An electron is on the floor of a rocket ship. The rocket accelerates
upwards. To an inertial observer the electron is seen to accelerate and
he observes EM radiation coming from it. To an observer in the rocket
the electron seems to be at rest in a gravitational field (not sure of
this claim as I am ignorant of GTR) and radiating. Is it true that an
electron at rest in the earth's gravitational field will radiate? And
if so, where does the energy come from?

The basic problem is trying to discuss "radiation" without really
specifying what one means by that term.

Let me instead discuss time-varying fields, as it is clear that if a
given observer does not observe a time-varying field then she observes
no radiation.

For the case of a charge at rest in a uniformly-accelerating rocket
(i.e. constant _proper_ acceleration) in Minkowski spacetime, and an
observer also at rest in the rocket, it is clear that the observer does
not observe a time-varying field. Just consider the instantaneously
comoving inertial frames of the observer at times t1 and t2 when
observations are made: the relationship between observer and charge in
such frames is always the same (the charge at its retarded time is
moving wrt the observer, but this motion is the same in every such
frame, as is the retarded position of the charge wrt observer).

For this rocket charge, an inertial observer will clearly measure a
time-varying field. Whether or not this is "radiation" depends in detail
on what one means by that.

As soon as the rocket's proper acceleration is not constant then the
rocket observer can observe a time-varying field, as will an inertial
observer. Whether or not this is "radiation" depends in detail on what
one means by that.

The equivalence principle implies that an observer at rest on the
surface of the earth will observe a constant field from a charge also at
rest on the surface (nearby), and hence no radiation.

I have been able to be definitive above, because time-varying fields are
unambiguous while "radiation" is not.
But I have avoided the question -=-

-=- of energy balance that is inevitable here. -=-

Google "radiation from uniformly accelerated charge" gives more
information (as usual, beware of crank sites).

Bottom line: this is a seemingly-simple question that does not have a
simple answer. Some answers are easy (a charge at rest on the earth does
not radiate to an observer at rest on the earth); others are not....

Tom Roberts