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It is said that a clock on a spaceship runs slower than one kept on earth.
The clock on earth can be taken to be at rest with respect to the events on earth.So, the clock on the spaceship is in motion with respect to the events on earth.Let the time interval between two events on earth be $t_1$ as measured by the clock on earth. Thus, the interval as measured by the clock on the spaceship will be $t_2=t_1/(1-v^2/c^2)^{1/2}$ [ v= speed of spaceship w.r.t. earth and c= speed of light], which makes $t_2>t_1$. So, shouldn't the clock on the spaceship run faster?

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  • $\begingroup$ is it possible you have the equation backwards? $\endgroup$ – Jim Aug 15 '16 at 13:14
  • $\begingroup$ well, all time intervals measured from a frame which is not at rest wrt events, i.e. the 2 events don't take place at the same place in the frame, are improper time intervals. Improper time intervals are larger than proper time intervals. If that knowledge is correct then the equation should be right. right? $\endgroup$ – D.K. Aug 15 '16 at 13:20
  • $\begingroup$ Improper time intervals of the same proper duration are larger than proper time intervals, yes. But that's a fancy way of saying that the improper time interval corresponding to the transform of a proper time interval would have a shorter proper duration $\endgroup$ – Jim Aug 15 '16 at 13:28
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    $\begingroup$ The accepted answer for this question addresses yours as well, I think: Twin paradox in case of two twins that don't meet $\endgroup$ – Michael Seifert Aug 15 '16 at 13:31
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The eqation you state is correct, however your interpretation is a bit off.

As you say the t's in the equation are time intervals between ticks on the clock. A larger time interval (or period) means less ticks on the space station's clock compared to the earth's clock, thus running slower, as seen by an observer on earth.

However an observer in the space station will see the clock on earth running slow compared to the clock in the space station.

Wikipedia does a pretty good job explaining time dilation

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"So, shouldn't the clock on the spaceship run faster?"

According to special relativity, the observer on the spaceship will discover that his clock runs faster. This obvious result is fatal for the theory but there are deep psychological reasons (accumulated over the last century) why refutation along this line of argument is impossible.

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  • $\begingroup$ I'd say the reasons are more mathematical and observational than psychological. Why do you say that special relativity says the clock runs faster? It doesn't say that. Furthermore, what psychological reasons could there be that inspire students newly learning the subject to not question the theory, if it were so obviously flawed. Furthermore, given the extraordinary amount of fame and wealth that would accompany proving relativity wrong, why would any sane scientist choose to ignore evidence against it? I think it is you who are unable to see the truth of the matter $\endgroup$ – Jim Aug 15 '16 at 16:13
  • $\begingroup$ "Why do you say that special relativity says the clock runs faster? It doesn't say that." It says that the observer on the spaceship will see stationary clocks running slower and his own (traveling) clocks running faster. $\endgroup$ – Pentcho Valev Aug 15 '16 at 18:20
  • $\begingroup$ Ah, I see what you mean. In the spacecraft frame, clocks on Earth would move slower and vice versa. That's the twin paradox. It has an easy to understand resolution. Look it up. If you still disagree, I'm sure someone can point you to the many other questions on this site where someone disagreed with the twin paradox solution and were corrected $\endgroup$ – Jim Aug 15 '16 at 18:23
  • $\begingroup$ "given the extraordinary amount of fame and wealth that would accompany proving relativity wrong" No, such people become unpersons: ebooks.adelaide.edu.au/o/orwell/george/o79n/chapter1.4.html George Orwell: "Withers, however, was already an unperson. He did not exist : he had never existed." $\endgroup$ – Pentcho Valev Aug 15 '16 at 18:35
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    $\begingroup$ I'm not going to ask what an unperson is. All I know is that a successful refutation of relativity would mean prizes and grant money forever for that person or group of people. You think Einstein is a big name? What do you think would happen to the guy that proves him wrong? Famous. Whatever you think, I promise you that a combination of the egos of scientists and the insistence of the scientific method would make it impossible to keep such an obvious refutation of relativity hidden $\endgroup$ – Jim Aug 15 '16 at 19:12

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