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I have a particle near two Schwarzschild black holes. Let the black holes remain at rest so that only the particle is moving for the observer. We are in a plane. I calculate the distance travelled by the particle in one frame for each black hole, using Schwarzschild solution. Now the problem is if it is possible to sum the velocities. We can name the variables $\Delta\tau_1$, $\Delta\tau_2$, $\Delta r_1$, $\Delta r_2$, $\Delta\phi_1$, $\Delta\phi_2$, and $\Delta t_1=\Delta t_2$, which are converted in Cartesian coordinates. Their values are known, but that I need is to find the values the "resultant" velocity if it is possible. How should I proceed?

If it is not possible, however is there a method to find the velocity of a particle near two Schwarzschild black holes if sufficient data are known?

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The Schwarzschild solution doesn't apply to a spacetime with two black holes, but numerical methods have been derived to find the metric tensor in a spacetime with two black holes. From the metric tensor, the motion of test particles can, in principle, be calculated. –  Jerry Schirmer Jan 11 at 22:00

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The Schwarzschild solution doesn't apply to a spacetime with two black holes, but numerical methods have been derived to find the metric tensor in a spacetime with two black holes.

From the metric tensor, the motion of test particles can, in principle, be calculated.

Beyond this, Cartensian coordinates will almost certainly not be definable in such a spacetime for an area larger than a neighborhood around a point.

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