Is there any formula, preferably in terms of energy, for the time dilation an object experiences taking both relativistic velocity and mass into account? I see both formulas frequently, but haven't been able to find a solution combining both. Do the time dilation factors just add or is it a more complicated solution?
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$\begingroup$ this is a hard problem, since it possibly involves gravitational fields varying at relativistic speeds, as well as the dependence of the velocity on the gravitational field. In principle, it should be possible to calculate this simply by integrating the proper time along a worldline through curved space, but the calculation will likely be tedious. $\endgroup$– paulinaCommented May 30 at 14:36
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$\begingroup$ I won't dupehammer this but I think it is covered by Are gravitational time dilation and the time dilation in special relativity independent? $\endgroup$– John RennieCommented May 30 at 15:09
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$\begingroup$ See also Is gravitational time dilation different from other forms of time dilation? $\endgroup$– John RennieCommented May 30 at 15:10
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Since you prefer a formula based on energy, the appropriate place to start would be the weak-field metric. Not all spacetimes have a notion of gravitational potential energy, but the weak-field metric does. This is given by $$ds^2=-c^2 d\tau^2 = -\left( 1+\frac{2 U}{c^2} \right) c^2 dt^2 + \left(1-\frac{2U}{c^2}\right)(dx^2+dy^2+dz^2)$$ where $U$ is the usual gravitational potential, $t$, $x$, $y$, and $z$ are the usual coordinates, $s$ is the spacetime interval, and $\tau$ is the proper time on a physical clock.
Now, we can divide by $-c^2 dt^2$ to get $$\frac{d\tau^2}{dt^2}=1-\frac{v^2}{c^2}+\frac{2 U}{c^2}(c^2+v^2)$$ Finally the time dilation is $\gamma=dt/d\tau$ so $$\gamma = \left( 1-\frac{v^2}{c^2}+\frac{2 U}{c^2}(c^2+v^2) \right)^{-1/2}$$
For more information about the weak field metric see here. This is the desired version including both energy and velocity. However, be aware that the weak field limit becomes less valid as $v$ approaches $c$ and as $U$ gets large. More general situations will need to be dealt with on a case-by-case basis as they will not generally have a notion of gravitational energy.
