Imagine that a rocket takes off, and inside the rocket is a ball on a scale. As the rocket accelerates, the ball “pushes down” on the scale – it has “weight.” The whole idea of equivalence, in this context, is that people in the rocket wouldn’t know (just from experiments they could do in the rocket) whether they were undergoing an acceleration or whether they were feeling the effects of a gravitational field.
So the rocket accelerates for a certain length of time, all of which the ball exhibits weight on the scale – evidence of the acceleration of the rocket. Suppose the rocket turns off its engine. The ball would no longer have “weight” – it would not push on the scale any more. No more work is being performed by the rocket.
Relative to the rocket, the ball has no more energy than before it took off. Someone on the rocket would conclude (mistakenly, as would someone holding a ball and believing he didn’t do any work) that “since the ball has no more energy now than when we started, we must not have done any work.” That is mistaken because it is using the wrong frame of reference relative to which it must be determined if the ball has more energy. It is the frame of reference from whence the rocket/ball started that the ball has more energy relative to. To people in that frame of reference, the ball and the rocket both have more energy than before it took off.
When we hold a ball, we are like the person on the rocket. We can hold it all day long, and even though it has weight (evidence of our doing work), it doesn’t seem to have any more energy than it did an hour or a day ago – within our reference frame. So we conclude that we have done no work. So who/what in our ball holding example is equivalent to the person where the rocket took off from, relative to which the ball does have more energy? It would be someone in free fall.
The longer you hold a ball in a gravitational field, the more energy it has relative to someone in freefall in the gravitational field. That is, the longer you hold the ball, the faster it is accelerating away from the free-falling person. But who is doing the accelerating – you, the earth, and the ball, or the free-faller? Although we, on the earth, learn to think of ourselves as being at rest, and falling things as being those which are accelerating, it isn’t the person in free-fall who is undergoing an acceleration. It is the ball – and you and the Earth - that is accelerating. Remember, feeling the force of a gravitational field is equivalent to the acceleration. As you hold the ball, it is moving away from the free-faller faster and faster.
Imagine someone standing on a trap door next to you as you hold the ball. You can hold the ball all the live-long-day, and it will not have any more energy from your friends’ perspective, just as it wouldn’t from yours. Suppose now that the trap door opened and your friend was to start a free fall. The ball you are holding, you, and the Earth would move away from/past him faster and faster – it would be accelerating. We tend to think that it is the falling person who is accelerating, but that is mistaken. It is we, the balls we hold, and the ground we are standing on that is accelerating. Remember – if the person falling had a ball on a scale it would not have weight – he would not be undergoing an acceleration. That is, no experiment he could do would show that he was either accelerating or was under the effect of a gravitational field. He might as well be very far removed from any massive body – he wouldn’t know the difference.
If the ball you are holding is undergoing an acceleration, then it should have more and more energy as time goes by, right? How can this energy be used by the free-falling person, showing that work has been done and that the ball (and you and the earth) have more energy? How can he measure the fact that you, the ball, and the Earth have more energy with each passing moment? Suppose that your falling friend was connected to a very long comb shaped device with paddles sticking out, so that one passes near the ball every ten seconds. If someone on the comb device wants to measure the kinetic energy of the ball as it passes by, he adjusts the next paddle slightly so it hits the ball. As the comb device falls faster and faster, the ball is moving faster and faster in the other direction, so that with each passing paddle (if it was to be made to hit the ball) there would be a greater ability to “do work” (impose a force on the paddle).
So who is undergoing an acceleration? Is it the ball you are holding (along with you, the earth,etc.), or the comb-device with your free-falling friend? If there was a ball on a scale on the comb device there would be no evidence of acceleration or of feeling the force of a gravitational field (the ball would not “weigh” anything on their scale, but would, like the rest of the comb device, be in free-fall). But the ball you are holding on your scale continuously has weight – evidence that it is undergoing an acceleration.
There is a tendency to think that the gravitation field is imparting more and more energy to your falling friend (doing work) as he accelerates away from you. But it is you who is doing work. Actually, it is you, the ground you’re standing on – the whole Earth – everything that stops you from following a geodesic – doing work – gaining more and more energy in your free-falling friends frame of reference. He is “standing still” in space-time – following a geodesic. So it’s not merely the ball that gets more and more energy – it is the whole structure of you, earth, etc. that is moving faster and faster relative to the free-falling friend, as all of it has more energy relative to the free-falling friend, and so is capable of doing more and more work in his frame of reference.
You and your friend were originally in the same frame of reference. When the trap door opened it wasn’t him that accelerated. Rather, it was him who stopped accelerating as he went into free-fall – his scale would stop registering weight of his ball on his scale showing that he was neither accelerating nor under the effect of a gravitational field. It would be equivalent to the rocket stopping it's acceleration. Imagine two rockets side by side with balls on scales. If one rocket stopped accelerating, it would appear to the one who was accelerating that he (the one who stopped accelerating) was falling. But it is really the one on the rocket that is still firing that is accelerating.
I know I’m going on and on about this, but I really am trying to make sense of it. I understand that within the context of the people holding the ball that the ball does not gain energy as it is being held. On the other hand, feeling the effects of a gravitational field is equivalent to acceleration, and it takes work to perform an acceleration. I think there is some subtle aspect of GR that I either don’t understand completely or I understand incorrectly.
Maybe I should have posted this in the general relativity section!