Does Gravity Affect Time Dilation on a Flat, Non-Rotating Earth?

[ScientificMind]

On a science test that I took the other day, one of the questions inspired me to ask a question of my own. The original question was a multiple choice question that went something like this: if someone shoots a gun horizontally and simultaneously drops another bullet, which one would hit the ground first? Of course, I chose the answer stating that they will both strike the ground at the same time, since they would both be accelerating downward at roughly 9.8 m/s/s, and it seemed fairly obvious that for this question I should assume that there are no obstacles or any change in terrain. Anyway, my question is this, since Einstein discovered that time slows down as something approaches the speed of light, would the bullet that was fired horizontally from the gun hit the ground second, at least from the point of view of a stationary observer, since the faster velocities would cause one second to be longer for the bullet that is moving faster than the other bullet? Also, I should probably mention that this is not a homework question, it was only inspired by one.

 

[bcrowell]

First let's think about this in Newtonian gravity. Let bullet A be the one fired from the gun, and B the one that's dropped. By switching into a frame of reference moving horizontally at a speed equal to the gun's muzzle velocity, we can interchange the roles of the bullets. Now A is the one falling straight down, and B is the one moving along a parabolic arc. This shows that the roles of the two bullets in Newtonian gravity are interchangeable, so they must hit the ground at the same time.

In the context of relativity, one thing that changes is that simultaneity is no longer frame-independent. If the bullets are released simultaneously in the gun's rest frame, and at different positions, then in some other frame they weren't fired simultaneously. There's a similar ambiguity with respect to the times when they hit the ground.

In both Newtonian mechanics and relativity, we're allowed to switch to a free-falling frame of reference. (This is the customary frame to use in relativity, but not in Newtonian mechanics.) In this frame, there is no gravitational field. (Or in the traditional Newtonian description you could say that a fictitious force is cancelling gravity.) The ground accelerates up toward the bullets and hits them. This description makes it seem that there can't be any difference between the two bullets except for the frame-dependence of simultaneity described above.

The argument about the free-falling frame of reference implicitly assumes special relativity, which is valid if the spacetime is flat. Contrary to what Einstein thought when he first created GR, SR is capable of describing accelerated frames of reference.

 

[A.T.]

and it seemed fairly obvious that for this question I should assume that there are no obstacles or any change in terrain.

You must also assume vacuum, and a flat, non rotating Earth.