Post-Newtonian expansion

In general relativity, post-Newtonian expansions are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which 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.

Diagram of the parameter space of compact binaries with the various approximation schemes and their regions of validity.

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 approximations was developed by Subrahmanyan Chandrasekhar and co-workers 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

${\displaystyle h_{\alpha \beta }=g_{\alpha \beta }-\eta _{\alpha \beta }\,.}$

To this end, one must choose a coordinate system in which the eigenvalues of ${\displaystyle h_{\alpha \beta }\eta ^{\beta \gamma }\,}$ 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:

${\displaystyle g^{\mu \nu }\approx \eta ^{\mu \nu }-\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \nu }+\eta ^{\mu \alpha }h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \nu }\,.}$

${\displaystyle {\sqrt {-g}}\approx 1+{\tfrac {1}{2}}h_{\alpha \beta }\eta ^{\beta \alpha }+{\tfrac {1}{8}}h_{\alpha \beta }\eta ^{\beta \alpha }h_{\gamma \delta }\eta ^{\delta \gamma }-{\tfrac {1}{4}}h_{\alpha \beta }\eta ^{\beta \gamma }h_{\gamma \delta }\eta ^{\delta \alpha }\,.}$

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 first simple case of the most common use of the PN expansion: Solving the general relativistic two-body problem, which includes the emission of gravitational waves.

Newtonian gauge

Main article: Newtonian gauge

In general, the perturbed metric can be written as[7]

${\displaystyle ds^{2}=a^{2}(\tau )\left[(1+2A)d\tau ^{2}-2B_{i}dx^{i}d\tau -\left(\delta _{ij}+h_{ij}\right)dx^{i}dx^{j}\right]}$

where ${\displaystyle A}$, ${\displaystyle B_{i}}$ and ${\displaystyle h_{ij}}$ are functions of space and time. ${\displaystyle h_{ij}}$ can be decomposed as

${\displaystyle h_{ij}=2C\delta _{ij}+\partial _{i}\partial _{j}E-{\frac {1}{3}}\delta _{ij}\Box ^{2}E+\partial _{i}{\hat {E}}_{j}+\partial _{j}{\hat {E}}_{i}+2{\tilde {E}}_{ij}}$

where ${\displaystyle \Box }$ is the d'Alembert operator, ${\displaystyle E}$ is a scalar, ${\displaystyle {\hat {E}}_{i}}$ is a vector and ${\displaystyle {\tilde {E}}_{ij}}$ is a traceless tensor. Then the Bardeen potentials are defined as

${\displaystyle \Psi \equiv A+H(B-E'),+(B+E')',\quad \Phi \equiv -C-H(B-E')+{\frac {1}{3}}\Box E}$

where ${\displaystyle H}$ is the Hubble constant and a prime represents differentiation with respect to conformal time ${\displaystyle \tau \,}$.

Taking ${\displaystyle B=E=0}$ (i.e. setting ${\displaystyle \Phi \equiv -C}$ and ${\displaystyle \Psi \equiv A}$), the Newtonian gauge is

${\displaystyle ds^{2}=a^{2}(\tau )\left[(1+2\Psi )d\tau ^{2}-(1-2\Phi )\delta _{ij}dx^{i}dx^{j}\right]\,}$.

Note that in the absence of anistropic stress, ${\displaystyle \Phi =\Psi }$.

See also

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

References

1. Kopeikin, S. (2004). "The speed of gravity in General Relativity and theoretical interpretation of the Jovian deflection experiment". Classical and Quantum Gravity. 21 (13): 3251–3286. arXiv:gr-qc/0310059. Bibcode:2004CQGra..21.3251K. doi:10.1088/0264-9381/21/13/010.
2. Chandrasekhar, S. (1965). "The post-Newtonian equations of hydrodynamics in General Relativity". The Astrophysical Journal. 142: 1488. doi:10.1086/148432.
3. 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. doi:10.1086/149003.
4. Chandrasekhar, S. (1969). "Conservation laws in general relativity and in the post-Newtonian approximations". The Astrophysical Journal. 158: 45. doi:10.1086/150170.
5. Chandrasekhar, S.; Nutku, Y. (1969). "The second post-Newtonian equations of hydrodynamics in General Relativity". Relativistic Astrophysics. 86.
6. Chandrasekhar, S.; Esposito, F.P. (1970). "The 2½-post-Newtonian equations of hydrodynamics and radiation reaction in General Relativity". The Astrophysical Journal. 160: 153. doi:10.1086/150414.
7. "Cosmological Perturbation Theory" (PDF). p. 83,86.

External links

• "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld
• Blanchet, Luc (2014). "Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries". Living Reviews in Relativity. 17 (1): 2. arXiv:1310.1528. Bibcode:2014LRR....17....2B. doi:10.12942/lrr-2014-2. PMC 5256563. PMID 28179846.