Gravity as Nonmetricity General Relativity in Metric-Affine Space (Ln,g)

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 LeviCivita 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 stressenergy 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, with transition between FRs 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 the gravitational field in a NIFR. We propose a generalization of GR that 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. PACS 04.20.-q, 02.40.-k, 04.20.Cv, 04.50.+h MSC: 53B05, 53B50, 53C20, 53C22, 53C80, 70G10 Introduction In the Riemannian space V4 of General Relativity (GR) two principal geometric objects, metric g and connection Γ, are linked through the requirement of metric homogeneity, i.e. the covariant derivative of metric vanishes identically: ∇g = 0. This condition assures that the length of a vector transported parallel in any direction remains invariant. Since GR was first formulated, metric g and the Levi-Civita connection Γ have been considered respectively as a potential and strength of the gravitational field. It is easy to see that the well-known difficulties, such as non-covariance of the energy-momentum pseudo-tensor of the gravitational field, that have plagued GR are directly related to the choice of noncovariant connection Γ (which is not a tensor) as the strength of a gravitational field. We will endeavor to demonstrate in the following that this need not be the case. In part I we show that in the presence of an arbitrary affine connection, the Einstein field equations lend themselves to a novel geometrical interpretation wherein the affine deformation tensor of the LeviCivita connection plays the role of a gravitational field. Furthermore, in the case of an affine connection with vanishing torsion, the gravitational field becomes the nonmetricity of spacetime. In this section we are not concerned with the nature of this auxiliary affine connection and can consider it as merely a convenient device. The fact that these results hold true for any auxiliary affine connection suggests that this geometric interpretation is merely a recasting of GR in a new light, which does not change the field equations or any