# Einstein–Infeld–Hoffmann equations

General relativity :
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$ :
Introduction
HistoryTimelineTests
Mathematical formulation :
Fundamental concepts
Equivalence principle
Special relativity
World line
Pseudo-Riemannian manifold :
Phenomena
Kepler problem
Gravitational lensing
Gravitational redshift
Gravitational time dilation
Gravitational waves
Frame-dragging
Geodetic effect
Event horizon
Singularity
Black hole
Spacetime
Spacetime diagrams
Minkowski spacetime
Einstein–Rosen bridge :
EquationsFormalisms
Equations
Linearized gravity
Einstein field equations
Friedmann
Geodesics
Mathisson–Papapetrou–Dixon
Hamilton–Jacobi–Einstein
Formalisms
ADM
BSSN
Post-Newtonian
Advanced theory
Kaluza–Klein theory
Quantum gravity :
Solutions
Schwarzschild (interior)
Reissner–Nordström
Einstein–Rosen waves
Wormhole
Gödel
Kerr
Kerr–Newman
Kerr–Newman–de Sitter
Kasner
Lemaître–Tolman
Taub–NUT
Milne
Robertson–Walker
Oppenheimer–Snyder
pp-wave
van Stockum dust
Weyl−Lewis−Papapetrou
Hartle–Thorne :
Scientists
Einstein
Lorentz
Hilbert
Poincaré
Schwarzschild
de Sitter
Reissner
Nordström
Weyl
Eddington
Friedmann
Milne
Zwicky
Lemaître
Oppenheimer
Gödel
Wheeler
Robertson
Bardeen
Walker
Kerr
Chandrasekhar
Ehlers
Penrose
Hawking
Raychaudhuri
Taylor
Hulse
van Stockum
Taub
Newman
Yau
Thorne
others :
Physics portal
 Category :
vte :
The Einstein–Infeld–Hoffmann equations of motion, jointly derived by Albert Einstein, Leopold Infeld and Banesh Hoffmann, are the differential equations describing the approximate dynamics of a system of point-like masses due to their mutual gravitational interactions, including general relativistic effects. It uses a first-order post-Newtonian expansion and thus is valid in the limit where the velocities of the bodies are small compared to the speed of light and where the gravitational fields affecting them are correspondingly weak.
Given a system of N bodies, labelled by indices A = 1, ..., N, the barycentric acceleration vector of body A is given by:
    ${\begin{aligned}{\vec {a}}_{A}&=\sum _{B\not =A}{\frac {Gm_{B}{\vec {n}}_{BA}}{r_{AB}^{2}}}\\&\quad+{\frac {1}{c^{2}}}\sum _{B\not =A}{\frac {Gm_{B}{\vec {n}}_{BA}}{r_{AB}^{2}}}\left[v_{A}^{2}+2v_{B}^{2}-4({\vec {v}}_{A}\cdot {\vec {v}}_{B})-{\frac {3}{2}}({\vec {n}}_{AB}\cdot {\vec {v}}_{B})^{2}\right.\\&\qquad\left.-4\sum _{C\not =A}{\frac {Gm_{C}}{r_{AC}}}-\sum _{C\not =B}{\frac {Gm_{C}}{r_{BC}}}+{\frac {1}{2}}(({\vec {x}}_{B}-{\vec {x}}_{A})\cdot {\vec {a}}_{B})\right]\\&\quad+{\frac {1}{c^{2}}}\sum _{B\not =A}{\frac {Gm_{B}}{r_{AB}^{2}}}\left[{\vec {n}}_{AB}\cdot (4{\vec {v}}_{A}-3{\vec {v}}_{B})\right]({\vec {v}}_{A}-{\vec {v}}_{B})\\&\quad+{\frac {7}{2c^{2}}}\sum _{B\not =A}{\frac {Gm_{B}{\vec {a}}_{B}}{r_{AB}}}+O(c^{-4})\end{aligned}}$
where:
    ${\vec {x}}_{A}$ is the barycentric position vector of body A
    ${\vec {v}}_{A}=d{\vec {x}}_{A}/dt$ is the barycentric velocity vector of body A
    ${\vec {a}}_{A}=d^{2}{\vec {x}}_{A}/dt^{2}$ is the barycentric acceleration vector of body A
    $r_{AB}=|{\vec {x}}_{A}-{\vec {x}}_{B}|$ is the coordinate distance between bodies A and B
    ${\vec {n}}_{AB}=({\vec {x}}_{A}-{\vec {x}}_{B})/r_{AB}$ is the unit vector pointing from body B to body A
    $m_{A}$ is the mass of body A.
    $c$ is the speed of light
    $G$ is the gravitational constant
    and the big O notation is used to indicate that terms of order c$^{−4}$ or beyond have been omitted.
The coordinates used here are harmonic. The first term on the right hand side is the Newtonian gravitational acceleration at A; in the limit as c → ∞, one recovers Newton's law of motion.
The acceleration of a particular body depends on the accelerations of all the other bodies. Since the quantity on the left hand side also appears in the right hand side, this system of equations must be solved iteratively. In practice, using the Newtonian acceleration instead of the true acceleration provides sufficient accuracy.