Geometrization Conditions for Perfect Fluids, Scalar Fields, and Electromagnetic Fields

TL;DR

This paper reviews and extends geometric conditions that determine when a spacetime metric corresponds to perfect fluids, scalar fields, or electromagnetic fields in general relativity, providing formulas to construct matter fields from the metric.

Contribution

It introduces unified, dimension-independent geometric conditions for identifying and constructing matter fields in Einstein's equations, including scalar fields and electromagnetic fields, with extensions to include cosmological constants and potentials.

Findings

Necessary and sufficient conditions for perfect fluids in any dimension.

Formulas for constructing scalar fields from the metric.

Conditions for electromagnetic fields including null and non-null cases.

Abstract

Rainich-type conditions giving a spacetime "geometrization" of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from the metric are unified and extended to arbitrary dimensions, to include a cosmological constant, and to include any self-interaction potential. Necessary and sufficient conditions on a four-dimensional spacetime metric for it to be an electrovacuum and formulas for constructing the electromagnetic field from the metric are generalized to include a cosmological constant. Both null and non-null electromagnetic fields are treated. A number of examples and applications of these results are presented.