Quote:
Originally Posted by grav
Two observers are moving toward each other with some relative speed between them. One then throws a ball to the other when they are at some distance from each other. These relative speeds are not great, so we probably don't need to bring relativity into this. Now, from the point of view of the person throwing the ball, considering that one stationary, then, the ball has travelled the original distance the two were separated by when the ball was thrown minus the distance the other has moved closer since then until the ball is caught, right? But according to the one catching the ball, the original distance the ball was thrown from is the distance the ball has travelled, regardless of how close the thrower has moved since then, isn't it? Which one is right? What is the actual distance the ball has travelled?
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Some have said "both are right."
I disagree.
Even in the darkest reaches of space, there is a velocity vector that's essentially zero. We know we don't experience that vector, as we're moving around on our planet, our planet is rotating, it's revolving around the sun, the sun is orbiting the Milky Way, and our galaxy is moving through space itself.
Just exactly how do we define the fact that our galaxy is moving through space, if velocity is nothing but relative? Simple - we take the mean of redshifts from all directions, calculate a net effective velocity, then subtract out the vectors for planetary rotation, orbit, and solar system orbit through the galaxy. The result is our galaxy's velocity.
Remove that, and you're at a position of zero velocity. Putting aside the proximity of mass which affects time, you're now experiencing the shortest duration of time compared to all points in the universe. It's true that regardless of the proximity of mass, you're now experiencing the shortest time rate at that particular point in space. If you were in motion, passing that point, your duration of a second would be slightly longer (relativistic effects).
I like to think of this as a "zero-velocity point," and there are an infinite number of such points throughout the universe, such that a chronometer which is stationed at that point experiences time at a rate faster than if the chronometer were transiting that point.
Thus, no, the distance the ball travelled isn't relative to the thrower or the catcher, each with two different but "accurate" answers. The only way this could happen is if you were to say, "relative to the thrower" or "relative to the catcher," neither of which accurately answers the question, "how far did the ball travel?"
The truth of the matter is that the ball travelled a distance relative to it's integrated velocity between being thrown and caught relative to the surrounding zero-velocity points divided by the duration of time as measured at those zero-velocity points along the route (not at the thrower, the catcher, or the ball).
Thus, the distance travelled by the home run hit by Banks was, in reality, quite a few miles...