# Paradox - Laws of physics are the same in all inertial reference frames vs Equivalence Principle (Pictures Added)

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:

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:

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?

• The ball moving sideways in the elevator will also experience time dilation (same amount as in scenario 2) – Michal Apr 30 '14 at 9:43
• Related question by OP: physics.stackexchange.com/q/110573/2451 – Qmechanic Jun 20 '15 at 10:20
• 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. – Kyle Kanos Jun 20 '15 at 13:01
• @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. – Kenshin Jun 20 '15 at 13:16
• 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). – Kyle Kanos Jun 20 '15 at 13:27