Theoretical Motivation For General Relativity - Electrodynamics in Curved Spacetime

Electrodynamics in Curved Spacetime

Maxwell's equations, the equations of electrodynamics, in curved spacetime are a generalization of Maxwell's equations in flat spacetime (see Formulation of Maxwell's equations in special relativity). Curvature of spacetime affects electrodynamics. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. The sourced and source-free equations become (cgs units):

,

and

where is the 4-current, is the field strength tensor, is the Levi-Civita symbol, and

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and the Ampère's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and Gauss's law for magnetism.

The electromagnetic wave equation is modified from the equation in flat spacetime in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

where the 4-potential is defined such that

.

We have assumed the generalization of the Lorenz gauge in curved spacetime

.