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On Hypermomentum in General Relativity II. The of

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 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 (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 (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 , the characteristics of Minkowski spacetime. of new , 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 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 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 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 ).

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 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 The Christoffel part of the connection in tetrad (time) and standards which can be related by coordinates (the well-known Ricci 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 of the tetrad vectors and 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 det(e/4) of the 4- 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 . 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. When tor-

The vanishing of Qijk specializes the (Li,g) to a sion is excluded, the most general geometrical frame-

Weyl spacetime with torsion Y4 . _ work compatible with the conformal and the projec- The vanishing of Qijk (i. e. of Qijk and Qi) defines tive structure is a W4 . Quite obviously, if torsion is

a Riemann-Cartan spacetime U4 . permitted one has a Y4 . The vanishing of Qijk and S'if defines a Weyl Apart from these restrictions, EPS require a new spacetime W4 . hypothesis: There shall be no "twin paradox of the The vanishing of Qijk and S'if (and thus of second kind." That is, two observers bearing iden- W if) reduces spacetime to the Riemannian space- tical clocks and rods shall not find, meeting each time V4 . Should {*•} then also vanish globally, the other after travelling via different spacetime paths, spacetime would be the Minkowski R4 . that their size scales are different and that their clocks tick at different rates. When no such effect is present, the geometry is found to simplify from a

to a V4. The corresponding reduction, when torsion is not eliminated by assumption, is from a

Y4 to a U4 .

Start once more with the manifold (L4,^). The connection and metric together comprise 64 plus 10 variables. In macroscopic physics, where is valid locally, we observe no "twin paradox of the second kind", so we take the restric- tion Qijk = 0 (40 equations) to be a valid hypo-

thesis here. The U4 geometry of the world has 34 degrees of freedom. Suppose, however, that the "second twin para- dox" can occur in the microscopic realm. Then the four components Qi would no longer be fixed, but would be determinable from some matter current (from the dilatation current, we expect). Perhaps

then the Y4 geometry (38 independent components) is a better picture for high energies. That this ge- ometry occurs in the local gauge theory of the Weyl , has already been shown by Bregman 10 and others. Fig. 1. Classification of (L4,<7) geometries according to the connection of Equation (3). If, in addition, we allow a nonvanishing Q^k, all 74 components of the affine geometry would be 4. On Projective and Conformal Concepts; determined by matter in the microscopic realm. the Geometry of the Physical World When we picture the effects of a Qijk on geometry Important for making the affine connection physi- in terms of rods and clocks, a number of very un- cal is the axiomatic point of view propounded by pleasant consequences come to light. The light cone F. "W. Hehl et al. • On Hypermomentum in General Relativity. II 527

is no longer invariant under parallel transport; it metric structure (L4,^), which allow an ortho- can be stretched and tipped; spacelike vectors normal tetrad to undergo all possible linear trans- (rods) after parallel transport become timelike vec- formations during parallel transport. Such a geom- tors (clocks), and so on. Clearly this sort of be- etry should provide the basis for dynamical defini- havior cannot be allowed for physical rods and tions of the Hilbert type. We expect the macroscopic clocks, and for observations in macroscopic physics. world to be adequately described by U4 geometry However, thought experiments with rods and clocks ("no second twin paradox in the large") or by a make little sense in the domain of elementary par- Y4 geometry in the scaling limit. That these speciali- ticles. The onset of quantum phenomena in this zations of an (L4,^) are adequate to describe phys- range makes the concept of a well defined light cone ics "in the large", should be seen to result from of zero thickness harder to justify (consider, for certain macroscopic properties of matter. We do not example, the problems with "light cone singulari- restrict a priori the (L4,^) geometry in the micro- ties" in quantum field theory). Similarly, the va- scopic world. lidity of a local Lorentz structure at these distances is hard to prove._Therefore we hold open the ques- tion of whether Q terms could be allowed in nature. We are grateful to Professor Peter Mittelstaedt The formalism, including the field equations, which and members of his relativity group, and to Profes- emerge will be mathematically consistent in any sor Ekkehart Kröner and Dr. Karl-Heinz Anthony case, and will embody all the simplest transforma- for interesting discussions. Furthermore, we thank Professor Jürgen Ehlers for a discussion on the EPS tions (linear ones) of vectors in spacetime. axiomatics. One of us (G.D.K.) would like to thank We sum up in a second working hypothesis: the Humboldt Foundation for the award of a fellow- spacetime possesses a general affine structure and a ship.

1 J. A. Schouten, Ricci Calculus, 2nd ed., Springer-Verlag, 6 F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Berlin 1954. Nester, Rev. Mod. Phys., July 1976 (in press). 2 C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravita- 7 P. von der Heyde, Lettr. Nuovo Cim. 14, 250 [1975]. tion, Freeman, San Francisco 1973. 8 E. Kröner, in A. Sommerfeld "Vorlesungen über theore- 3 E. Schrödinger, Space-time Structure, reprinted with corr., tische Physik", Vol. 2, Chap. 9; 5th ed., Akad. Verlagsges., Cambridge Univ. Press 1960. Leipzig 1964. 4 A. Trautman, Ann. N. Y. Acad. Sei. 262, 241 [1975]. 9 J. Ehlers, F. A. E. Pirani, and A. Schild, Festschrift for 5 F. W. Hehl, P. von der Heyde, and G. D. Kerlick, Phys. Synge (L. O'Raifeartaigh ed.), Oxford Univ. Press 1972. Rev. D 10,1066 [1974]. 10 A. Bregman, Progr. Theor. Phys. 49, 667 [1973].