On Hypermomentum in General Relativity III. Coupling Hypermomentum to Geometry Friedrich W. Hehl, G. David Kerlick *, and Paul von der Heyde Institut für Theoretische Physik der Universität zu Köln (Z. Naturforsch. 31a, 823-827 [1976]; received May 12, 1976) In Part 11 of this series we presented the notion of the material hypermomentum current and motivated its introduction into general relativity. In Part II 2 we showed that a general, linearly connected manifold with symmetric metric (L4, g) is the appropriate geometrical framework for such an introduction. The present paper completes the picture by giving dynamical definitions for energy-momentum and hypermomentum for a minimally coupled material Lagrangian. We derive and discuss the field equations of a new metricaffine gravitational theory which embodies these notions. 1. Dynamical Definition of Hypermomentum obtained by varying with respect to the traceless nonmetricity tensor. In this way we have exploited We now propose the dynamical definition of the analogy between Eqs. (1.8) and (II.5). hypermomentum as the variation of the material One can show by means of the Rosenfeld identi- Lagrangian density with respect to the affine con- ties satisfied by the material Lagrangian that the nection in the space (L4,g). That is, we set** hypermomentum dynamically defined by Eq. (1) is indeed the same quantity that arises from the ca- eA f : - - , (1) nonical definition where the minimally coupled material Lagrangian 3£ J* = - (5) density £(ip,Vip) depends on the metric g^ and the 3(3 iv) connection J1*- as independent variables. We postu- where / j? are the representations of the generators late that the tensor A kji of general coordinate transformations. The anti- so defined is in fact the symmetric part and the trace with respect to j and hypermomentum tensor described in Part /. k of Eq. (5) are the well-known canonical defini- Using the decomposition (II.3) of the connec- tions of the spin angular momentum current and tion, the definition (1) is easily shown to be equiv- the (intrinsic) dilatation current in special rela- alent to the separate relations for spin tivity, respectively. In anholonomic coordinates, the J£ xkji.=J[kj]i = e-1 (2) quantities f ]l represent the generators of the general •öMijk linear group GL (4, R) of which the Lorentz group defined by varying £(1p, 3rp, g, 3g, M, Qi, Q) with and the dilatations are subgroups. respect to the (generalized) contortion tensor, for This last remark seems to preclude the definition the dilatation current of a canonical hypermomentum tensor for spinor fields, since GL (4, R) has no spinor representations. (3) For the time being, at least, we exclude spinor fields from our consideration if we deal with ge- obtained by varying with respect to the Weyl vector, ometries more general than a Y4 . and for the traceless proper hypermomentum The definition (1) and it corollaries (2, 3, 4) AW: -1 gk> Ai = - 2 e"1 <5£/<5Qijk (4) provide a consistent procedure for linking the no- tion hypermomentum, enunciated and given a phys- Reprint requests to Prof. Dr. F. W. Hehl, Institut für ical interpretation in Part I, with the geometry Theoretische Physik der Universität zu Köln, Zülpicher Str. 77, D-5000 Köln 41. (L4, g) proposed in Part II. Our remaining task is * Alexander von Humboldt Fellow. the construction of a consistent gravitational theory ** Symbols not defined here are defined in Parts I and II or which incorporates these notions and accords with in Hehl, von der Heyde, Kerlick, and Nester 3. We have our experience. We shall thus require that the new e: = (— det gij) V* and k: = S Jt G/c* where G is Newton's gravitational constant. theory which results will, like the U4 theory, reduce Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. 824 F. W. Hehl et al. • On Hypermomentum in General Relativity. III to Einstein's theory in the realm of macroscopic in an (L4, g) and primarily determining the non- physics. metricity in terms of a new source vki\ It is easy to Recall that the metric condition Qijk = 0 is pos- rewrite the second and third field equations in the tulated from the very beginning in the U4 theory. more suggestive, unified form We now propose to drop this postulate and to see to what extent it can be derived (at least in some (10) 2c <5n- " limit) from the unconstrained metricaffine theory. .here Jkji _ ykji _ ukji _ 2. Field Equations (11) We now complete this dualistic field theory by Up to this point we have left the gravitational field introducing an action function for the gravitational Lagrangian V undetermined; its choice is the sub- field expressed in geometrical variables. Since we ject of the next section. would like to compare the theory which results with the U4 theory and with general relativity, we 3. Gravitational Field Lagrangian choose, instead of the independent variables (gxj, Ffj), the equivalent set of independent quantities The obvious first guess for a gravitational field (g^, S'if, Qijk)- As Fig. 1 of Part II clearly shows, Lagrangian is the curvature scalar in an (L4, g) it is the vanishing of the nonmetricity Qijk which since the U4 theory and Einstein's theory result characterizes a U4 and the vanishing of torsion Sjj* from this choice in the cases of U4 and V4 geome- together with the vanishing of Qijk which charac- tries, respectively. The corresponding density R can terizes the Riemannian spacetime \\ of general be reduced to an effective first-order Lagrangian relativity. We take as the gravitational field Lagran- and varied with respect to metric and connection as gian density a scalar density V which depends on computed in 3. The field equations which result are g, S, Q, and their derivatives. The variational prin- the pair ciple for the interacting system of matter and gravi- cm-kW-.-^tor) (12) tational field is then dg-ii l SR V(g,S,Q)+L(xp,Vxp,g) d4x = 0. (6) (13) 2 Jc 2 e dr% The gravitational field equations are the Euler- where G'7 is the Einstein tensor of an (L4, g), Toli Lagrange equations of this variation with respect to differs from by a divergence, and P}Jl is a ge- the geometrical variables, ometrical tensor computed in Ref. 3 whose prop- 1 •6V 2 d£.(g,S,Q) erties are noted in the Appendix of this article. ol] (7) Particularly troublesome among the properties of k e ö9ü e dgij P'i?1 is that its trace P vanishes identically due to i by 1 b£.(g, S,Q) the projective invariance of the curvature scalar. (8) k = /'* 2 k e SS'if e dS'if The vanishing of P\f! implies via Eq. (13) the inconsistent relation 1 ÖV 2 d£(g,S,Q) vkji. _ _ (9) k e bQijk Ö Q,ijk J!'=0, (14) The first field eq. (7) results from variation with inconsistent because the dilatation current, a physi- respect to the metric tensor and is a generalization cal current which does not vanish in general, is here of Einstein's equation of general relativity. The required to vanish. The projective invariance of R source for the metric is the (L4, g) generalization also means that the Weyl vector Q; is left undeter- of Hilbert's metric energy-momentum tensor o''. mined. The second field eq. (8) resembles the second field The symmetric traceless part of Eq. (13) equation of the U4 theory which relates torsion to P$>=kÄÜ (15) the spin energy potential tensor here also gen- eralized to an (L4, g). The third equation is com- is mathematically consistent and can be resolved to pletely new, making its appearance for the first time give Q in terms of A: F. W. Hehl et al. • On Hypermomentum in General Relativity. III 825 4. The Dilatation Current in a Y4 and Elemen- Quk = k[-2 (Aijk - Ajki + Akij) + gjk A-{] , (16) tary Particle Physics but the antisymmetric part In Part I we mentioned that the dilatation current P[kh =k xkf (17) is conserved exactly in the high-energy "scaling limit" of elementary particle physics. This current contains terms which depend on Qi, so that it is Jk contains both an intrinsic part Ak whose canoni- not possible to determine the torsion uniquely in cal definition is the trace of Eq. (5) and an orbital terms of the spin. Similarly, the Einstein-like Eq. part constructed from the canonical energy momen- (12) for the metric contains (^-dependent terms. tum tensor 2lk. Thus, in special relativity, We note in passing that the projective invariance Jk = Ak + xl2lk. (18) of Jl and the inconsistent eq. (14) already cause trouble in a Y4 theory with scalar curvature La- Note that in the (L4,g) framework the symmetric grangian. The other field equations of that theory part of Zlk obeys the relation = rolk. The are obtainable by setting A = 0 in Eqs. (12) and special relativistic conservation law for Jk is Tk]k (17). = 6'k, where &k is the "soft trace" of the "im- proved" energy-momentum tensor that goes to zero The projective invariance of R has been known in the scaling limit (compare Callan, Coleman, and since Weyl's time (see for example Schouten4).