The Einstein–Infeld–Hoffmann equations of motion, jointly derived by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
,
Leopold Infeld
Leopold Infeld (20 August 1898 – 15 January 1968) was a Polish physicist who worked mainly in Poland and Canada (1938–1950). He was a Rockefeller fellow at Cambridge University (1933–1934) and a member of the Polish Academy of Sciences.
Ea ...
and
Banesh Hoffmann
Banesh Hoffmann (6 September 1906 – 5 August 1986) was a British mathematician and physicist known for his association with Albert Einstein.
Life
Banesh Hoffmann was born in Richmond, Surrey, on 6 September 1906. He studied mathematics and ...
, 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
In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of ...
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:
:
where:
:
is the barycentric position vector of body A
:
is the barycentric velocity vector of body A
:
is the barycentric acceleration vector of body A
:
is the coordinate distance between bodies A and B
:
is the unit vector pointing from body B to body A
:
is the mass of body A.
:
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
:
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.
[Standish, Williams. Orbital Ephemerides of the Sun, Moon, and Planets, Pg 4. ]
References
Further reading
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Differential equations
General relativity
Albert Einstein
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