Post-Newtonian expansion explained

In general relativity, post-Newtonian expansions (PN expansions) are used for finding an approximate solution of Einstein field equations for the metric tensor. The approximations are expanded in small parameters that express orders of deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of effective field theories. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.

Expansion in 1/c²

The post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter that creates the gravitational field, to the speed of light, which in this case is more precisely called the speed of gravity.^[1] In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity. A systematic study of post-Newtonian expansions within hydrodynamic approximations was developed by Subrahmanyan Chandrasekhar and his colleagues in the 1960s.^{[2] [3] [4] [5] [6]}

Expansion in h

Another approach is to expand the equations of general relativity in a power series in the deviation of the metric from its value in the absence of gravity.

h _{\alpha} = g _{\alpha} - η _{\alpha} . To this end, one must choose a coordinate system in which the eigenvalues of h _{\alpha} η ^{\beta} all have absolute values less than 1.

For example, if one goes one step beyond linearized gravity to get the expansion to the second order in h:

g ^{\mu} ≈ η ^{\mu} - η ^{\mu} h _{\alpha} η ^{\beta} + η ^{\mu} h _{\alpha} η ^{\beta} h _{\gamma} η ^{\delta} . \sqrt{ - g} ≈ 1 + \tfrac 12 h _{\alpha} η ^{\beta} + \tfrac 18 h _{\alpha} η ^{\beta} h _{\gamma} η ^{\delta} - \tfrac 14 h _{\alpha} η ^{\beta} h _{\gamma} η ^{\delta} . Expansions based only on the metric, independently from the speed, are called post-Minkowskian expansions (PM expansions).

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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.^[7]

Uses

The first use of a PN expansion (to first order) was made by Albert Einstein in calculating the perihelion precession of Mercury's orbit. Today, Einstein's calculation is recognized as a common example of applications of PN expansions, solving the general relativistic two-body problem, which includes the emission of gravitational waves.

Newtonian gauge

See main article: Newtonian gauge. In general, the perturbed metric can be written as^[8]

ds ^{2} = a ^{2} (\tau) \left[( 1+2 A)d\tau ^{2} - 2 B _{i} dx ^{i} d\tau - \left( \delta _{ij} + h _{ij} \right)dx ^{i} dx ^{j\right]} where A , B _{i} and h _{ij} are functions of space and time. h _{ij} can be decomposed ash _{ij} = 2 C \delta _{ij} + \partial _{i} \partial _{j} E - (1)/(3) \delta _{ij} \Box ^{2} E + \partial _{i} \hat{E} _{j} + \partial _{j} \hat{E} _{i} + 2 \tilde{E} _{ij} where \Box is the d'Alembert operator, E is a scalar, \hat{E} _{i} is a vector and \tilde{E} _{ij} is a traceless tensor.Then the Bardeen potentials are defined as\Psi \equiv A + H(B - E'), + (B + E')', \Phi \equiv - C - H(B - E') + (1)/(3) \Box E where H is the Hubble constant and a prime represents differentiation with respect to conformal time \tau .

Taking

B = E =0 (i.e. setting \Phi\equiv - C and \Psi\equiv A ), the Newtonian gauge is ds ^{2} = a ^{2(\tau)} \left[ ( 1 + 2 \Psi) d\tau ^{2} - ( 1 - 2 \Phi)\delta _{ij} dx ^{i} dx ^{j\right]} .

Note that in the absence of anisotropic stress,

\Phi = \Psi .

A useful non-linear extension of this is provided by the non-relativistic gravitational fields.

See also

• Coordinate conditions
• Einstein–Infeld–Hoffmann equations
• Linearized gravity
• Parameterized post-Newtonian formalism

External links

• "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld
• 10.12942/lrr-2014-2. 28179846. 5256563. 1310.1528. 2014LRR....17....2B. Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries. 2014. Blanchet. Luc. Living Reviews in Relativity. 17. 1. 2. free .
• Clifford . M. Will . On the unreasonable effectiveness of thepost-Newtonian approximation ingravitational physics . . 108 . 15 . 5938–5945 . 10.1073/pnas.1103127108 . free . 2011 . 1102.5192 .

Notes and References

1. Kopeikin, S. . Sergei Kopeikin . The speed of gravity in General Relativity and theoretical interpretation of the Jovian deflection experiment . Classical and Quantum Gravity . 2004 . 21 . 13 . 3251–3286 . 10.1088/0264-9381/21/13/010 . gr-qc/0310059 . 2004CQGra..21.3251K. 13998000 .
2. Subrahmanyan Chandrasekhar . Chandrasekhar . S. . 1965 . The post-Newtonian equations of hydrodynamics in General Relativity . The Astrophysical Journal . 142 . 1488. 10.1086/148432 . 1965ApJ...142.1488C .
3. Subrahmanyan Chandrasekhar . Chandrasekhar . S. . 1967 . The post-Newtonian effects of General Relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the MacLaurin spheroids . The Astrophysical Journal . 147 . 334. 10.1086/149003 . 1967ApJ...147..334C .
4. Subrahmanyan Chandrasekhar . Chandrasekhar . S. . 1969 . Conservation laws in general relativity and in the post-Newtonian approximations . The Astrophysical Journal . 158 . 45. 10.1086/150170 . 1969ApJ...158...45C . free .
5. Subrahmanyan Chandrasekhar . Chandrasekhar . S. . 1969 . Nutku . Y. . The second post-Newtonian equations of hydrodynamics in General Relativity . Relativistic Astrophysics . 86. 55 . 10.1086/150171 . 1969ApJ...158...55C . free .
6. Subrahmanyan Chandrasekhar . Chandrasekhar . S. . 1970 . Esposito . F.P. . The 2½-post-Newtonian equations of hydrodynamics and radiation reaction in General Relativity . The Astrophysical Journal . 160 . 153. 10.1086/150414 . 1970ApJ...160..153C . free .
7. 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.
8. Web site: Cosmological Perturbation Theory . 83,86 . 2016-08-10 . 2016-08-26 . https://web.archive.org/web/20160826054737/http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf . dead .