I am currently reading Wald's General Relativity and a result of section 4.3 stomped me. Part of Maxwell's equation in GR may be written as $$\nabla^a F_{ab} = \nabla^a \nabla_a A_b - R^a_b A_a= 0.$$

Now, let us look for wave solutions, $A_a = C_a e^{iS}$, where we assume that the derivatives of the amplitude $C_a$ are "small" (geometric optics approximation), so that terms like $\nabla^a\nabla_aC_b$ may be neglected. We then obtain the condition that $$\nabla^aS \nabla_a S = 0.$$ In other words, the surfaces of constant phase $S$ are null and thus, by a well-known result, $k_a = \nabla_a S$ is tangent to a null geodesic. This just means that light travels on null geodesics. However, to get there, one had to assume that derivatives of $C_a$ were "small". In other words, the general case seems to suggest that in general, the surfaces of constant phase are not necessarily 0 and thus not null.

I, however, though, that light would always travel along null geodesics, without any kind of approximation?


Saying that light travels along some curve (geodesic or not) is always an approximation of light propagation called geometrical optics. This approximation would fail when characteristic length scales on which the properties of light propagation vary become comparable with the wavelength of this light. Without this approximation EM radiation is a wave for which no single trajectory could be defined.

So, for example, if the wavelength of EM radiation propagating in a black hole background becomes comparable with the Schwarzschild radius of that black hole wave effects (such as diffraction patterns forming) that cannot be described by geometrical optics become noticeable. See e.g. this answer by Chiral Anomaly and links in it.

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    $\begingroup$ +1, The only thing I would also add is that light in media also does not travel along null geodesics, but generally on time-like orbits. (People tend to forget that e.g. astrophysical black holes tend to have a decent amount of plasma around them.) $\endgroup$ – Void Sep 13 '19 at 13:45

"Light" versus Electromagnetic disturbances. As A.V.S. points out there is a difference. However I'd add to this that in the general theory of partial differential equations the method of characteristics leads to the same geometry for a general solution. There is a difference between a ray theoretic description of light or sound which involves using the divergence of a ray bundle to estimate the amplitude, and a description of the propagation of a surface of initial data through space-time. The bi-characteristics of hyperbolic PDEs are null geodesics. You can look up derivations in Courant and Hilbert.


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