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Suppose two pointed masses are given in space. Suppose further that one of the masses has a given velocity at (local) time 0. Is there a way to compute its position in a future time?

Neglecting general relativity, I will simply compute an integral, but with general relativity, we see that the metric of the space changes with time, so I need to compute an integral with respect to a measure that changes along time.

Can this be done? If so, how?

Thank you!

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If the moving mass is small enough, you can do this using the geodesic equation,

$$\frac{\mathrm{d}^2x^\lambda}{\mathrm{d}t^2} + \Gamma^{\lambda}_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}t}\frac{\mathrm{d}x^\nu}{\mathrm{d}t} = 0$$

This is essentially the general relativistic equivalent of Newton's second law: it's a differential equation that governs how a test particle's position changes in time. The connection coefficients $\Gamma^{\lambda}_{\mu\nu}$ (also called Christoffel symbols) can be calculated from the metric. So if the metric is known, even if it's time-dependent, you can calculate the connection coefficients in your desired reference frame, plug them in, and find a solution to the geodesic equation that tells you how the particle will move. There may not be an analytic solution, but in all but the most extreme cases you can either solve the equation numerically, or make some approximation that might make it analytically solvable.

If the moving particle is not small enough to be considered a test particle, then the situation becomes more complicated because the metric, and thus the connection coefficients, will depend on the motion of the test particle itself. So you wind up with a coupled system of three equations: the geodesic equation for the moving particle, the geodesic equation for the other particle, and the Einstein equation

$$G^{\mu\nu} = 8\pi T^{\mu\nu}$$

which tells you how the metric changes in response to the motion of the two particles. In this case it's highly unlikely that you could find an analytic solution, but you could potentially still use a numeric differential equation solver, at least for some range of time. (All the equations are nonlinear so it's likely that your solution would lose accuracy quickly.)

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