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.

Post-minkowskian vs Post-newtonian expansions

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]

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.[2]

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

References

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  1. ^ Damour, Thibault (2016-11-07). "Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory". Physical Review D. 94 (10): 104015. arXiv:1609.00354. Bibcode:2016PhRvD..94j4015D. doi:10.1103/PhysRevD.94.104015. ISSN 2470-0010. S2CID 106399287.
  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. arXiv:1908.01493. Bibcode:2019JHEP...10..206B. doi:10.1007/JHEP10(2019)206. ISSN 1029-8479. S2CID 199442337.
  3. ^ Bertotti, B. (1956-10-01). "On gravitational motion". Il Nuovo Cimento. 4 (4): 898–906. Bibcode:1956NCim....4..898B. doi:10.1007/BF02746175. ISSN 1827-6121. S2CID 120443098.