The Determination of the Metric by the Weyl and Energy-Momentum Tensors

TL;DR

This paper explores how specifying the Weyl and energy-momentum tensors constrains the space-time metric, showing that, generally, the metric is uniquely determined up to a conformal factor.

Contribution

It demonstrates that the metric tensor is uniquely determined by the Weyl and energy-momentum tensors under most conditions, except for special cases.

Findings

The metric is generally uniquely determined up to a conformal factor.

Special conditions are required for non-uniqueness.

The Weyl and energy-momentum tensors strongly constrain the metric.

Abstract

This brief paper investigates the consequences for the metric tensor of space-time when the Weyl tensor (in its conformally invariant form) and the energy-momentum tensor is specified. It is shown that, unless rather special conditions hold, the metric is uniquely determined up to a constant conformal factor.