In
general relativity, post-Newtonian expansions (PNexpansions) are used for finding an approximate solution of
Einstein field equations for the
metric tensor. The approximations are expanded in small parameters that express orders of deviations from
Newton's law of universal gravitation. 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/c2
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, which in this case is more precisely called the speed of gravity.[1] 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 his colleagues in the 1960s.[2][3][4][5][6]
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
eigenvalues of all have absolute values less than 1.
For example, if one goes one step beyond
linearized gravity 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).
0PN
1PN
2PN
3PN
4PN
5PN
6PN
7PN
1PM
( 1
+
+
+
+
+
+
+
+
...)
2PM
( 1
+
+
+
+
+
+
+
...)
3PM
( 1
+
+
+
+
+
+
...)
4PM
( 1
+
+
+
+
+
...)
5PM
( 1
+
+
+
+
...)
6PM
( 1
+
+
+
...)
Comparison table of powers used for PN and PM approximations in the case of two non-rotating bodies.
0PN corresponds to the case of Newton's theory of gravitation. 0PM (not shown) corresponds to the
Minkowski flat space.[7]
^Chandrasekhar, S. (1967). "The post-Newtonian effects of General Relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the MacLaurin spheroids". The Astrophysical Journal. 147: 334.
Bibcode:
1967ApJ...147..334C.
doi:
10.1086/149003.