Why don't two observers' clocks measure the same time between the same events?

Question

Person A in reference frame A watches person B travel from Star 1 to Star 2 (a distance of d). Of course, from person B's reference frame, he is at rest and is watching Star 2 traveling to him.

Now we know from the principle of relativity, each one will measure the other one’s clock as running slower than his own.

Let’s say that Person A measures Person B’s speed to be v, and that Person A measures 10 years for person B to make it to Star 2. Let’s also say that person B is moving at the speed so that the Gamma Factor is 2. This means person A observes person’s B’s clock to have elapsed a time of 5 years.

Now let’s look at this from Person B’s perspective:

Person B observes Star 2 approaching (and Star 1 receding from) him also at speed v. Since the two stars are moving, the distance between them is length contracted (after all, if there were a ruler in between the stars, the moving ruler would be contracted) by a factor of 2. Since person B measures the initial distance to Star 2 to be d/2 and its speed v, he calculates the time to Star 2’s arrival to be 5 years. Since he observes person A’s clock as running slow (since Person A is moving also at speed v), when Star 2 arrives, he measures Person A’s clock to have elapsed a time of 2.5 years.

Do you see why I’m confused? Person A measures Person B’s elapsed time to be the same as Person B measures Person B’s elapsed time (both 5 years), but Person B does not measure Person A’s elapsed time to be the same as Person A measures Person A’s elapsed time (Person B get’s a measurement of 2.5 years while Person A measured 10 years). This is asymmetrical, which probably means it is wrong. But I’m not sure what the error is.

I suspect if I had done this correctly, each person should measure his own elapsed time to be 10 years and measure the other’s elapsed time to be 5 years. This would be symmetrical and would make the most sense, but again, I can’t seem to justify how person B wouldn’t measure his trip time to be 5 years.

What's my mistake?

Answer 1

Everything you have said describing the situation in your question is correct; Person A and Person B disagree about how much time elapses on person A's clock between the two events. (The first event is Person B leaving Star 1 and the second event is Person B arriving at Star 2) This is not a logical contradiction. It stems from the relativity of simultaneity and the fact that the time between two events is different in different reference frames.

The time between two events is minimized when the spatial separation between them is zero because the interval

$$\Delta s^2 = \Delta t^2 - \Delta x^2$$

is invariant (the same for everyone). Person B therefore perceives the minimum possible time between the two events, which is 5 years. Person A perceives some spatial separation between the events, and so perceives a longer time between them (10 years).

We can use this information to work out the speed $v$. For Person A, $\Delta t^2 - \Delta x^2 = 5^2$ because that's the answer for Person B, and it must be the same for A. We know $\Delta t^2 = 100$, so

$$100 - \Delta x^2 = 25$$

or

$$ \Delta x = \sqrt{75} = 5\sqrt{3}$$

$v$ is then

$$v = \frac{\Delta x}{\Delta t} = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$$

The situation is not symmetric with respect to A and B because A is not moving relative to the stars, but B is. The existence of the stars breaks the symmetry. A symmetric situation would be if A and B start at their own stars, then meet in the middle.

Another symmetric scenario would be to let B start moving away from A. When A's clock reads 10 years, ask her what B's clock reads. When B's clock reads 10 years, ask him what A's clock reads. In that case, both would say that the other's clock reads 5 years.

So, if the setup of the problem is symmetric with respect to A and B, their answers should be, also. Because this problem does not have that symmetry, the answers A and B give do not have the symmetry.

Finally, you might be concerned that Person A thinks the time between the two events is 10 years, but according to Person B, Person A's clock reads only 2.5 years elapsed. This is due to the relativity of simultaneity. According to Person B, he is arriving at Star 2 and checking Person A's clock simultaneously. Those events have a big spatial separation, though. According to Person A, they are not simultaneous. Person A thinks Person B has checked her clock too soon.

Answer 2

Layman's answer here. First we need to clean up the thought experiment a bit.

In relativity texts it's common for these kind of thought experiments to use observers stationed at the points in question, so that the travel delay of light can be ignored. As part of the setup of the thought experiment Person A would have someone whose clock is synchronized with A's clock stationed at Star 2; let's call that someone A2. When you say "Person A measures 10 years for person B to make it to Star 2", a relativity text would take this to mean that B notes that A2's clock shows 10 years elapsed when B passes by A2. And when you say "for person B to make it to Star 2 ... person A observes person’s B’s clock to have elapsed a time of 5 years", a relativity text would take this to mean that A2 notes that B's clock elapsed 5 years when B passes by A2. (A relativity text could assume that both A's and B's clock were reset to 0 when they passed each other, as was A2's clock in A/A2's frame.)

Now we wonder, if A2's clock runs at half-speed as measured in B's frame, how could A2's clock elapse double the time that B's clock did, by when B makes it to A2? Wouldn't A2's clock elapse half the time (2.5 years) instead? The answer is that A2's clock didn't elapse double the time that B's clock did; it indeed elapsed half, 2.5 years. If B had a helper B2 at rest in B's frame and passing by A2 at the moment in B's frame that B passes by A, B2 would note that A2's clock is not at 0 but rather at 7.5 years. The difference between t=0 and t=7.5 is explained by relativity of simultaneity. As recorded by observers at rest with respect to B's frame and recorded simultaneously in that frame, nearby clocks at rest with respect to A's frame (and synchronized in A's frame) show greater elapsed times the further away they are in B's direction of motion.