Einstein's equations with an embedding-dependent energy-momentum tensor for the compactified Minkowski time space and their relationship to the conformal action of SO(4,4) on S3 x S3

Abstract

For a pseudoriemannian 4-manifold isometrically embedded in a pseudoriemannian 6-manifold an embedding-dependent energy-momentum tensor is proposed, as well as an embedding-dependent cosmological constant. Their behavior is examined under the conformal transformations. It is specialized to the case of S1 x S3 embedded in S3 x S3, where the latter has the metric tensor in the conformal class of the one inherited from sum from 4i=1 (dxi)2 - sum from 8i=5(dxi)2 on R8. Einstein's equations for the restricted metric are shown to be satisfied if and only if the scalar curvature of S3 x S3 vanishes on S1 x S3.