But doesn't a clock in an accelerating spaceship run at the same rate
no matter where in the ship you put it?
Remarkably, the answer is, even in the context of SR, no.
It turns out that acceleration of an extended object is quite subtle.
That is to say, we can't meaningfully speak of the acceleration of an extended object.
Essentially, the 'front' (top?) of the spacecraft accelerates less than the 'back' (bottom?) of the spacecraft if the spacecraft is not to stretch and eventually fail structurally.
Thus, the clocks at the front (top) run faster than the clocks at the back (bottom) as would be the case for clocks at rest at different heights in a gravitational potential.
This is actually well known and best understood in the context of Rindler observers.
Note that Rindler observers with smaller constant x coordinate are
accelerating harder to keep up! This may seem surprising because in
Newtonian physics, observers who maintain constant relative distance
must share the same acceleration. But in relativistic physics, we see
that the trailing endpoint of a rod which is accelerated by some
external force (parallel to its symmetry axis) must accelerate a bit
harder than the leading endpoint, or else it must ultimately break.
Now, this isn't meant to answer your general question but, rather, to address the particular question quoted at the top.
Your argument is true classically also. You see, the effect of Gravitation close to earth and that of acceleration in flat spacetime is same. But this equivalence goes way past just the elevator experience, where all the observer feeling is a state of weightlessness or equal weight. While introducing GR Einstein took this equivalence farther and said that there are no laws of physics that can distinguish between an accelerated frame and a stationary frame in gravitational field. This goes for all laws including Electro-magnetism. So yes, if you see a charged particle in accelerated frame it should radiate. This can then be extrapolated and we can say that a charged particle which is stationary in a gravitational field should also radiate.
Best Answer
No, it wouldn't. The two situations are experimentally indistinguishable. That's one of the points of this thought experiment.
An even more important point (at least to me) is that it highlights a problem with the Newtonian concept of an inertial frame. Suppose a mad scientist alien teleports you to an elevator car. You're weightless. The alien tells you that you might be orbiting a star, or you might be in one of those huge voids in space. You need to use the local physics experiments piled up in a corner of the elevator car to determine which situation applies. Can you do it? (The answer is no.)
Next the alien teleports you to another elevator car in which you feel Earth normal gravity. The alien now tells you that you might be stationary on the surface of a non-rotating planet, or you might be in a spaceship in otherwise empty space accelerating linearly at 9.8 m/s2. Once again, you need to use the local physics experiments to determine which situation applies. Can you do it? (The answer is once again no.)
Newtonian mechanics tells us that there's a big difference between the orbiting elevator car and the elevator car in a void between galaxies. The first is non-inertial, the second is inertial. But there's no way to distinguish between the two using local experiments. General relativity is consistent with the experimental results; it says that both situations constitute local inertial frames.
Newtonian mechanics also tells us that there's a big difference between the elevator car sitting on the surface of a planet and the elevator car in an accelerating spacecraft. The first is inertial, the second is non-inertial. Once again there's no way to distinguish between the two using local experiments. General relativity is once again consistent with the experimental results; it says that in these cases, both situations constitute non-inertial frames.