Physics: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 (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 | + | [math]\displaystyle{ v^2 }[/math] | + | [math]\displaystyle{ v^4 }[/math] | + | [math]\displaystyle{ v^6 }[/math] | + | [math]\displaystyle{ v^8 }[/math] | + | [math]\displaystyle{ v^{10} }[/math] | + | [math]\displaystyle{ v^{12} }[/math] | + | [math]\displaystyle{ v^{14} }[/math] | + | ...) | [math]\displaystyle{ G^1 }[/math] |
| 2PM | ( 1 | + | [math]\displaystyle{ v^2 }[/math] | + | [math]\displaystyle{ v^4 }[/math] | + | [math]\displaystyle{ v^6 }[/math] | + | [math]\displaystyle{ v^8 }[/math] | + | [math]\displaystyle{ v^{10} }[/math] | + | [math]\displaystyle{ v^{12} }[/math] | + | ...) | [math]\displaystyle{ G^2 }[/math] | ||
| 3PM | ( 1 | + | [math]\displaystyle{ v^2 }[/math] | + | [math]\displaystyle{ v^4 }[/math] | + | [math]\displaystyle{ v^6 }[/math] | + | [math]\displaystyle{ v^8 }[/math] | + | [math]\displaystyle{ v^{10} }[/math] | + | ...) | [math]\displaystyle{ G^3 }[/math] | ||||
| 4PM | ( 1 | + | [math]\displaystyle{ v^2 }[/math] | + | [math]\displaystyle{ v^4 }[/math] | + | [math]\displaystyle{ v^6 }[/math] | + | [math]\displaystyle{ v^8 }[/math] | + | ...) | [math]\displaystyle{ G^4 }[/math] | ||||||
| 5PM | ( 1 | + | [math]\displaystyle{ v^2 }[/math] | + | [math]\displaystyle{ v^4 }[/math] | + | [math]\displaystyle{ v^6 }[/math] | + | ...) | [math]\displaystyle{ G^5 }[/math] | ||||||||
| 6PM | ( 1 | + | [math]\displaystyle{ v^2 }[/math] | + | [math]\displaystyle{ v^4 }[/math] | + | ...) | [math]\displaystyle{ G^6 }[/math] | ||||||||||
| 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 Minkowsky 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
- ↑ Damour, Thibault (2016-11-07). "Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory". Physical Review D 94 (10): 104015. doi:10.1103/PhysRevD.94.104015. ISSN 2470-0010. Bibcode: 2016PhRvD..94j4015D.
- ↑ 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. doi:10.1007/JHEP10(2019)206. ISSN 1029-8479. Bibcode: 2019JHEP...10..206B.
- ↑ Bertotti, B. (1956-10-01). "On gravitational motion" (in en). Il Nuovo Cimento 4 (4): 898–906. doi:10.1007/BF02746175. ISSN 1827-6121. Bibcode: 1956NCim....4..898B. https://doi.org/10.1007/BF02746175.
 |  |
