Post-Minkowskian expansion explained

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 (which must be negligible compared to that of light) and the gravitational constant, 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]

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

Notes and References

1. Damour. Thibault. 2016-11-07. Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory. Physical Review D. 94. 10. 104015. 10.1103/PhysRevD.94.104015. 1609.00354. 2016PhRvD..94j4015D. 106399287. 2470-0010.
2. Bern. Zvi. Cheung. Clifford. Roiban. Radu. Shen. Chia-Hsien. Solon. Mikhail P.. Zeng. Mao. 2019-08-05. Black Hole Binary Dynamics from the Double Copy and Effective Theory. Journal of High Energy Physics. 2019. 10. 206. 10.1007/JHEP10(2019)206. 1908.01493. 2019JHEP...10..206B. 199442337. 1029-8479.
3. Bertotti. B.. 1956-10-01. On gravitational motion. Il Nuovo Cimento . en. 4. 4. 898–906. 10.1007/BF02746175. 1956NCim....4..898B. 120443098. 1827-6121.