Geometrical structure of gravitation and matter fields

Abstract

Einstein's requirement of a unified geometrical description of gravitational fields and their matter sources is shown to become possible (at least for certain matter sources) by relaxing his other requirement of a minimal interaction of gravitation with matter. Arguments are presented to demonstrate that Schrodinger's discovery of pair creation by gravitational fields and the associated effects of virtual pairs make the relaxation of the latter requirement inevitable in order to obtain a complete macroscopic description (which needs no separate insertion to take account of averaged quantum effects). The gravitational field equations in case of a nonminimal interaction need higher derivatives of the metric than the second. The author's gauge theory on the manifold of the anti-de Sitter groupSO(3, 2) with the subgroupSO(3, 1) (proper Lorentz group) as gauge group and the factor space of the two group manifolds as space-time manifold gives rise to a Yang-Mills field which can be interpreted to be composed of Riemannian curvature and a tensor formed out of torsion. Einstein's equations with a cosmological member are satisfied by the Cartan-Killing metric on the group manifold so that the generalization to a Kaluza-Klein theory results in a minimal disturbance of the group symmetry. The separation of the Yang-Mills field results in a part of its energy-momentum tensor becoming purely Riemannian; this part may be interpreted to be due to the contribution of virtual matter, whereas the part with torsion is due to real matter and its interaction with curvature. The Yang-Mills field equations have a third-order derivative purely metric part, which is equivalent to the field equations suggested by Yang (in the latter, however, torsion should be inseparably present and has been ignored). The torsion part is the “matter source” of this term and it is tempting to relate it to elementary particle spin. The theory can be regarded as a gauge theory of space-time geometry. It needs generalizations to geometrize matter with an energy-momentum tensor of nonvanishing trace. The equations, however, already considerably modify the problem of gravitational collapse. Further developments should serve to eliminate the “absurdity of relativity”—the collapse to a point (of which Einstein himself never became convinced).