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

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 elec- tromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equa- tions are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Ge- ometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and for- mulas for constructing the scalar field from the metric are unified and extended to arbitrary dimensions, to include a cosmological con- stant, 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 electromag- netic 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