Understanding time dilation

Question

I have a hard time understanding time dilation and special relativity; each explanation seems to contradict the other, and don't explain the apparent paradoxes they cause.

Say clock A orbits clock B at a very high speed. According to the explanations I've heard, B would perceive A as ticking slower than B, because of time dilation. But since A has equal right to claim to be stationary, it should observe B as orbiting A, thus ticking slower than A. If at any moment the clocks' time would be measured, would their respective elapsed time be different? In other words, is one clock actually slower than the other? This seems unreasonable, since the respective situations of the clocks are identical.

If the answer is no, what exactly happens if the clocks suddenly stopped orbiting each other and become stationary relative to each other? If each clock has perceived the other as ticking slower for a while, would the clocks instantly jump to the same time? This also seems unreasonable, since orbiting each other for a longer time then would imply a different outcome when stopping, even if the events of stopping are identical. How can this be resolved?

Answer 1

The proper time between two events is given by $\Delta \tau^2=\Delta t^2-\Delta x^2/c^2$. The proper time measures the amount of time that passes on the clock of an observer that travels in a straight line between those two events. You can break up the curved path of the orbiting clock into many infinitesimally small straight line segments. You can integrate all those segments to get the total proper time for the orbiting clock using the following formula $$\tau=\int\mathrm dt\, \sqrt{1-\dot x^2/c^2}$$ In other words: plug in a path $x(t)$ of an observer and this formula will give you the time that has passed on this observers clock. If you calculate this for $A$ and $B$ you will see that $B$, the inertial observer, will have the largest proper time, i.e. its clock will have ticked the fastest. If two observers both pass through two spacetime points, the observer that travelled in a straight line will always have the largest proper time. To remember this you can use the following mnemonic. An observer at rest only travels in time, while an observer that is moving at certain speed is moving more slowly through time because he now also has to move through space. For an inertial observer, you can always move to a frame where this observer is at rest, so the proper time is the largest.

Answer 2

1. B is in a fixed inertial frame. In that frame, A's clock ticks slowly.

2. A (the orbiter) is in a constantly changing inertial frame --- say A1 at a given moment M1 and A2 at a moment M2 shortly thereafter. In both frames A1 and A2, B ticks slowly.

3. In frame A1, A itself ticks slowly at moment M2 and in frame A2, A itself ticks slowly at moment M1. At any given moment, A says that A usually ticks slowly; it just happens to be ticking normally right at this very moment. Also, in those various frames, A at various moments started off out of synch by different amounts.

4. To keep track of things from A's viewpoint requires accounting for the constant changes of frame which result in constant changes in both A's own tick rate at various points along the orbit and its original offset at various points along the orbit. That can be a little complicated.

5. But this much is easy: B has a single unchanging inertial frame, and in that frame A ticks slowly. Therefore A falls behind B, and continues to fall farther behind as time goes on.

Answer 3

You might find it less confusing if you abandon any notion of clocks ticking at different rates.

Time dilation refers to the fact that the time between two events that occur in one place in an inertial frame is less than the time between the same events in any other inertial frame in which they occur in two different places.

Suppose you sit in your chair and count ten seconds on your watch. In the rest frame of passing muons, your ten second interval might be 100 seconds long, owing to the time dilation effect- in the muon frame, you are travelling at nearly the speed of light, so the start and end of your count occur at two widely separated places.

Now notice that it is the length of the interval that differs between the two frames- in one it is ten seconds and in the other it is one hundred. A second is a second, and good clocks accurately measure seconds. So when your watch records only ten seconds for an interval that in the muon frame is one hundred seconds, it is not because your watch is 'ticking slowly'- it is ticking correctly. What is different is the actual duration of the interval between the two frames.

Hopefully you can now be in a position to see why time dilation can be reciprocal. A person sitting in the muon frame could count ten seconds on their watch, while the same interval in your frame- where the start and end are very far apart spatially- would be one hundred seconds long.

What I have said so far applies to inertial frames. For non-inertial frames the physics is much more complicated, but the underlying principle is that the duration between two events is dependent on the path between them, and in general the effects are not symmetrical in the way that they are between two inertial frames. This is the reason for the famous twin 'paradox'- which is not a paradox, because there is no symmetry in the overall arrangement (one twin remains in a single rest frame through-out, while the other does not).