In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
for the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. The approximations are expanded in small parameters which express orders of deviations from
Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of
effective field theories. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.
Expansion in 1/''c''2
The post-Newtonian approximations are
expansions in a small parameter, which is the ratio of the velocity of the matter that creates the gravitational field, to 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 ...
, which in this case is more precisely called the ''
speed of gravity
In classical theories of gravitation, the changes in a gravitational field propagate. A change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational field which it produces. In ...
''. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to
Newton's law of gravity. A systematic study of post-Newtonian expansions within hydrodynamic approximations was developed by
Subrahmanyan Chandrasekhar and co-workers in the 1960s.
Expansion in ''h''
Another approach is to expand the equations of general relativity in a power series in the deviation of the metric from its
value in the absence of gravity
:
To this end, one must choose a coordinate system in which the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of
all have absolute values less than 1.
For example, if one goes one step beyond
linearized gravity
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects ...
to get the expansion to the second order in ''h'':
:
:
Expansions based only on the metric, independently from the speed, are called
post-Minkowskian expansions (PM expansions).
Uses
The first use of a PN expansion (to first order) was made 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 ...
in calculating the
perihelion precession of Mercury's orbit. Today, Einstein's calculation is recognized as a first simple case of the most common use of the PN expansion: solving the
general relativistic two-body problem, which includes the emission of
gravitational wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s.
Newtonian gauge
In general, the perturbed metric can be written as
: