Einstein–Infeld–Hoffmann equations

Definition

Consider $N \ge 2$ point masses with masses $m_1, \dots, m_N > 0$, positions $x_a(t) \in \R^3$, velocities $v_a = \dot x_a$ and accelerations $a_a = \ddot x_a$, in harmonic coordinates with coordinate time $t$. Write $r_{ab} = \abs{x_a - x_b}$ and $n_{ab} = (x_a - x_b)/r_{ab}$ for $a \ne b$, let $G$ be the gravitational constant and $c$ the speed of light. Fix an index $a \in \set{1, \dots, N}$. The Einstein–Infeld–Hoffmann equations are $$\begin{aligned} a_a = {} & \sum_{b \ne a} \frac{G m_b\, n_{ba}}{r_{ab}^2} \Big[ 1 - \frac{4}{c^2} \sum_{d \ne a} \frac{G m_d}{r_{ad}} - \frac{1}{c^2} \sum_{d \ne b} \frac{G m_d}{r_{bd}} + \frac{\abs{v_a}^2}{c^2} + \frac{2\abs{v_b}^2}{c^2} - \frac{4\, v_a \cdot v_b}{c^2} \\ & \qquad - \frac{3}{2c^2} (n_{ab} \cdot v_b)^2 + \frac{1}{2c^2} (x_b - x_a) \cdot a_b \Big] \\ & + \frac{1}{c^2} \sum_{b \ne a} \frac{G m_b}{r_{ab}^2} \big[ n_{ab} \cdot (4 v_a - 3 v_b) \big] (v_a - v_b) + \frac{7}{2c^2} \sum_{b \ne a} \frac{G m_b\, a_b}{r_{ab}}. \end{aligned}$$ All terms of order $c^{-4}$ are dropped. So in the $c^{-2}$ terms each $a_b$ may be replaced by its Newtonian value $\sum_{d \ne b} G m_d n_{db}/r_{bd}^2$.