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Post-Minkowskian expansion

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Approximation technique in general relativity
Post-minkowskian vs Post-newtonian expansions

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).

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

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

References

  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.
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