Rainich Conditions in (2+1)-Dimensional Gravity

TL;DR

This paper derives Rainich-like conditions for (2+1)-dimensional gravity coupled to electromagnetism, enabling a geometric characterization of solutions similar to the (3+1)-dimensional case, and applies these to topologically massive gravity.

Contribution

It provides the first formulation of Rainich conditions in (2+1) dimensions, extending geometric characterization to lower-dimensional gravity theories.

Findings

Derived Rainich conditions for non-null and null electromagnetic fields in (2+1) dimensions.

Extended these conditions to topologically massive gravity.

Obtained plane-fronted wave solutions using the new conditions.

Abstract

In (3 + 1) spacetime dimensions the Rainich conditions are a set of equations expressed solely in terms of the metric tensor which are equivalent to the Einstein-Maxwell equations for non-null electromagnetic fields. Here we provide the analogous conditions for (2 + 1)-dimensional gravity coupled to electromagnetism. Both the non-null and null cases are treated. The construction of these conditions is based upon reducing the problem to that of gravity coupled to a scalar field, which we have treated elsewhere. These conditions can be easily extended to other theories of (2 + 1)-dimensional gravity. For example, we apply the geometrization conditions to topologically massive gravity coupled to the electromagnetic field and obtain a family of plane-fronted wave solutions.