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However I am not satisfied with the answer. Let's try to be clear and use precise notations. Let's consider a non-charged and non-rotative sphere of mass M, an observer A associated with an inertial referential R very far away from the massive body, and an observer B associated with a non-inertial referential R' close to the massive body. If B is at rest with respect to the massive body, the Schwarzschild metric gives:

$$ {d \tau '}^{2} = \left(1 - \frac{r_\mathrm{s}}{r} \right)\,dt^2 $$

with:

  • $r_{s} = 2 G M/c^{2}$, G being the gravitationnal constant
  • $\tau '$ the proper time of B
  • $t$ the coordinate time associated with B seen from A
  • r is the radial coordinate (here constant)

Here are my questions:

  1. Do we agree that here $t$ is the coordinate time associated with B (seen from A) and not the proper time of A even if $dr=0$ and $d\phi = 0$ ? I am asking this question because in special relativity, when we take $dx=0$, we get $dt=d\tau$
  2. What is the relationship between $d\tau$ (with $\tau$ the proper time of A) and $dt'$ (with $t'$ the coordinate time associated with A seen from B) ?
  3. Is there a relationship between $d\tau$ (proper time of A) and $d\tau'$ (proper time of B)?

Thank you!

François Ritter
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You're talking about things like "B seen from A" and "A seen from B," which doesn't make sense in general relativity. In GR, we don't have global frames of reference, only local ones. When we say that there is time dilation for A relative to B by a factor $k$, we mean that if A sends signals to B or vice versa, and these signals propagate at $c$, then they are received in a form in which their time-variation is compressed or expanded by the factor $k$, exactly as we would observe if it were a Doppler effect in SR.

Do we agree that here t is the coordinate time associated with B (seen from A) and not the proper time of A[...]

Coordinates are not associated with observers, so there is no concept of a corodinate time associated with B. A coordinate time in the Schwarzschild coordinate system is just the coordinate time in that coordinate system.

Because A happens to be very far away, and is at rest, and because the Schwarzschild coordinates are constructed in a certain way, it is true that A's proper time is the same as the $t$ coordinate change between events at A's position. So this is the other way around compared to the way you said it.

Is there a relationship between $d\tau$ (proper time of A) and $d\tau'$ (proper time of B)?

There is such a relationship if they are exchanging signals. For example, if A sends a signal, waits time $d\tau$, and then sends another signal, then B will receive those signals separated by a time interval $d\tau'$.

  • Thank you for your comment. From what I read, are you ok that in this case: $${d \tau '}^{2} = \left(1 - \frac{r_\mathrm{s}}{r} \right),d\tau^2$$ with $\tau$ the proper time of A and $\tau'$ the proper time of B? – François Ritter Oct 15 '18 at 19:35
  • @FrançoisRitter: Only if you interpret it as in the paragraph of my answer that says "There is such a relationship..." In your comment, it sounds like you might still be conceptualizing this as if it were a transformation between two different frames of reference, which is not what it is. –  Oct 15 '18 at 23:37