Einstein’s Pseudo-Tensor in <i>n</i> Spatial Dimensions for Static Systems with Spherical Symmetry

Abstract

It was noted earlier that the general relativity field equations for static systems with spherical symmetry can be put into a linear form when the source energy density equals radial stress. These linear equations le a d to a delta function energymomentum tensor for a point mass source for the Schwarzschild field that ha s vanishing self-stress, and whose integral therefore transforms properly under a Lorentz transformation, as though the particle is in the flat space-time of special relativity (SR). These findings were later extended to n spatial dimensions. Consistent with this SR-like result for the source tensor, Nordstrom and independently, Schrodinger, found for three spatial dimensions that the Einstein gravitational energy-momentum pseudo-tensor vanishe d in proper quasi-rectangular coordinates. The present work shows that this vanishing holds for the pseudo-tensor when extended to n spatial dimensions. Two additional consequences of this work are: 1 ) the dependency of the Einstein gravitational coupling constant κ on spatial dimensionality employed earlier is further justified; 2 ) the Tolman expression for the mass of a static, isolated system is generalized to take into account the dimensionality of space for n ≥ 3 .