Einstein–Infeld–Hoffmann equations

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Timeline Tests 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
Equations Formalisms 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 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
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman 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
v t e

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:

${\displaystyle {\begin{aligned}{\vec {a}}_{A}&=\sum _{B\not =A}{\frac {Gm_{B}{\vec {n}}_{BA}}{r_{AB}^{2}}}\\&{}\quad {}+{\frac {1}{c^{2}}}\sum _{B\not =A}{\frac {Gm_{B}{\vec {n}}_{BA}}{r_{AB}^{2}}}\left[v_{A}^{2}+2v_{B}^{2}-4({\vec {v}}_{A}\cdot {\vec {v}}_{B})-{\frac {3}{2}}({\vec {n}}_{AB}\cdot {\vec {v}}_{B})^{2}\right.\\&{}\qquad {}\left.{}-4\sum _{C\not =A}{\frac {Gm_{C}}{r_{AC}}}-\sum _{C\not =B}{\frac {Gm_{C}}{r_{BC}}}+{\frac {1}{2}}(({\vec {x}}_{B}-{\vec {x}}_{A})\cdot {\vec {a}}_{B})\right]\\&{}\quad {}+{\frac {1}{c^{2}}}\sum _{B\not =A}{\frac {Gm_{B}}{r_{AB}^{2}}}\left[{\vec {n}}_{AB}\cdot (4{\vec {v}}_{A}-3{\vec {v}}_{B})\right]({\vec {v}}_{A}-{\vec {v}}_{B})\\&{}\quad {}+{\frac {7}{2c^{2}}}\sum _{B\not =A}{\frac {Gm_{B}{\vec {a}}_{B}}{r_{AB}}}+O(c^{-4})\end{aligned}}}$

where:

${\displaystyle {\vec {x}}_{A}}$ is the barycentric position vector of body A

${\displaystyle {\vec {v}}_{A}=d{\vec {x}}_{A}/dt}$ is the barycentric velocity vector of body A

${\displaystyle {\vec {a}}_{A}=d^{2}{\vec {x}}_{A}/dt^{2}}$ is the barycentric acceleration vector of body A

${\displaystyle r_{AB}=|{\vec {x}}_{A}-{\vec {x}}_{B}|}$ is the coordinate distance between bodies A and B

${\displaystyle {\vec {n}}_{AB}=({\vec {x}}_{A}-{\vec {x}}_{B})/r_{AB}}$ is the unit vector pointing from body B to body A

${\displaystyle m_{A}}$ is the mass of body A.

${\displaystyle c}$ is the speed of light

${\displaystyle G}$ 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.^[1]

References

Further reading

• Einstein, A.; Infeld, L.; Hoffmann, B. (1938). "The Gravitational Equations and the Problem of Motion". Annals of Mathematics. Second series. 39 (1): 65–100. Bibcode:1938AnMat..39...65E. doi:10.2307/1968714. JSTOR 1968714.
• Kovalevsky, Jean; Seidelmann, P. Kenneth (2004). Fundamentals of Astrometry. New York: Cambridge University Press. p. 173. ISBN 0521642167.
• Landau, Lev; Lifshitz, Evgeny (1971). The classical theory of fields. Oxford: Pergamon Press. p. 337.

This relativity-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e

This page was last edited on 6 January 2023, at 03:25