A Gravitational Field with a Curious Geometrical Property

Abstract

IT is well known1 that there are fourteen independent absolute scalar differential invariants of the second order associated with the gravitational metric, It can be argued that the vanishing of all the invariants need not imply the vanishing of all the twenty independent components of the curvature tensor Rhijk. On the other hand, it may be pointed out that, when the invariants vanish, there are fourteen differential equations to be satisfied by the ten gµv components ; and it is not at all obvious that a gravitational field with a Riemannian metric exists for which the fourteen invariants vanish. We report here the existence of such a gravitational field described by the metric for which the surviving components of the energy-momentum tensor satisfy the relations, each being equal to The above metric with the conditions (3) may be compared to the gravitational field corresponding to a directed flow of radiation as given by Tolman2. For (2), the conformal curvature tensor vanishes and Tµv has the structure of the electromagnetic energy-momentum tensor ; these circumstances together being responsible for the vanishing of the complete set of invariants.

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References

1. 1

Thomas, T. Y., "Differential Invariants of Generalized Spaces", 183 (1934).

2. 2

Tolman, R. C., "Relativity, Thermodynamics and Cosmology", 273 (1934).

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