Suppose you have with you in your box two marbles. Can you use these in an experiment to discover if you are being mechanically accelerated, or stationary on the surface of the Earth? Surely you can, if you have a sufficiently sensitive measuring instrument.
Release your marbles simultaneously from the top of the box. They will fall to the bottom. If you are being accelerated, their trajectories will be parallel, but if you are on the surface of a planet their trajectories will converge on the centre of the planet, so they will converge as they fall.
The reason that they will converge is that the gravitational attraction is from the equivalent of a point source. If you are on a sufficiently large planet the point will be far enough away that you can't detect the convergence.
Alternatively, if you release one from the top of the box and one from waist height, they will maintain that separation until the first one hits the floor, under acceleration. Under gravity, however, the lower marble will fall faster than the upper one, so the separation will increase.
The reason here is close to the same as for the first case. Since the force exerted on the marbles varies according to the inverse square law the force on the marbles is different based on the distance from the center of the planet.
Even in terms of SR it would be possible to distinguish between being in deep space and free falling towards a planet. If you released your marbles at the same elevation, with a horizontal separation, in deep space they would maintain that separation, but in free fall towards a planet they would slowly move together, thus warning you of a nasty crash in your future.
Have I got this wrong, or is the equivalence principle just a convenient approximation?
If you were in free fall the marbles wouldn't move with respect to the box. They would be falling at the same speed you were. For the illustration of the equivalence principle we are assuming that the box is on the surface of the planet.
The equivalence principle isn't an approximation, it is a simplification. It is the simplest statement of the principle. For real world situations you have to account for all the factors that aren't included in the statement, such as the fact that gravity generally comes from an approximately point source. It is the same as for F=Ma. There is no real case where only one force is acting on an object. So for the equivalence principle you have to do the sum of all the forces. For GR (you said SR, but I think you meant GR) you don't really talk about forces. You have to talk about perturbations in the gravitational field, but the idea is the same.
Bill Gill
