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Currently, I compute the force between two gravitational interacting particles in a simulation with $n$ bodies according to

$$F = G\frac{m_1m_2}{r^2}.$$

Doing this, however, assumes that all bodies in the simulation interact instantly with each other meaning that I assume the speed of light to be infinity.

Let's assume I want so simulate the formation of galaxy clusters where the propagation velocity of light plays an essential role. How would one incorporate the fact that the propagation speed of light is finite into an $n$-body simulation? Any ideas?

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You could use the Einstein-Infeld-Hoffman equations, derivable from General Relativity, which have Newton’s inverse square force as their dominant term but also include first-order relativistic corrections.

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    $\begingroup$ What does $C$ stand for in the Einstein-Infeld-Hoffman equations? A third body? $\endgroup$
    – Gilfoyle
    Commented Oct 26, 2019 at 17:45
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    $\begingroup$ Yes. The B and C indexes run over the other particles, to compute their effect on the particle with index A. So it is very interesting that when you include general-relativistic corrections, not only is the force not simply inverse-square, but it is no longer a simple superposition of forces between each pair of particles. This is because GR is nonlinear. $\endgroup$
    – G. Smith
    Commented Oct 26, 2019 at 17:58

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