Gravity vs. Velocity Time Dilation

Question

Can some one give me the unified formula for Gravity Time Dilation and Velocity Time Dilation.

Answer 1

The unified formula used in General Relativity is $$d\tau=\sqrt{\sum_{\mu=0}^3\sum_{\nu=0}^3 g_{\mu\nu}dx^\mu dx^\nu},$$ which by Einstein's notation (summation over doubly repeating indices is implicit) is also written as $$d\tau=\sqrt{g_{\mu\nu}dx^\mu dx^\nu}.$$ Here $d\tau$ is the proper time "felt" or measured by particle moving in the spacetime, for some infinitesimal segment of the world line; $dx^\mu=(dx^0,dx^1,dx^2,dx^3)=(dt,dx,dy,dz)$ are coordinate components of the displacement along that world line, and $g_{\mu\nu}$ (16 numbers) is the metric tensor for the specific curved spacetime where the motion takes place. The metric tensor both shows the shape of the curved spacetime and the way the soordinate system is drawn onto it. For the longer part of the world line, you should integrate that formula over the world line, so it becomes $$\tau=\int_L\sqrt{g_{\mu\nu}dx^\mu dx^\nu}.$$

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If we divide the formula by $dt$, we get (latin indices are summed only over the 1...3 range, so they imply only spatial coordinates): $$\frac{d\tau}{dt}=\sqrt{g_{00}+(g_{0i}+g_{i0})v^i+g_{ij}v^i v^j}.$$ $d\tau/dt$ is the time dilation factor with respect to the coordinate $t$. For the flat spacetime and Cartesian coordinates, $g_{00}=1$, $g_{11}=g_{22}=g_{33}=-1$ and all other $g_{\mu\nu}$'s are zero. For the spherical non-rotating gravitating body (Schwarzschild metric) and polar coordinates, $$g_{tt}=1-\frac{2GM}{r},\quad g_{rr}=\frac{1}{1-\frac{2GM}{r}},\quad g_{\theta\theta}=r^2,\quad g_{\varphi\varphi}=r^2\sin^2\theta,$$ and all other $g_{\mu\nu}$'s are zero.

Besides that, GR considers very general spacetimes, so the general formula is the only one used everywhere.

Answer 2

The time dilation due to velocity and due to spacetime curvature can't be separated. Both are derived from the metric. There isn't a general formula for this because it depends on the metric in question. For example in my answer to the question A clock in freefall I calculate the time dilation for an observer falling from infinity towards a black hole, but the resulting equation applies only to that situation and isn't general.

Generally speaking the metric gives you an expression for the proper time, and the time dilation is normally taken to be the ratio of proper time to coordinate time, $d\tau/dt$. If you can give us some idea of the system you're looking at we can explain how to calculate this ratio.