Einstein-Infeld-Hoffmann equation

The Einstein-Infeld-Hoffmann equation is an equation of motion that was jointly developed by Albert Einstein , Leopold Infeld and Banesh Hoffmann . It is a differential equation that approximately describes the kinetics of a system of point masses under mutual gravitational attraction, taking into account general relativistic effects. It uses a post-Newtonian extension of the first order and is therefore valid in areas in which the speeds of the masses are small compared to the speed of light and the gravitational fields that act on them are correspondingly weak.

For a system of masses denoted by the indices , the barycentric acceleration vector of the body is given by: ${\displaystyle N}$ ${\displaystyle A=1,\dotsc ,N}$ ${\displaystyle A}$

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

The following applies:

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

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

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

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

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

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

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

${\displaystyle G}$is the gravitational constant .

The first term on the right corresponds to the Newtonian gravitational acceleration on . Newton's equation of motion is obtained in the limit value . ${\displaystyle A}$ ${\displaystyle c\to \infty }$

The acceleration of a certain body depends on the accelerations of all other bodies. Since the acceleration vector appears on both sides of the equation, the system of equations must be solved iteratively. In practice, however, Newton's equation of motion is sufficient to achieve sufficient accuracy.

application

The Einstein-Infeld-Hoffmann equation is used in determining the International Celestial Reference System (ICRF). For this purpose, the ephemeris of the planets is calculated by integrating the equation, which results in the dynamic implementation of the ICRF.

literature

• Albert Einstein, L. Infeld, B. Hoffmann: The Gravitational Equations and the Problem of Motion . Annals of Mathematics Second series 39 (1): pp. 65-100, 1938.
• Jean Kovalevsky, P. Kenneth Seidelmann: Fundamentals of Astrometry , New York: Cambridge University Press. S. 173., 2004.

Individual evidence

1. ^ Standish, Williams: Ephemerides of the Sun, Moon, and Planets , p. 4.