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On Hypermomentum in General Relativity I. the Notion of Hypermomentum

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On Hypermomentum in General Relativity I. the Notion of Hypermomentum Band 31 a Zeitschrift für Naturforschung Heft 2 On Hypermomentum in General Relativity I. The Notion of Hypermomentum Friedrich W. Hehl, G. David Kerlick *, and Paul von der Heyde Institut für Theoretische Physik der Universität zu Köln (Z. Naturforsch. 31a, 111-114 [1976]; received December 27, 1975) In this series of notes, we introduce a new quantity into the theory of classical matter fields. Besides the usual energy-momentum tensor, we postulate the existence of a further dynamical attribute of matter, the 3rd rank tensor Aift of hypermomentum. Subsequently, a general rela- tivistic field theory of energy-momentum and hypermomentum is outlined. In Part I we motivate the need for hypermomentum. We split it into spin angular momentum, the dilatation hypermomentum, and traceless proper hypermomentum and discuss their physical meanings and conservation laws. 1. Spacetime as a 4-Dimensional Elastic e o'i: = 2 d£/dgij is a mathematical expression of Continuum this elastic property of spacetime. We begin, then, by accepting the notion of space- The vacuum is a complicated physical entity: it time as a 4-dimensional differential manifold X4 can be polarized (the Lamb shift), some of its and by accepting the existence of dynamical defini- symmetries can be broken or break spontaneously, tions (like Hilbert's) as an expression of the elastic its metric field gtj (i, /',... = 0, 1, 2, 3) represents reaction of spacetime to an imposed matter distribu- the gravitational potential, and through it propagate tion. all other physical fields. In many ways it seems to be at least as complex a construct as the mechanical aether of the last century. Spacetime is well repre- 2. Hyperstress in 3-Dimensional Polar Continua sented by a mathematical continuum (the differential Consider first a 3-dimensional elastic continuum. manifold ** X4) at least down to the scale of 10~14 Cut a very small cavity out of an ordinary elastic cm or so. This continuum, in the light of Einstein's continuum and impose a prescribed stress distribu- theory of gravitation, the general theory of relativity tion on the surface of the cavity (one could realize (GR), is neither static nor rigid, but reacts elas- this physically by inserting a larger piece of foreign tically to its matter distribution. (Just as one does material into the cavity). It has long been known in GR, we shall always regard the matter distribu- (see, for example, Love2) that, to a first approxi- tion as foreign to the geometry of spacetime itself.) mation, the action of the stresses can be represented, Let us conduct a thought experiment, and dis- when the net force vanishes, by double forces with tribute matter in a flat Minkowski spacetime R4 . and without moments. By means of these tools it is According to GR, the flat R4 deforms into the Rie- straightforward to build up a so-called "center of mannian spacetime V4. The spacetime thus deformed rotation" out of the double forces with moments, becomes flat once more when we remove the matter, and a "center of dilatation" out of the double forces hence the conception of spacetime as a gravita- without moments. A point defect in a crystal repre- tionally elastic medium. Let the matter distribution sents a realization of a center of dilatation, for ex- be characterized by its Lagrangian density £. Then ample. These notions can be sharpened to the defi- Hilbert's definition of the energy-momentum tensor nition of an elastic dipole. Recall the definition of an electric dipole as a Reprint requests to Prof. Dr. F. W. Hehl, Institut für Theoretische Physik der Universität zu Köln, Zülpicher limiting case of the situation where two equal and Straße 77, D-5000 Köln 41. opposite charges + q and — q are separated by a distance Ax3. The distance Ax" tends to zero, but the * Alexander von Humboldt fellow. a ** Our mathematical notation is taken from Schouten charges are increased so that the product j q | • | Ax J e: = (det gij) Symmetrization and antisymmetrization remains constant throughout the process. In an exact of indices are denoted by () and [], respectively, i, j . analogy, let us now imagine two equal and opposite = 0, 1, 2, 3; a, ß .. = 1, 2, 3. <5=variational derivative, Vk = covariant derivative. forces + fß and — whose points of application are 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. 112 F. W. Hehl, G. D. Kerlick, and P. v. d. Heyde • On Hypermomentum in General Relativity. I 124 separated by the vector Axa. Then we define the As we might expect, a usual classical point con- elastic dipole moment according to tinuum cannot bear hyperstress. Only if we allow for extra degrees of freedom (independent rotations, Z)a/?: = lim Ax? (Ax/ = const) . (1) for example) which characterize so-called polar con- Here and hereafter, we regard these elastic dipoles tinua3-6, can we appreciate the nature of these as intrinsic objects, not reducible into pairs of stresses. forces. Then we call D^ a "hyperforce" (see Jaun- Briefly summarized, a polar continuum is a clas- zemis3, Kröner4, Truesdell-Toupin5, Truesdell- sical point continuum to each point of which is Noll6). attached a triad of vectors. Such mathematical con- tinua have been used, for instance, in dislocation theory, in the description of molecular crystals with rotational degrees of freedom and in the description of liquid crystals and anisotropic fluids. In geome- try, these structures are principal fibre bundles of the general linear group GL(3,R) or its unimodular subgroup 0(3,R) over a base space X3 . (Inciden- tally, a 4-dimensional continuum of this type with Fig. 1. An elastic dipole. group 0(3,1) was used by Einstein and others for the spacetime of his teleparallelism theory.) Now take a step familiar from continuum me- Roughly, without going into too much detail, we chanics: Smooth out these hyperforces into a con- can say that the ordinary stress tensor deforms tinuous distribution represented by the "hyperstress" the underlying point continuum, and that the tensor This corresponds, in the above descrip- hyperstress tensor Aa^y deforms the triads attached tion of a crystal, to a continuous distribution of to the points. The action of stress and hyper- point defects (compare Kroner's "extra matter"7'8; stress are not unrelated. The generalized conditions see also Bilby et al. 9, Anthony 10). Consider a sur- of equilibrium (conservation laws), patterned after face element dAy. The hyperforce acting through polar continuum mechanics, are of the form [see, that element is then e. g., Mindlin11, Eq. (4.1)] 7 dZ)^ = Aafo dAy, (2) VY2A = 0; VY + O'P = , (4), (5) a just as the force acting through dAy is given by where O P is a symmetric stress tensor. The anti- 3 d/3 = dAy, where is the ordinary (force) symmetric part of Eq. (5), Vy t ^ = is the stress tensor. well-known equilibrium condition for moments in We shall now try to picture the action of hyper- continuum mechanics. We can formulate the equi- stress in 3 dimensions. First, we decompose A librium conditions (4, 5) with precision only after into its antisymmetric part, its trace, and its sym- we have specified the continuum in detail. We shall metric traceless part with respect to its first two in- do so for a spacetime continuum in Part II. The dices a and ß. (This corresponds to the similar de- price of introducing hypermomentum is a more composition of the hyperforce dDaß.) We obtain complicated continuum. More intricate statics re- (n = number of dimensions, here 3) quires a correspondingly more intricate geometrical Jaßy = xaßy + (gaß/n)Jr + A^ . (3) description. The antisymmetric part r2^: could be called rotatory hyperstress, its conventional name is spin 3. Momentum and Hypermomentum of Matter moment stress (or torque stress). The symmetric in a 4-Dimensional Spacetime Continuum part AWr is the proper hyperstress (without mo- ment) , it can be resolved into the dilatational hyper- The concepts of the previous section will now be stress Aj: = Aa'ay_ and the traceless proper hyper- generalized to spacetime. The analogy between the stress Aa?P with A^ = 0; = 0. This may be 3-stress tensor of continuum mechanics and the compared with the highly suggestive Fig. 2 in Mind- 4-momentum current (stress-energy or energy-mo- lin mentum) of spacetime has been known since the 114 F. W. Hehl, G. D. Kerlick, and P. v. d. Heyde • On Hypermomentum in General Relativity. I 113 early days of special relativity. The relativistic gen- the divergence relation that Ak is conserved only if eralization of the 3-spin moment stress to the the trace of (-2— Q)a vanishes, that is in the 4-dimensional spin angular momentum current "scaling limit" in elementary particle physics. is also well established 12 and is fundamental to the The conservation laws for the dilatation current introduction of spin and torsion into GR (Refs. 13 and for the angular momentum current confirm part and 14, and the literature cited therein). The equi- of our working hypothesis. What remains is to librium conditions of continuum mechanics for demonstrate the existence of the remaining invariant forces and moments find their appropriate generali- part of hypermomentum, namely the traceless prop- zation in the conservation laws of momentum and er hypermomentum angular momentum.
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