Post-Minkowskian expansion

In physics, precisely in the general theory of relativity, post-Minkowskian expansions (PM) or post-Minkowskian approximations are mathematical methods used to find approximate solutions of Einstein's equations by means of a power series development of the metric tensor.

Unlike post-Newtonian expansions (PN), in which the series development is based on a combination of powers of the velocity $v$ (which must be negligible compared to that of light $c$ ) and the gravitational constant $G$ , in the post-Minkowskian case the developments are based only on the gravitational constant, allowing analysis even at velocities close to that of light (relativistic).^[1]

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	0PN		1PN		2PN		3PN		4PN		5PN		6PN		7PN
1PM	( 1	+	$v^{2}$	+	$v^{4}$	+	$v^{6}$	+	$v^{8}$	+	$v^{10}$	+	$v^{12}$	+	$v^{14}$	+	...)	$G^{1}$
2PM			( 1	+	$v^{2}$	+	$v^{4}$	+	$v^{6}$	+	$v^{8}$	+	$v^{10}$	+	$v^{12}$	+	...)	$G^{2}$
3PM					( 1	+	$v^{2}$	+	$v^{4}$	+	$v^{6}$	+	$v^{8}$	+	$v^{10}$	+	...)	$G^{3}$
4PM							( 1	+	$v^{2}$	+	$v^{4}$	+	$v^{6}$	+	$v^{8}$	+	...)	$G^{4}$
5PM									( 1	+	$v^{2}$	+	$v^{4}$	+	$v^{6}$	+	...)	$G^{5}$
6PM											( 1	+	$v^{2}$	+	$v^{4}$	+	...)	$G^{6}$
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.^[2]

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One of the earliest works on this method of resolution is that of Bruno Bertotti, published in Nuovo Cimento in 1956.^[3]

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Post-Minkowskian expansion

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