Post-Newtonian expansion

Post-Newtonian expansions in general relativity 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.

Expansion in 1/c²

The post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of matter, forming the gravitational field, to the speed of light, which in this case is better 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.^[2]

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

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

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

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

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

Hybrid expansion

Sometimes, as with the parameterized post-Newtonian formalism, a hybrid approach is used in which both the reciprocal of the speed of gravity and masses are assumed to be small.

Uses

The first use of a PN expansion (to first order) was made by Einstein in calculating the perihelion precession of Mercury's orbit; today it 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

In general, the perturbed metric can be written as^[3]

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

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

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

where $\Box$ is the d'Alembertian operator, $E$ 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 $H$ is the Hubble constant and differential with respect to conformal time $\tau$ is denoted with a prime. Then (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, $\Phi=\Psi$.

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. Kopeikin, S., Efroimsky, M., Kaplan, G. (2011). Relativistic Celestial Mechanics of the Solar System. Wiley-VCH. p. 860. ISBN 978-3-527-40856-6.
3. "Cosmological Perturbation Theory" (PDF). p. 83,86.

External links

• "On the Motion of Particles in General Relativity Theory" by A.Einstein and L.Infeld
• "Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries" by Luc Blanchet