# Low frequency electromagnetic waves in General Relativity

I am becoming familiar with the Geometric Optics approximation in General Relativity which (to summarise) says that EM waves follow null geodesics under the geometric optics approximation. In the context of General Relativity the Geometric Optics approximation amounts to the assumption that:

$\lambda << R$ where $\lambda$ is the wavelength and R is some measure of curvature (~ 100 meters near Black Holes).

So under this approximation EM waves follow null geodesics. Additionally the "photon" model can make sense too, and we can say "photons follow null geodesics". But what about the "general case"? In the general case the order of curvature is the same as (or less) than that of the wavelength. So the usual statements do not obviously apply. Where is the full case discussed and what implications are there?

To give some extreme examples for those who have not thought about this before:

1. In a Schwarzchild model at R=2M photons can get into circular null orbits. But clearly this provides an IR cut-off as they would be in a Bohr-like orbit with a max wavelength $\lambda_{max}=4\pi M$.

2. The usual "equivalence principle" may fail as the local trivialisation cannot extend far enough into the curvature region (and $\lambda$ is larger still). So the equivalence laboratory can detect such waves, but cannot measure their behaviour and prove that they follow geodesics (local straight lines).

I am struggling to find any coherent account of this, just odd facts and claims.

EDIT In large part this question is answered by the comment from Genneth, and the links related to this Wikipedia article: Maxwell's equations in curved spacetime.

• One thing that I would add is that the intensity of your beam must be small enough that you can ignore the self-gravity of the light beam (a changing intensity ray of light amounts to a changing stress-energy tensor, which should then have some of its energy and momentum carried away by gravitational radiation, which would then cause it to deviate from background geodesics). I can't think of a real-life case where the gravitational self-force of light is physically important, but theoretically, it would be a nonzero effect. – Jerry Schirmer Feb 9 '11 at 21:59
• Classically (not quantum) Maxwell's equations can be written in a (general) covariant form: $dF=0$, $d\star F=j$, where $F$ is the electromagnetic 2-form and $j$ is the current. The Hodge star $\star$ hides the metric (and more generally, constitutive material properties). – genneth Feb 10 '11 at 9:59

## 1 Answer

Roy, indeed, geometric optics with trajectories of light rays is only applicable when $\lambda\ll R_{curvature\,radius}$. The beginning of your question suggests that you are aware of this condition.

However, the rest of your question is all about your refusal of the condition. So it's hard to understand whether you actually understand the condition or not.

In particular, your conclusion 1 seems to be an artifact of the negligence of the condition because it clearly makes no sense to talk about "orbits" of photons whose wavelength is longer than the black hole radius.

Also, I don't know why you would say 2. The equivalence principle always applies locally - it's how it's defined - and this principle is 100% valid in the real world. You can't disprove the equivalence principle by violating its assumptions.

The right way to treat light and electromagnetic waves outside the regime of validity of geometric optics is to treat them as waves, i.e. using field theory. This is what you may be asking although I haven't really understood your question. So one has to learn Maxwell's equations. If the term "Maxwell's equations" sounds new to you, then it could have been a bit counterproductive and illogical to study general relativity (before Maxwell's equations) but don't panic, I did the same mistake 25 years ago. ;-)

Classical electromagnetic waves in general relativity are described by the Einstein-Maxwell system; quantum radiation - including the concept of photons - requires you to study quantum electrodynamics on a curved background. Quantum field theories on curved backgrounds bring some novelties such as particle production - different Hamiltonians and different ground states of these Hamiltonians.

• I am familiar with Maxwell's equations but not so in terms of the covariant form $dF=0$ mentioned by Genneth. In particular the range of solutions of said equations. The gravitational restriction on geometric optics is similar to the corresponding flat space restriction, but again it will introduce new features. I guess I just need a text to help discuss this further. – Roy Simpson Feb 10 '11 at 10:42