On Hypermomentum in General Relativity II. The Geometry of Spacetime Friedrich W. Hehl, G. David Kerlick *, and Paul von der Heyde Institut für Theoretische Physik der Universität zu Köln (Z. Naturforsch. 31a, 524-527 [1976] ; received March 29, 1976) In Part I** of this series we have introduced the new notion of hypermomentum A'ik as a dynamical quantity characterizing classical matter fields. In Part II, as a preparation for a general relativistic field theory, we look for a geometry of spacetime which will allow for the accomodation of hypermomentum into general relativity. A general linearly connected spacetime with a metric (L4, g) is shown to be the appropriate geometrical framework. 1. Einstein's General Relativity and U4 Theory longest lines with respect to the metric) coincide with autoparallel curves (straightest lines with re- The success of relativistic quantum field theory spect to the connection). The Christoffel connection in describing fundamental processes continues to in GR can be interpreted physically as a "guiding affirm the validity of Minkowski's flat spacetime R4 field" because the motion of structureless non- (notation of Schouten1) as a model of reality, at rotating test particles (which can be derived from least where gravitational fields are not so strong as the field equations) turns out to be autoparallel with to dominate. respect to the connection. Einstein adopted a spacetime for general relativity After the appearance of GR, mathematicians (GR) which retains locally, in the neighborhood of stimulated by Einstein's work developed a plethora a point, the characteristics of Minkowski spacetime. of new geometries, among which the Riemannian V4 This Riemannian spacetime V4 derives all of its is a special case (compare Schouten 1 and Schrödin- properties from the metric tensor g^ (i, /'... = 0, 1, ger 3). ln the last few years, the most general affine 2,3; see Part I for notation). Interpreted physi- space compatible with a Riemannian metric (the cally, the metric tensor represents infinitesimal Riemann-Cartan space U4) was successfully applied spacetime intervals as measured by standard clocks to an extension of Einstein's GR. In the gravitational and rods (for a detailed description see Misner, theory of that spacetime, spin angular momentum is Thorne, and Wheeler 2, Ch. 13). coupled to torsion just as energy-momentum is Levi-Civita offered an additional interpretation of coupled to the metric. We take this extended gravi- the Riemannian space in terms of parallel transport tational theory, the so-called U4 theory, for granted of vectors. In addition to the metric structure, a V4 and refer to the literature 4-6 for the available evi- has an affine structure as well. The Christoffel sym- dence. bol Motivated by the considerations on hypermomen- {*} —9klA$[hdagbc] (1) tum in Part I, we will investigate a still more general is the affine connection in a V4 . Here, the permuta- geometry, namely the general affine geometry L4 tion tensor Af-f has been defined by with metric g. In this framework, which we denote by (L4, <7), the metric and the affine structures are decoupled, as are their physical interpretations. Because of the relationship (1), the affine proper- ties of a V4 can be expressed in terms of the metric 2. The General Affine Space with Metric (L4, g) tensor. There is no independent affine structure in a V4; everything is subordinate to the metric ("no Allow the 64 components of the affine connection affine freedom"). The extremal curves (shortest or rfj to be completely arbitrary, instead of being given by Equation (1). Then prescribe the (sym- * Alexander von Humboldt fellow. metric) metric tensor field g^ of signature + 2 (ten ** Z. Naturforsch. 31a, 111 [1976], independent components) and let these two fields Reprint requests to Prof. Dr. F. W. Hehl, Institut für (rfj, gij) be completely unrelated. What structures Theoretische Phvsik der Universität zu Köln, Zülpicher Straße 77, D-5000 Köln 41. can we extract from these assumptions? F. "W. Hehl et al. • On Hypermomentum in General Relativity. II 525 Without using the metric, we can split the con- dependent fields. Therefore (4 a), (4 b), and (4 c) nection rfj into its symmetric and its antisymmetric are equivalent for the description of the (h4,g). part. The symmetric part T\tj) is still an affinity, The set Equation (4 c) will turn out to be particu- transforming inhomogeniously under general co- larly convenient: the generalized contortion M^k ordinate transformations. The antisymmetric part couples to the spin of matter, whereas the non- Sif: = transforms as a tensor and has 24 in- metricity Qijk measures the deviations from U4 dependent components (Cartan's torsion). The tor- theory. Let us now analyze the affine connection of sion measures the non-commutativity of covariant an (L4,g) by means of a moving orthonormal derivatives on scalar fields (and the closure failure tetrad. of infinitesimal parallelograms). The geometrical importance of the metric gtj sug- 3. The Moving Tetrad r gests that its covariant derivative Q^ : = — V,- gjk The presence of a hyperbolic metric on an X4 (40 independent components) should be an impor- allows us to define locally at each point an (an- tant quantity in an (L4,^). This tensor, which we holonomic) orthonormal tetrad (frame) of vectors shall call nonmetricity, measures the failure of the eu which satisfy e„ • e, = g/iv = diag ( — 1, +1, +1, connection to preserve the metric during parallel + 1) at every point (/.i, v ... = 0, 1, 2, 3). When transport. The vanishing of Quk in an (L4,(/) is the the X4 is an L4 , the parallel transport of tetrads metric condition which defines a U4 geometry. In provides a suggestive visualization of the properties the corresponding U4 theory4-6, we regard the of the connection in Equation (3). We follow vanishing of Qijk as a reflection of physical reality: here essentially the presentation of Kröner 8. on a macroscopic scale, there exist global length The Christoffel part of the connection in tetrad (time) and angle standards which can be related by coordinates (the well-known Ricci rotation coeffi- parallel displacement. In the neighborhood of each cients) , causes only a rotation of the frames during point, spacetime is Euclidean with respect to metric parallel transport with respect to the given tetrad 7 and (anholonomic) connection . system. The tensor Mtjk = — Mjkj causes an addi- It will be useful to split the nonmetricity into a tional, independent rotation of the frame. Thus the trace Qj and a traceless part Qijk: tensor M^k suggests itself as the appropriate vari- Qijk = Qi9jk + Qijk • (2) able for defining dynamical spin in an (L4,#). The vector part of the nonmetricity (better known Here, Qt : = Qi}/4 and Q;/ = 0 . as the Weyl vector) causes a uniform change of With these definitions of torsion and nonmetricity scale in a parallelly transported tetrad. Relative at hand, we can express the connection in terms of lengths of the tetrad vectors and angles between g, Q, and 5 by means of the following identity: them remain unaltered. Such a geometrical variable rl= gkl Atlll 9 be - 9cd S-ai + I Qa 9bc + iQabc] suggests itself as a counterpart to the dilatation = iii) + {£•} - Mif +1 Qif . (3) current. The definitions of W'if and of the generalized con- The traceless part of the nonmetricity Qlf causes tortion tensor M^ = — in terms of S'if and a shear of the tetrad vectors which does not preserve Qijk can be read off from (3) and (1). In the case angles or relative lengths, but which does preserve the volume det(e/4) of the 4-parallelepiped whose where Qijk = 0, the generalized contortion reduces sides are given by the tetrad vectors. The dynamical to the contortion of U4 theory. counterpart that we expect to arise here is the By means of (3), we have separated the affinity traceless proper hypermomentum defined in Part I. and the tensor part of the connection. The affinity We express our results symbolically: part {ij} is expressed in terms of the metric tensor. So it is a matter of convenience which of the follow- tetrad connection of the (L4,#) (16x4) ing alternatives we use for specifying the (L4,^) : ~ contortion * (6x4) (rfj, 9a) » (9ij . ^if, Qijk) (4 a, b, c) © (relative) dilatation (1x4) (5) or (gij, Mijk, Qijk) etc. © (relative) shear (9x4) In contrast to (4 a), the sets (4 b, c) comprise only * Instead of "relative rotation", we use the more convenient tensor fields. In each case we can prescribe 74 in- term "contortion". 526 F. W. Hehl et al. • On Hypermomentum in General Relativity. II The term relative should remind us that these de- Ehlers, Pirani, and Schild (EPS)9. In this approach, formations compare two tetrads at different points the primitive concepts from which the geometry of of an (LIn Part III the tetrad deformations of spacetime is to be deduced are the paths of light Eq. (5) will be related to the corresponding dy- rays and of structureless, massive, non-rotating test namical quantities in Eq. (8) of Part I. particles. The paths of light rays determine a con- In the following section we will investigate the formal structure (light cone at each point) on a conditions imposed on a geometrical framework by differentiate manifold. The paths of massive par- the requirement that it leads to a consistent physical ticles determine a set of autoparallel curves (a pro- interpretation. To facilitate this discussion, we clas- jective structure). EPS discovered that these struc- sify the specializations of the spacetime manifold tures taken together already determine the differenti- (L4, g) as follows (see also Figure 1) : ate and affine structure of spacetime.