theory. The key p oint consists in the feature of a metric gravitational eld whose canonical momenta on the constraint space are equal to zero [13].