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I understand that the formula for time dilation is given as

$$T = T_0\gamma = \frac{T_0}{\sqrt{1-v^2/c^2}}$$

Where T is moving with velocity v seen from $T_0$. Though, this is when an event is occuring, since the value for $T$ would in any case of $v>0$ be greater than $T_0$, meaning that the event takes longer time for $T$. However, in a situation like the twin paradox, if we ignore acceleration and all that, $T_0$ should be greater than $T$, since the time would tick slower for $T$. So what is the exact formula for such a situation, how is it derived, and is it true that abovementioned formula is only true for an event happening?

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  • $\begingroup$ You seem to have missed the starting problem that leads up to the twin paradox: for two unaccelerated observer they both see the other's clocks running slowly. (This is resolved by examining the situation in terms of the relativity of simultaneity and seeing that the two observers necessarily compare different pairs of events in space time events in space time in reaching their conclusions). $\endgroup$ – dmckee May 27 '18 at 19:54
  • $\begingroup$ See physics.stackexchange.com/questions/237958/…, physics.stackexchange.com/a/111089/520 and others $\endgroup$ – dmckee May 27 '18 at 19:58
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In your case when the twin on the spaceship travels with constant speed, the twins will see each other age at the same rate. Both of them could say that the other one is traveling relative to him, so the other one should age slower. But they age at the same rate in this case. This is because speed is simmetrically relative.

The aging difference comes when the traveling twin gets to the point of return. There, he has to accelerate and decelerate. That is the way for him to return. At that point, because acceleration is absolute as per GR, the traveling twin will age slower.

The reason for that is that as per GR, the object (his spaceship) is accelerating when turning, and that effect is the same as a gravitational field, and in a stronger (stronger then earth) gravitational field, the traveling twin will slow down in the time dimension, he will age slower (compared to the twin on earth).

Why that is happening is because the four speed vector's magnitude has to stay c. If the traveling twin accelerates, his spatial vectors will change, and because the magnitude of the four speed vector has to stay c, the time component will decrease to compensate. So the spaceship and the traveling twin will move in the time dimension slower at the point of return, and so he will age slower then the twin on earth.

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