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According to the equivalence principle, no experiment should exist that one can perform to determine whether one is in an accelerating elevator, or in a gravitational field. I will outline two scenarios that will differ depending on whether you are in an elevator or in a gravitational field, and thus provide an experiment one can determine to differentiate the two.

Scenario 1:

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Suppose I am standing in an elevator which is accelerating upwards at g and also suppose I am holding one ball in each hand.

Now with my right hand, I do nothing but release the ball, but with my left hand, I throw the ball perfectly horizontally, with a velocity v.

Now we know that in this situation, the elevator will strike both balls simultaneously because the vertical velocity of both balls is equal to 0 and it is only the lift that is moving up.

Scenario 2:

enter image description here

This time, suppose I am standing on the Earth, and acceleration due to gravity is exactly g. Now again I simply release the ball in my right hand, but throw the ball in my left hand perfectly horizontally, with a horizontal velocity v. Imagine I measure that the ball on the right falls to the ground after 1 second.

Now as shown in the answers to this question, as a result of time dilation, we measure the moving ball on the left striking the ground after not 1 second, by $\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ seconds.

That is, the ball on the left in this scenario will take longer to hit the ground.

Summary:

Therefore, if I experience a "gravitational pull" I can determine if it is from a gravitational field, or due to an accelerating elevator, by throwing one ball out horizontally, and dropping another. If they hit the ground at the same time I am in a lift, otherwise I am in a gravitational field.

How can this apparant violation of equivalence principle be resolved?

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    $\begingroup$ The ball moving sideways in the elevator will also experience time dilation (same amount as in scenario 2) $\endgroup$ – Michal Apr 30 '14 at 9:43
  • $\begingroup$ Related question by OP: physics.stackexchange.com/q/110573/2451 $\endgroup$ – Qmechanic Jun 20 '15 at 10:20
  • $\begingroup$ You can tell the difference between the two frames with two balls & a good measuring device, provided you are sufficiently far away: drop both balls, if they move closer together, then you're in a gravitational field. $\endgroup$ – Kyle Kanos Jun 20 '15 at 13:01
  • $\begingroup$ @KyleKanos, yes but the equivalence principle assumes a constant gravitational field and thus you cannot invoke tidal forces. My example will work in an infinitesimally small area of space where the gravitational field is constant. $\endgroup$ – Kenshin Jun 20 '15 at 13:16
  • $\begingroup$ The two balls would move closer due to the separation of the balls being attracted to the gravitational potential, e.g. from $\ddot{x}_1=-\nabla\phi(x_1)$ and $\ddot{x}_2=-\nabla\phi(x_2)$ (define $\eta=x_2-x_1$ and then Taylor expand). $\endgroup$ – Kyle Kanos Jun 20 '15 at 13:27

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