The Eastwood-Singer gauge in Einstein spaces

TL;DR

This paper explores the Eastwood--Singer gauge in Einstein spaces, solving associated wave equations in de Sitter spacetime, and demonstrates exponential decay of solutions under certain conditions, advancing understanding of electrodynamics in curved spacetime.

Contribution

It provides explicit solutions to scalar and vector wave equations in Einstein spaces using the Eastwood--Singer gauge, including in de Sitter spacetime, and analyzes their decay properties.

Findings

Scalar wave equation admits exponential decay solutions in de Sitter space.

Explicit solutions for the vector wave equation with time-dependent potential are obtained.

The gauge condition leads to a fourth-order scalar equation with well-behaved solutions.

Abstract

Electrodynamics in curved spacetime can be studied in the Eastwood--Singer gauge, which has the advantage of respecting the invariance under conformal rescalings of the Maxwell equations. Such a construction is here studied in Einstein spaces, for which the Ricci tensor is proportional to the metric. The classical field equations for the potential are then equivalent to first solving a scalar wave equation with cosmological constant, and then solving a vector wave equation where the inhomogeneous term is obtained from the gradient of the solution of the scalar wave equation. The Eastwood--Singer condition leads to a field equation on the potential which is preserved under gauge transformations provided that the scalar function therein obeys a fourth-order equation where the highest-order term is the wave operator composed with itself. The second-order scalar equation is here solved in de Sitter spacetime, and also the fourth-order equation in a particular case, and these solutions are found to admit an exponential decay at large time provided that square-integrability for positive time is required. Last, the vector wave equation in the Eastwood-Singer gauge is solved explicitly when the potential is taken to depend only on the time variable.