# 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

## 2 Answers

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.

• Yes, you can derive the first expression from a particle following a geodesic, so is it possible to do the same with the second expression? Is there a geodesic associated with the configuration space of the electromagnetic field? – Larry Harson Apr 6 '14 at 15:43
• I wouldn't think so. The geodesic equation is an ODE where the independent variable can be taken as proper time (for a massive particle). Maxwell's equations are PDE, and to interpret them as some geodesic you would need something like the proper time (or an affine parameter) for the EM field. – Robin Ekman Apr 6 '14 at 16:27
• What about using the proper time of the center of electromagnetic momentum frame? – John McVirgooo Apr 6 '14 at 18:15

The geodesic hypothesis pertains to "test particles", not to fields. However, given that it does pertain to photons as such, and photons are the quanta of the EM field, you could say that, yes, the geodesic hypothesis assumes how the quanta of the EM field evolves in spacetime.