A theory of classical spacetime (I)—foundations

Abstract

Despite its beauty and grandeur the theory of GR still appears to be incomplete in the following ways:

• (1) It cannot accommodate the asymmetric total energy momentum tensor whose asymmetry has been shown to exist in the presence of electromagnetism.

• (2) The law of angular momentum balance as an exact equation is not an automatic consequence of the field equations as is the case with the law of linear momentum balance.

• (3) The four degrees of arbitrariness left by the contracted second Bianchi identity makes a unique solution of the field equations unattainable without extra (unphysical) postulates.

To answer the challenge posed by the above assertions we propose in this paper to complete Einstein's theory by postulating the principle fibre bundle P[M,SU(2)] for the underlying geometry of the 4-dimensional spacetime, where the structure group SU (2) is the real representation of the special complex unitary group of dimension 2; SU (2) leaves concurrently invariant the metric form dS²=g_{α β}dx^αdx^β and the fundamental 2-form ϕ=(1/2₁)a_αβdx^αΛdx^β defined globally on M. The Einstein equation defined in terms of the SU(2)-connection is imposed on the spacetime manifold together with the Maxweil inhomogeneous equation as the supplementary condition where the electromagnetic tensor is identified with a contracted form of the curvature tensor. The result is a set of 16 functionally independent equations to the 16 unknown field variables (g_αβ a_αβ. Moreover, the law of angular momentum balance is just/the skew-symmetric part of the generalized Einstein equation where the spin angular momentum tensor is shown directly proportional to the torsion tensor.