By Alexander Poltorak
General Patent Corporation
75 Montebello Rd., Suffern, NY 10901-3740 USA
E-mail: apoltorak@generalpatent.com

Abstract

In this paper we propose a new geometric interpretation for General Relativity (GR). It has always been presumed that the gravitational field is described in GR by a Levi-Civita connection. We suggest that this may not necessarily be the case. We show that in the presence of an arbitrary affine connection, the gravitational field is described as nonmetricity of the affine connection. An affine connection can be interpreted as induced by a frame of reference (FR), in which the gravitational field is considered. This leads to some interesting observations, among which: (a) gravity is a nonmetricity of space-time; (b) the affine curvature of space-time induced in a noninertial FR contributes to the stress-energy tensor of matter as an additional source of gravity; and (c) the scalar curvature of the affine connection plays the role of a “cosmological constant”. It is interesting to note that although the gravitational field equations are identical to Einstein’s equations of GR, this formulation leads to a covariant tensor (instead of the pseudotensor) of energy-momentum of the gravitational field and covariant conservation laws. We further develop a geometric representation of FR as a metric-affine space (Ln,g), with transition between FR represented as affine deformation of the connection. We show that the affine connection of a NIFR has curvature and may have torsion. We calculate the curvature for the uniformly accelerated FR. Finally, we show that GR is inadequate to describe gravitational field in a NIFR. We propose a generalization of GR, which describes gravity as nonmetricity of the affine connection induced in a FR. The field equations coincide with Einstein’s except that all partial derivatives of the metric are replaced by covariant derivatives with respect to the affine connection. This generalization contains GR as a special case of the inertial FR.

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