TITLE:
Einstein’s Pseudo-Tensor in n Spatial Dimensions for Static Systems with Spherical Symmetry
AUTHORS:
Frank R. Tangherlini
KEYWORDS:
Field Equations; Point Particle; Dimensionality of Space; Einstein’s Pseudo-Tensor
JOURNAL NAME:
Journal of Modern Physics,
Vol.4 No.9,
September
30,
2013
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 lead to a
delta function energymomentum tensor for a point mass source for the
Schwarzschild field that has 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 vanished 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.