# Frames of reference of Maxwell's Equations

The Maxwell's Equations are one of the most famous sets of equations physics have ever known. But just as different sets of equations are applicable to different frames of reference, where are Maxwell's​ Equations applicable? To be more specific:

• Are Maxwell's Equations valid in only inertial frames of reference?

• If not, then how can we reformulate the equations so that it is valid in any accelerating frame with an arbitrary acceleration?

• Presumably one could add fictitious electric and magnetic fields in much the same way one adds fictitious forces in the usual case - just pretend the objects generating the field are accelerating with the acceleration of the frame. Nothing about the laws themselves has to change. – probably_someone Jun 28 '17 at 3:55
• That's the whole problem! Adding fictious fields are not proving to be that easy as it seems , since there is a difference between fields and forces. But if the laws could be reformulated to incorporate an acceleration term, it would be much more insightful – Yuzuriha Inori Jun 28 '17 at 3:59
• Maybe this is helpful: arxiv.org/pdf/1009.3968.pdf – probably_someone Jun 28 '17 at 4:17

As they are conventionally written Maxwell's equations are valid only in inertial frames of reference in flat spacetime. This is because the derivatives in the equation are not covariant derivatives and therefore don't apply when the coordinate system is curved.

It is possible to write Maxwell's equations in arbitrary coordinate systems though it gets somewhat complicated. The trick is to note that Einstein's equivalence principle tells us that acceleration is locally indistinguishable from gravity, and therefore the treatment of Maxwell's equations in accelerating frames is the same as formulating them in curved spacetime.

In principle all we need to do is replace all physical quantities by tensors, and replace normal derivatives by covariant derivatives. However the process of doing this makes the equations look very different. The details are described in the Wikipedia article Maxwell's equations in curved spacetime. Specifically note that the introduction to this article states:

The electromagnetic field also admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not Cartesian. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates.

So this approach is just as useful for curved (e.g. non-inertial) coordinates in flat spacetime as it is for curved spacetimes.

• If you fix your answer (which is minimal), I'll give you the bounty. – PiKindOfGuy Feb 19 '19 at 5:20
• @PiKindOfGuy what extra info are you looking for? The note in your bounty says the first bulleted question doesn't have a direct answer, but the direct answer is simple Yes. Unless you rewrite the equations in tensor form they only apply to inertial frames in flat spacetime. – John Rennie Feb 19 '19 at 5:29
• Except you didn't even say "yes". You just answered the second question. At least that's how it reads. – PiKindOfGuy Feb 19 '19 at 5:32
• @PiKindOfGuy is that clearer? I would wait the full seven days before awarding any bounty since the bounty might attract someone prepared to write a more extended description of how to do electrodynamics in a curved spacetime. – John Rennie Feb 19 '19 at 5:45
• @PiKindOfGuy I would say if you aren't responsible enough to reward your bounties then you shouldn't put them up in the first place, yes? – Aaron Stevens Feb 19 '19 at 6:12