# Does a charge in a gravitational field make a magnetic field?

I know that general relativity unifies the electric charge with the magnetic field via relative motion. So if I accelerate a charge then anyone in another inertial frame will see a magnetic field. Einstein also tells us that the effects of acceleration are indistinguishable from the effects of gravity. Does this mean that a charge in a gravitational field will likewise be seen as a magnetic field?

• This is an interesting question – although I'm sure that a moving inertial frame would view an electric field as a magnetic field, I don't know if this translates exactly to non-inertial (accelerating) frames. Feb 7, 2019 at 5:14
• Just to clarify: "moving inertial frame would view an electric field as a magnetic field" is explained by special relativity, without the need of general relativity (which considers acceleration and gravity). Feb 7, 2019 at 7:15
• You may find the following discussion by Kevin Brown interesting: Does a uniformly accelerating charge radiate? According to Kevin Brown the question whether a uniformly accelerating charge will radiate remains unresolved. Kevin Brown's discussion gives me the impression that our current theories are not equipped to address this issue. (Which of course is contrary to what one would expect.) Feb 7, 2019 at 21:42

The answer is no if the gravitational field is static. I'm not sure what will happen if the field is not static.

Let me quote a paragraph from Wolfgang Rindler's Relativity - Special, General, and Cosmological (2nd edition, ISBN 44198567324), Sec.1.15 (P.22-23):

Consider the following notorious paradox: an electric charge is at rest on the surface of the earth. By conservation of energy (or just by common sense!), it will not radiate. And yet, relative to an imagined freely falling cabin around it, that charge is accelerating. But charges that accelerate relative to an IF radiate. Why doesn’t ours? Again, consider a charge that is fixed inside an earth-orbiting space capsule. Now, circularly moving charges do radiate, and one cannot imagine how the earth’s gravitational field could change that. But relative to the freely falling space capsule the charge is at rest, and charges at rest in an inertial frame do not radiate. Where is the catch? Much has been written on these paradoxes, but the proper solution seems to have been first recognized by Ehlers: It is necessary to restrict the class of experiments covered by the EP to those that are isolated from bodies or fields outside the cabin. In the case of the charges discussed above, their electric field extends beyond the cabin and is, in fact, ‘anchored’ outside; since radiation is a property of that whole field, it follows that these ‘experiments’ lie outside the scope of the EP.

In the above case, we can see that not all effects of acceleration are indistinguishable from the effects of gravity.

Another example is that gravitational field felt by an object may be non-uniform (e.g. gravitational attraction from the Earth felt by a satellite), while acceleration does not.

• Please define "EP" too many anagrams floating around in my head. Thanks Feb 7, 2019 at 14:20
• "EP" means "equivalence principle" here. Feb 7, 2019 at 15:10
• This only covers static gravitational field. But what about charge at rest in a stationary but not static field? Feb 7, 2019 at 18:39
• I think you are right, so I've modified my answer a bit. Feb 8, 2019 at 6:52