2 frames (A & B)are moving away from each other: A's clock will be slower than B, but B will be slower than A?

Question

If two Relativistic Frames (A&B) are moving from each other, then from B's frame, the clock in Frame A will appear to be running slower. However, from A's frame, the clock in Frame B will appear to be running slower. How can it be that both clocks are running slower?

Answer 1

There are many ways to approach this problem. First, a quick and easy way:



Let's agree that Alice is in frame A and Bob is in frame B.



If they both go on a trip and come back at the same time, as measured by Chuck who stayed home, then based on special relativity we can all agree that whomever was traveling closer to the speed of light on their trip will have a clock that's behind everyone else's (ignore accelerations for simplicity). So, if Alice traveled faster then her clock would be behind even Bob's clock. Based on this you must logically conclude that Alice would have observed Bob's clock ticking FASTER, not slower THAN HER OWN if she somehow viewed him in transit. Of course, Chuck would have seen both their clocks running slowly, but Alice's more slowly than Bob's.



Some things are relative but you have some invariants, namely spacetime and the speed of light. Even if observers don't agree when things happened, the events are still fixed, in one position in spacetime. Einstein said that individually space and time are relative but you still have an absolute from which to measure, and that's spacetime. All entities are moving through spacetime at the speed of light (c). That's right, even you, but most of that velocity is directed through time. If an object were to have a zero velocity in space then its speed through time (the 4th dimension) is c. As you increase your speed through space some of your speed through time is diverted (but the vector total is always c) and thus time appears to slow down for you as observed by someone at rest in space.

The point being that there IS an absolute from which to measure so if Bob were traveling slower than Alice then Alice would see Bob's clock ticking faster than hers but slower than the stationary clock.



I didn't quote any sources but most of my current conception of relativity came from an excellent book: "The Fabric of the Cosmos" by Brian Greene.

Answer 2

There is no reason Alice and Bob have to meet in the same place to compare times. I usually don't press my eye against my watch to measure the time. While they are traveling apart from eachother, they can send a timestamp back and forth and both Alice and Bob will observe the other's clock running slow. This does seem puzzling, but only because you are implicitly assuming that there is some form of absolute time. The rate at which a clock runs is entirely dependent on your velocity with respect to it. A is running slow to an observer in B, but not in general. Both clocks arent running "slower than eachother" because you have to pick a frame to measure both their rates.

Answer 3

If A and B were to actually compare times, then they would have to be brought back together at some point. For this to happen, one of them will have to accelerate in the direction of the other. This acceleration destroys the symmetry of the problem. It turns out that the one which does the acceleration will experience less time progression in total than the one who doesn't. The difference can be calculated as a simple path-integral, and contrary to popular belief (even amongst some physicists who should know better) does not require general relativity.



If you like, I can write down the equations for you.

Answer 4

"It's all in the mind, ya know"



It's all relative. Depending on where you're standing, time is different. The supposed paradox you're seeing here gets resolved once both relativistic frames come back into sync (i.e. stop moving away from each other and start moving together again).



At least, that's my understanding of it.