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I have some doubts about getting the time-dilation concept right, so I would like to suggest a scenario and see if my conclusions are right or wrong.

Person A is on Earth, Person B is on a spaceship. Both are on Earth at $t=0$. Person B leaves Earth and heads for planet X. When Person B reaches planet X his clock reads $t_B$, and person A's clock reads $t_A$.

Now, simplified time dilation formula is $$t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}} $$ where $t_0$ is the proper time.

My thoughts on this:

Person B's time is the only proper time because only in his frame the two events (leaving Earth/arriving at planet X) happened in the same place - they have the same $x$-coordinate. So, by this formula, Person A measures longer time compared to Person B's clock.

Now, if my thoughts are right (?) I am troubled with this statement:

Moving clocks run slower.

From Person B's perspective, Person A is moving. But, he does not see this clock run slower. What am I concluding wrong? Should all clocks moving relative to us run slower? Is there a situation in which Person B (moving away from Earth) would see Person A's clock run slower? Maybe when looking at an event that happened and ended on Earth?

Sorry about many questions, but this all bugs me a lot.

I kind of think that all problems with conceptually understanding dilation arise when comparing one single event across two frames, if the event happened and ended only in one of these frames.

Maybe this statement has sense only when comparing two events: one in frame A, and the same event (like a heartbeat) in frame B?

Help me wrap this up!

Jakov
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2 Answers2

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If person A on Earth looks at his own clock A', he sees his proper time, because he is at rest with respect to his own clock. Similarly, if person B on the rocket looks at his own clock B', he sees his proper time as well. But if person A looks at the other guy's clock B', he will see it being slower, because it is moving with respect to him. Similarly, if person B on the rocket looks at the clock A' on Earth, he will see it being slower as well.

Photon
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  • Your statement "Similarly, if person B on the rocket looks at the clock A' on Earth, he will see it being slower as well." I know that this is a basic premise of SR, but has it ever been actually tested? We all know that the clocks on GPS satellites are adjusted against Earth time because of their speed and altitude. But any other satellite, or an intercontinental missile, that receives their position information from the GPS satellite has their clock adjusted against Earth time, and not against the GPS satellite time. So has this ever been tested? – foolishmuse May 06 '22 at 17:40
  • I don't know, to be honest, but I assume that all these relativistic effects are taken into account, otherwise the accuracy would be miserable. – Photon May 07 '22 at 05:23
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So $A$'s clock is going to tick off a time of

$$t_A=t_0$$

which is the proper time between the two relevant events because $A$ is at rest.

As $B$ approaches the planet, he has been moving at $\beta$ in the unprimed frame, so his clock approaches:

$$ t'_B = \frac{t_A}{\gamma} $$

where $\gamma$ is the Lorentz factor.

Since $B$ is at rest in the primed frame, $B$ sees $A$'s clock running slow, and thus says:

$$ t'_A=\frac{t'_B}{\gamma}=\frac{t_A}{\gamma^2} $$

Note that $B$ is still moving w.r.t. to $A$.

At the planet, $B$ stops and is now reenters the unprimed frame of $A$. Their clocks now both run at the same speed, and they both agree on the simultaneity of events. $B$'s clock reads $t_0/\gamma$, while $A's$ clock reads $t_0$.

JEB
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  • If unprimed refers to times measured in the unprimed frame ($A$ is stationary) then shouldn't it be: $$t_B=t_A/\gamma$$ instead of $t_B'=t_A/\gamma$? – Jakov May 06 '22 at 18:35
  • @Jakov $t_B = L/\beta$, where $L$ is the distance traveled. That's how long it takes for $A$ to see it get to the planet, and it equals $t_A$, since $t_A$ is time $A$ see it to get to the planet. $t_B'$ is the time elapsed on $B$'s clock, which is $L'/\beta=(L/\gamma)/\beta=t_A/\gamma$ – JEB May 07 '22 at 01:42