Einstein’s Pseudo-Tensor in *n* 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 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.

Cite this paper

F. Tangherlini, "Einstein’s Pseudo-Tensor in*n* Spatial Dimensions for Static Systems with Spherical Symmetry," *Journal of Modern Physics*, Vol. 4 No. 9, 2013, pp. 1200-1204. doi: 10.4236/jmp.2013.49163.

F. Tangherlini, "Einstein’s Pseudo-Tensor in

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[1] F. R. Tangherlini, Nuovo Cimento, Vol. 27, 1963, pp. 636-651.

[2] F. R. Tangherlini, Physical Review Letters, Vol. 6, 1961, pp. 147 -149.

[3] W. H. Panofsky and M. Phillips, “Classical Electricity and Magnetism,” 2nd Edition, Addison-Wesley, Reading, 1962.

[4] F. Rohrlich, “Classical Charged Particles,” 3rd Edition, World Scientific, Singapore, 2007.

[5] G. Nordstrom, Proceedings of Amsterdam Academy of Sciences, Vol. 20, 1918, pp. 1238-1245.

[6] E. Schrodinger, Physikalische Zeitschrift, Vol. 19, 1918, pp. 4-7.

[7] F. R. Tangherlini, Nuovo Cimento, Vol. 26, 1962, pp. 497-524.

[8] A. Papapetrou, Proceedings of the Royal Irish Academy, Vol. A51, 1947, pp. 191-204.

[9] F. R. Tangherlini, Nuovo Cimento, Vol. 38, 1965, pp. 153-171.

[10] R. C. Tolman, “Relativity, Thermodynamics, and Cosmology,” Clarendon Press, Oxford, 1934.

[11] P. Ehrenfest, Proceedings of the Amsterdam Academy of Sciences, Vol. 20, 1917, pp. 200-209.

[12] P. Ehrenfest, Annals of Physics (Leipzig), Vol. 61, 1920, pp. 440-446.

[13] G. J. Whitrow, British Journal for the Philosophy of Science, Vol. 6. 1955, pp. 13-31.

[14] G. J. Whitrow, “The Structure and Evolution of the Universe,” Harper Torchbooks, New York, 1959, pp. 199-201.