On Hypermomentum in 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 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 . 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) 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 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 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- . 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 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 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 , 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). Jackiw8). Hence the conservation law for Ak is Trautman5 (see also Kopczynski6) encountered this difficulty in his derivation of the U4 theory. He VkAk=6kk-ykk. (19) proved that one must assume, in addition to the A form similar to (19) can be obtained from the vanishing of what we call the vanishing of identities satisfied by the material Lagrangian in a the Weyl vector Qi. This can be seen from the Y4 . [Recall from Eq. (3) that a Y4 is the appro- more detailed examination of P'k) in the Appendix, priate framework for incorporating the dilatation see in particular Equation (26). Trautman did not, current via a dynamical definition.] In a Y4 with a however, provide any physical interpretation for minimally coupled material Lagrangian, we find the quantity A(k^1 or argue for its existence. from taking the trace of the canonical and metric Sandberg 7 has proposed the modification of the energy momentum that the conservation law matter Lagrangian in an (L4, g) so that it too is for Ak can be expressed as projectively invariant, but this procedure seems physically unjustified to us. We rather expect that i9ij VÄ (e gV Ak) =e(&£- 2?) , (20) the difficulty will be resolved by replacing the * gravitational field Lagrangian by an expression where X"k: = Vk + 2 Skj reduces to a divergence on quadratic in the curvature tensor or by adding qua- vector densities. Here we have set dratic terms to R. e & = e o{j + 2 VA. (e rkW)) . (21) We would propose as a general guideline for Since a gravitational theory in a Y4 should arise modifying the field Lagrangian that the theory from local gauge invariance with respect to the Weyl which results should reduce to the U4 theory upon group (the Poincare group plus dilatations, eleven the vanishing of and to Einstein's theory parameters in all), we look for this procedure to k L when T i is also set equal to zero. An ad hoc modi- suggest a satisfactory Y4 field Lagrangian. Local fication, in line with these principles but otherwise gauge theories for the Weyl group are derived and physically unmotivated, is to add a term (e/2a)QiQi discussed in Bregman 9 and Charap and Tait10, for to R. In such a model theory, Eqs. (12), (15), and instance. (17) retain their general form, but the "bad" eq. (14) is replaced by the "good" equation Ql = akAl. 5. Proper Hypermomentum in General Relativity, Conclusions * Another possible escape from this difficulty might be a "dynamical breaking" of the projective sym- We have found a consistent means for linking up metry of the field Lagrangian if this could be ar- the concept of hypermomentum with the most gen- ranged in a physically reasonable way. eral metricaffine geometry (L4,g). The dynamical 826 F. W. Hehl et al. • On Hypermomentum in General Relativity. III definitions which provide that link are both inter- been sought). However, a proper hypermomentum nally consistent and physically suggestive. need not to be a conserved current in order to be tak- Unfortunately, the simplest generalization of the en seriously. It is certainly possible to construct a field Lagrangian of general relativity to an (L4, g) canonical proper hypermomentum current for a runs into trouble because of its projective invari- massive vector field, and, if such a field turns out to ance. It is easy enough to construct a new Lagran- be elementary (i. e. not reducible into more elemen- gian which is not beset by these difficulties and tary spinor fields), this can be taken as an argument which has the correct limits, but a natural choice in support of the hypermomentum hypothesis. for this new Lagrangian has yet to be found. In Part II we sounded a cautionary note about

A gravitational theory based on an (L4, g) and the causal properties of an (L4, g) connection in embodying the canonical and dynamical definitions which Q is nonvanishing. We can now make the of hypermomentum should arise as the local gauge following comments: If acausal phenomena occur, theory of the general affine group GA (4, R) over they will probably be confined within matter, as spacetime with an additional local Minkowski struc- exemplified by the field eq. (16), or at least within ture. The general affine group is the semidirect such a short range that quantum mechanical "smear- product of the linear group GL (4, R) with the trans- ing out" of the light cone can still be expected. lations and includes the Weyl group as a subgroup. Therefore we expect the motion of test particles out- The translational gauge potentials in such a local side matter to be perfectly normal. gauge theory will be the (4 x 4) tetrad components Whether proper hypermomentum exists is an and the GL (4, R) potentials will be the (4x16) open question subject in principle to experimental connection coefficients in anholonomic coordinates. verification. Independent of this question, the for- The formalism used by the authors in Ref. 3 can malism developed in this series of nothes has already be extended to this more general group by enlarging been a valuable aid to ordering our knowledge of the set of group generators and providing them the relation between affine and metric structures in with physical interpretations. general relativity. We have been led to the new concept of hyper- momentum by analogies with Appendix and by arguments from geometry and classical field The geometrical Tensor P'^1 theory rather than by the known symmetries of elementary particles. The notion of the general af- Explicit computation of the variation of the fine group, like that of the Poincare group, origi- (L4, g) curvature scalar density with respect to the nates from the concept of local frames of vectors connection yields 3 and their transformations rather than from anything P'f-rtf-d&Qn+tfiQS (22) in microphysics. Thus we may well ask whether we can extend these concepts to the microphysical do- where T 'if : = S'if+ 2 (5^ Sjji is the modified torsion main. Certainly invariance with respect to the Poin- tensor, QI'. = QUJ4 the Weyl vector, and Q* the care group is well established in elementary particle traceless nonmetricity tensor. The symmetric and physics by experiment, and dilatation invariance in antisymmetric parts of (22) with respect to the the high-energy scaling limit seems to be well ful- indices i and j are given by filled. The other nine elements of GA(4, R) which P(ij) = ~ 2 Q(ij)+1 <3?£ Q-j)I, (23) generate shearlike distortions do not seem to lead to any observed exact symmetries or conserved cur- P[ij] = Tijk - tfiQj] - § Q{ij]+ 2 d[iQlj]i rents in particle physics (though perhaps it is fair = -Mifo-dfiMfa + WM-fa. (24a,b) to say that such symmetries and currents have not The trace Pjf1 vanishes identically, so the sym- metric part (23) can be expressed entirely in terms * In Part I the following misprints should be corrected: of the traceless nonmetricity tensor Q. From Eq. Page 112, 1st column, 3rd line from below, read uA°ßyn (instead of Aarß); page 113, line before Eq. (6), insert (24 a) the U4 limit = Tijk can be easily read "laws" after conservation. Erratum: Page 113, 2nd col- off; then, in Eq. (24 b), the (generalized) contor- umn, first two lines, read .. the divergence relation that tion M'ij degenerates to K'ik. It is now a straight- the total dilatation current is conserved only if the trace of Gij vanishes, that is in the ...". forward matter to compute the relations between F. W. Hehl et al. • On Hypermomentum in General Relativity. III 827

the symmetries of Pijk and the geometry of the The starred affinity *T in (27) is projectively

(L4, g) rhanifold as given in Table 1 (see also invariant and defined by Schrödinger 11 and Trautman 5). *T%,=T%-\b)St. (28)

(L4 , g) with Note that for the connection in (27) =5 (?[i<5fl. vanishing (25) contortion Therefore either (?j = 0 or S'ik ==0 leads to the Christoffel connection of a V4 . P(ij)k=0oQijk = Qi gjk y4 (26)

Y4 with We would like to acknowledge helpful discussions Pijk M)<>r£={f.}-HQi<5* vanishing contor- (27) tion or v4(*r) with Prof. P. Mittelstaedt. One of us (G.D.K.) ex- presses his thanks to the Humboldt Foundation for Table 1. Symmetries of Pijk and spacetime geometries. the award of a fellowship.

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