Maxwell's Equations in Curved Spacetime - Nonlinearity of Maxwell's Equations in A Dynamic Spacetime

Nonlinearity of Maxwell's Equations in A Dynamic Spacetime

When Maxwell's equations are treated in a background independent manner, that is, when the spacetime metric is taken to be a dynamical variable dependent on the electromagnetic field, then the electromagnetic wave equation and Maxwell's equations are nonlinear. This can be seen by noting that the curvature tensor depends on the stress-energy tensor through the Einstein field equation

where

is the Einstein tensor, G is the gravitational constant, g_ab is the metric tensor, and R (scalar curvature) is the trace of the Ricci curvature tensor. The stress-energy tensor is composed of the stress-energy from particles, but also stress-energy from the electromagnetic field. This generates the nonlinearity.