# Is there an analogue of a geodesic for the evolution of the electromagnetic field? [duplicate]

For a charged particle moving in free space, we can say from the homogeneity of space-time, that it moves along a geodesic.

Is there an analogous principle for the evolution of the electromagnetic field in space-time?

• It's amazing what has been asked before :-) – John Rennie Apr 6 '14 at 15:19
• @JohnRennie that question is asking if electromagnetic forces affect space-time like gravity, causing particles to follow geodesics. I'm asking about the evolution of the electromagnetic fields themselves, not the particles affected by them. – Larry Harson Apr 6 '14 at 15:34
• If you like this question, you may also enjoy reading this Phys.SE post. – Qmechanic Apr 6 '14 at 18:10

That a free particle follows a geodesic follows from the principle of least action and taking the action as $$S = \int d\tau \frac{m}{2} g_{\mu\nu} \dot{x}^\nu\dot{x}^\mu$$ (really just the generalization of the action of a free particle as just the kinetic energy).
Similarly you can derive the equations of motion for the electromagnetic field from the principle of least action applied to the action $$S = -\frac{1}{4}\int d^4 x F_{\mu\nu}F^{\mu\nu}$$ where $F_{\mu\nu}$ is the electromagnetic field strength tensor. But this generates precisely Maxwell's equations.