Post-Newtonian expansion

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]

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.

${\displaystyle h_{\alpha \beta }=g_{\alpha \beta }-\eta _{\alpha \beta }\,.}$

To this end, one must choose a coordinate system in which the eigenvalues of ${\displaystyle h_{\alpha \beta }\eta ^{\beta \gamma }\,}$ 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:

${\displaystyle g^{\mu \nu }\approx \eta ^{\mu \nu }-\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \nu }+\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \nu }\,.}$

${\displaystyle {\sqrt {-g}}\approx 1+{\tfrac {1}{2}}h_{\alpha \beta }\eta ^{\beta \alpha }+{\tfrac {1}{8}}h_{\alpha \beta }\eta ^{\beta \alpha }h_{\gamma \delta }\eta ^{\delta \gamma }-{\tfrac {1}{4}}h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \alpha }\,.}$

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	+	${\displaystyle v^{2}}$	+	${\displaystyle v^{4}}$	+	${\displaystyle v^{6}}$	+	${\displaystyle v^{8}}$	+	${\displaystyle v^{10}}$	+	${\displaystyle v^{12}}$	+	${\displaystyle v^{14}}$	+	...)	${\displaystyle G^{1}}$
2PM			( 1	+	${\displaystyle v^{2}}$	+	${\displaystyle v^{4}}$	+	${\displaystyle v^{6}}$	+	${\displaystyle v^{8}}$	+	${\displaystyle v^{10}}$	+	${\displaystyle v^{12}}$	+	...)	${\displaystyle G^{2}}$
3PM					( 1	+	${\displaystyle v^{2}}$	+	${\displaystyle v^{4}}$	+	${\displaystyle v^{6}}$	+	${\displaystyle v^{8}}$	+	${\displaystyle v^{10}}$	+	...)	${\displaystyle G^{3}}$
4PM							( 1	+	${\displaystyle v^{2}}$	+	${\displaystyle v^{4}}$	+	${\displaystyle v^{6}}$	+	${\displaystyle v^{8}}$	+	...)	${\displaystyle G^{4}}$
5PM									( 1	+	${\displaystyle v^{2}}$	+	${\displaystyle v^{4}}$	+	${\displaystyle v^{6}}$	+	...)	${\displaystyle G^{5}}$
6PM											( 1	+	${\displaystyle v^{2}}$	+	${\displaystyle v^{4}}$	+	...)	${\displaystyle G^{6}}$
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]

The first use of a PN expansion (to first order) was made by Albert Einstein in calculating the perihelion precession of Mercury's orbit. Today, Einstein's calculation is recognized as a common example of applications of PN expansions, solving the general relativistic two-body problem, which includes the emission of gravitational waves.

In general, the perturbed metric can be written as[8]

${\displaystyle ds^{2}=a^{2}(\tau )\left[(1+2A)d\tau ^{2}-2B_{i}dx^{i}d\tau -\left(\delta _{ij}+h_{ij}\right)dx^{i}dx^{j}\right]}$

where ${\displaystyle A}$, ${\displaystyle B_{i}}$ and ${\displaystyle h_{ij}}$ are functions of space and time. ${\displaystyle h_{ij}}$ can be decomposed as

${\displaystyle h_{ij}=2C\delta _{ij}+\partial _{i}\partial _{j}E-{\frac {1}{3}}\delta _{ij}\Box ^{2}E+\partial _{i}{\hat {E}}_{j}+\partial _{j}{\hat {E}}_{i}+2{\tilde {E}}_{ij}}$

where ${\displaystyle \Box }$ is the d'Alembert operator, ${\displaystyle E}$ is a scalar, ${\displaystyle {\hat {E}}_{i}}$ is a vector and ${\displaystyle {\tilde {E}}_{ij}}$ is a traceless tensor. Then the Bardeen potentials are defined as

${\displaystyle \Psi \equiv A+H(B-E'),+(B+E')',\quad \Phi \equiv -C-H(B-E')+{\frac {1}{3}}\Box E}$

where ${\displaystyle H}$ is the Hubble constant and a prime represents differentiation with respect to conformal time ${\displaystyle \tau \,}$.

Taking ${\displaystyle B=E=0}$ (i.e. setting ${\displaystyle \Phi \equiv -C}$ and ${\displaystyle \Psi \equiv A}$), the Newtonian gauge is

${\displaystyle ds^{2}=a^{2}(\tau )\left[(1+2\Psi )d\tau ^{2}-(1-2\Phi )\delta _{ij}dx^{i}dx^{j}\right]\,}$.

Note that in the absence of anisotropic stress, ${\displaystyle \Phi =\Psi }$.

A useful non-linear extension of this is provided by the non-relativistic gravitational fields.

• Coordinate conditions
• Einstein–Infeld–Hoffmann equations
• Linearized gravity
• Parameterized post-Newtonian formalism

• "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld Archived 2012-03-08 at the Wayback Machine
• Blanchet, Luc (2014). "Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries". Living Reviews in Relativity. 17 (1): 2. arXiv:1310.1528. Bibcode:2014LRR....17....2B. doi:10.12942/lrr-2014-2. PMC 5256563. PMID 28179846.