If you have two clocks moving toward each other and they both stop when they meet does it point to a paradox?

Question

Say they both decelerate at the same rate also. If clock A sees clock B as slower than itself and clock B see clock A as slower than itself due to relativity isn't this a paradox because when they meet (as far as my understanding is concerned) they will each see the other as being slower? Obviously I am missing something here but I do not know what.

Answer 1

You might find it helpful to consider the Doppler effect. Imagine two police cars approaching head on. The occupants in each car will hear the siren of the other car as going faster than their own, owing to the Doppler effect, but when they come to a halt and meet the sirens will sound the same. Something similar happens with the approaching clocks. When they meet, they will be ticking at the same rate.

Assuming the set-up was properly symmetrical, the elapsed times on the clocks would also agree. Let's suppose the clocks were travelling at such a speed that each thought the other was time dilated by 50%, and when they meet each clock shows 10 hours since the start of their journey. You will ask yourself, how can they both show the same time if each thought the other was running at half speed? The answer is that they disagree on when their respective journeys started. In the frame of each clock, the clock's journey started ten hours ago while the other clock's journey started 20 hours ago. All the main effects of SR, such as time dilation and length contraction, boil down to a lack of synchronisation at a distance.

Answer 2

In a lot of popular descriptions of the twin's paradox, they will use a dialogue that goes something like this. "Observer A sees B's clock tick slower, and B sees A's clock tick slower? Which one is right? It turns out they are both right!" I would argue, they are both wrong!

They both must be ticking at the same rate and the elapsed times are equal when they meet. The problem is, when they use the Lorentz transformation, there is a not meaningful comparison of clock rates. It is a bit like two observers that are some distance apart comparing their ruler sizes optically. They think the other's ruler is shorter, but when they bring them together, they are the same size. However, this does not mean time dilation is an illusion and the twin's paradox demonstrates two travelers can have different elapsed times and that time dilation can have real consequences. How is this resolved? The answer is to sum up and compare the spacetime intervals of the moving clocks. All observers will then agree on which clock has the greatest elapsed time.

This is a bit like drawing two routes on a map and measuring which path is the longest. To sum up path segments on an x,y map you can calculate the Pythagorean distance of each segment using $\sqrt{\Delta x^2 + \Delta y^2}$. In relativity, time does not add up in the same way as spatial distances, and we have to use $\sqrt{\Delta t^2 - \Delta x^2/c^2}$ to calculate the invariant spacetime interval of each segment. In your example, the spacetime intervals are identical, so the elapsed proper times are identical, and this will be true from any observer's point of view.

Trying to compare elapsed times of two spatially separated clocks that have relative motion using the time dilation equation instead of the spacetime interval, is like trying to compare the lengths of two rods in a photograph when one rod is near the camera and the other is far away. Best to always use the spacetime interval.