# Evidence for electrodynamics in curved spacetime

Field theories in curved spacetime is usually formulated by integrating their Lagrangian over the curved spacetime. For example, for electrodynamics, we have the action

$$S = \int d^4x \left( -\frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} \sqrt{-g} + A_\alpha J^\alpha \right)$$

It can also be straightforwardly coupled to gravity. The equation of motion can then be obtained using Hamilton's principle.

While it is a natural framework from a theoretical point of view, I am not aware of any experimental / observational evidence supporting results obtained from such a formulation.

Is there any empirical evidence for electrodynamics in curved spacetime?

For the purpose of this question, only classical EM is concerned, although evidence for QED in a curved spacetime (if any) would be even better.

This question is partly inspired by What is the most compelling evidence of General Relativity in the presence of matter and energy?

• Related: physics.stackexchange.com/q/78600/50583 The question is now: Has anyone observed, for example, Hawking radiation? The main predictions seem to all relate to black hole dynamics, which is experimentally a bit...difficult to access. – ACuriousMind Feb 11 '15 at 22:34
• @ACuriousMind I don't think Hawking radiation has been observed, and I agree that it'd be difficult to do so. However, I think classical EM in a curved spacetime is a weaker assertion than QED in curved spacetime, and may be easier to verify. – Chenfeng Feb 11 '15 at 22:44
• Maybe I'm just being dumb, but since when is the term $-A_\alpha\partial_\beta F^{\alpha\beta}$ in the EM Lagrangian? – Ryan Unger Feb 11 '15 at 23:37
• Since light is EM radiation, gravitational lensing can be seen as another example of electrodynamics in curved spacetime. – Paul Feb 11 '15 at 23:49
• Begs the question: "Is there a measurable effect from the Sun's gravitational field on its magnetic field?" – Keith Feb 12 '15 at 0:18

Classical electrodynamics is certainly studied in curved spacetimes to understand real phenomena. What better place for gravity and electromagnetism to work together than the ionized, magnetized plasma surrounding an accreting black hole?

In particular, we observe quasars with extremely powerful relativistic jets. Quasars are the supermassive black holes at the centers of galaxies when they are accreting matter and emitting copious quantities of light. Much of the emitted energy is in the form of jets, and it is natural to ask how the energy of the system is converted into this form. The most commonly believed answer is the Blandford–Znajek process, in which the coupling of a magnetic field to a rotating black hole extracts the rotational energy of the black hole itself.

The original paper and those that follow it go into much more detail, but the simplest approach is to assume the plasma continuum has infinite electrical conductivity (ideal magnetohydrodynamics) and negligible mass (force-free). Magnetic fields are "frozen" into such a fluid, and the dragging of this fluid through the ergosphere leads to the effect.

Indeed the entire field of general relativistic magnetohydrodynamics (GRMHD) is based on coupling electrodynamics (and the evolution of fluids) to curved spacetime. Sometimes this is a one-way coupling to a stationary spacetime, as is sufficient for studying systems where the stress-energy is dominated by a nearby black hole. In other cases, such as studying core-collapse supernovae or neutron star mergers, the matter/EM field affect the dynamical evolution of spacetime itself. Thus I'd say a broad swath of high-energy astrophysics is using (and therefore testing) electrodynamics on a curved spacetime on a daily basis.

A very simple example of electromagnetism in curved spacetime is the observed bending of light due to gravitational fields. Usually this is presented as the statement that "photons follow null geodesics." This statement can be derived in a geometric optics approximation to Maxwell's equations in curved spacetime (i.e. it is not just an additional postulate in GR). Assume that the potential has the form

$$A_\mu(x) = \epsilon_\mu(x) A(x) e^{i \phi(x)},$$

where the polarization $\epsilon_\mu (x)$ and amplitude $A(x)$ are slowly varying compared to the phase $\phi(x)$. Then from Maxwell's equations (and an appropriate gauge condition for $A_\mu$) you can derive that the wavefront $\nabla_\mu \phi$ is a null vector that is also geodesic.

The fact that we have observed gravitational lensing in many, many telescope images (like this happy guy) is a confirmation of electrodynamics in curved spacetime.