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I'm trying to understand something about time dilation on orbiting bodies in General Relativity. I was watching the Matt Parker video General Relativity: Top 05 mishaps and he seems to calculate the time dilation of the GPS satellite as ($r_s$ is the Schwartzchild Radius) :

$t_0 = t_f*\sqrt {1-r_s/r}$

The article in Wikipedia Gravitational Time Dilation under Circular Orbit is:

$t_0 = t_f*\sqrt {1-(3/2)*(r_s/r)}$

My logic says the orbiting satellite is in free-fall, so does not feel the effects of gravity

$t_0 = t_f$

engineercliff
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  • Not sure how you deduced that $t_0=t_f$ simply because the satellite is orbiting or in free-fall. Can you elaborate? – joseph h Mar 01 '21 at 03:17
  • I think they call that the equivalence principle? If you are weightless, you can't tell if you are in freefall or far away from any gravity source. – engineercliff Mar 01 '21 at 04:32
  • The equivalence principal says inertial mass and gravitational mass are the same and does not mean that the time observed locally is equivalent to that for a distant observer, which is what you are saying. So again, how did you come to that equality? – joseph h Mar 01 '21 at 06:08
  • I guess I misunderstood the equivalence principle. I thought it meant that any experiment would have equivalent results. Which of the other 2 is correct? – engineercliff Mar 01 '21 at 23:15
  • Is your question which of the above two equations is correct? – joseph h Mar 02 '21 at 01:38
  • Right. There are three equations, you say the last one is incorrect. Which of the other two is? Maybe you could format this as an answer, rather than having a long comment string? – engineercliff Mar 02 '21 at 01:46
  • I haven't watched the video, though the first equation is time dilation for a clock at rest in space, while the second one is for a clock orbiting earth. Two distinct cases, which is why they are different. It is still not clear what you mean by $t_0=t_f$ since this implies that the amount of time elapsed for the observer on earth is that same as that for the clock far away. Which means no time dilation to begin with. – joseph h Mar 02 '21 at 03:20

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