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Imagine Alice is near a massive black hole and Bob is on the Earth.

Obviously their gravitational fields are different for each other. In other words, their spacetime is flat for themselves but curved for each other. Bob sees spacetime of Alice is curved. And Alice sees spacetime of Bob is curved. Thus time should run slower for each other.

However, in reality it is said that from Alice's point of view Bob's time is running faster, and from Bob's point of view Alice's time is running slower.

If this is true, how can we say that gravity is relative?

Update:

Based upon questions asked in comments. Here is update: Both Alice and Bob will agree that spacetime is more curved in Alice's vicinity. Thus Speed of one clock (because of it's gravity) can be measured regardless of gravity of the observer. Unlike velocity, which is dependent of velocity of observer, because velocity is relative.

Gravity is not relative to observer's gravity. Why do we say that gravity is relative?

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    $\begingroup$ You have a number of misconceptions here. Firstly curvature is not relative in the way you suggest - both Alice and Bob will agree that spacetime is more curved in Alice's vicinity - and secondly time dilation is not simply related to curvature anyway. $\endgroup$ Commented Apr 9 at 15:27
  • $\begingroup$ You are mixing the twin paradox of Kinetic Time Dilation in Special Relativity with Gravitational Time Dilation in General Relativity. They are two different things. Both people would agree that Alice ages slower than Bob. $\endgroup$ Commented Apr 9 at 15:58
  • $\begingroup$ "Obviously their gravitational fields are different for each other. " What did you think this means, and what made you think it was obvious? $\endgroup$
    – WillO
    Commented Apr 9 at 16:03
  • $\begingroup$ @foolishmuse Actually yes I am coming from there only. I thought the paradox should exists here as well if gravity is relative just as in velocity is relative. $\endgroup$ Commented Apr 9 at 16:06
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    $\begingroup$ @Wookie I got the point you made. Speed of one clock (because of it's gravity) can be observed and told independently. Unlike velocity of object which is dependent of velocity of another object (observer) because velocity is relative. With the of example you given it is clear that gravity is not relative to observer's gravity. That's my question was: how can we say that gravity is relative? $\endgroup$ Commented Apr 9 at 18:25

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The time dilation due to the curvature of spacetime is not symmetric between observers in the same way as their relative velocity is.

In Special Relativity, the time dilation factor $\gamma = (1-v^2/c^2)^{-1/2}$ involves the speed. It does not matter whether the velocity is $\pm v$, the time dilation factor is the same. Therefore both observers would claim that the other's clock is running slower.

That symmetry does not apply to the gravitational time dilation that arises from observers being at different positions with respect to a gravitational potential. All that matters here is the (signed) difference in potential and both observers agree that Bob is at a higher potential than Alice and therefore his clock runs relatively more quickly in comparison.

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  • $\begingroup$ Yes, we don't measure any time dilation except the one between the observers at relative positions to the gravitational potential. $\endgroup$
    – Wookie
    Commented Apr 9 at 17:25
  • $\begingroup$ I understand it fully and many thanks for answer. I am still not getting why then gravity is said to be relative entity when time dilation and other effects caused by it can be calculated independently regardless of observers' gravity $\endgroup$ Commented Apr 12 at 5:10
  • $\begingroup$ @user1976551 Time dilation is relative. You can only compare two clocks. And the difference in their apparent rates depends where the observers are. "All that matters here is the (signed) difference in potential." $\endgroup$
    – ProfRob
    Commented Apr 12 at 6:13

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