In physics, non-relativistic general relativity is an approximate approach to modeling gravity based on applying effective field theory. Effective field theory treats gravitational interactions between point particles, adapting techniques developed for quantum field theory. The first systematic treatment was by Walter D. Goldberger and Ira Rothstein in 2006. The approach lead to a systematic application of Feynman diagrams to higher order post-Newtonian expansions and a Kaluza-Klein like decomposition of general relativity. The primary application is gravitational waves from inspiraling compact objects like black holes.
Effective field theory
In the post-Newtonian approximation for a two body gravitational system, like a pair of inspiralling black holes, different physical effects dominate at different length scales. The black hole itself has a characteristic internal structure radius, its Schwarzschild radius, r s {\displaystyle r_{s}} . A pair of black holes have a length scale, their separation distance, r {\displaystyle r} . As long as r ≫ r s {\displaystyle r\gg r_{s}} the orbital velocity, v {\displaystyle v} , will be small compared to the speed of light and Newtonian gravity will be a good approximation. The motion of the black holes generates gravitational waves with characteristic wavelength, λ ≈ v / r {\displaystyle \lambda \approx v/r} . In effective field theory (EFT), the full problem is solved in two stages. In the first stage, the gravitational field of each individual black hole is represented by the field of point particles out to the orbital radius r {\displaystyle r} . In this stage, the physics of a Schwarzschild black hole and its gravitational radiation field is matched to a point particle and its radiation field. The details of the black hole are summarized or integrated into parameters of the point-particle field. In the second stage, the bound state of two point particles is matched to bound state potential modes and long range radiation modes of an effective field for a composite object with a size r {\displaystyle r} . The radiation from this two-particle bound state is identified with gravitational waves as long as their wavelength is long compared with the distance between the particles, r ≪ λ {\displaystyle r\ll \lambda } .
Kaluza-Klein like decomposition One result from the application of effective theory to general relativity was a decomposition of general relativity into several non-relativistic gravitational fields similar to the model proposed by Kaluza-Klein theory. Within general relativity (GR), Einstein's relativistic gravity, the gravitational field is described by the 10-component metric tensor. In a completely non-relativistic limit 9 fields can be ignored leaving only a single component Newtonian gravitational potential characteristic of Newtonian gravity. The concept of non-relativistic gravitational fields attempts to give physical interpretation to these nine fields. A reader who is familiar with electromagnetism (EM) will benefit from the following analogy. In EM, one is familiar with the electrostatic potential ϕ EM {\displaystyle \phi ^{\text{EM}}} and the magnetic vector potential A →
EM {\displaystyle {\vec {A}}{}^{\text{EM}}} . Together, they combine into the 4-vector potential A μ EM ↔ ( ϕ EM , A →
EM ) {\displaystyle A_{\mu }^{\text{EM}}\leftrightarrow (\phi ^{\text{EM}},{\vec {A}}{}^{\text{EM}})} , which is compatible with relativity. This relation can be thought to represent the non-relativistic decomposition of the electromagnetic 4-vector potential. Indeed, a system of point-particle charges moving slowly with respect to the speed of light may be studied in an expansion in v 2 / c 2 {\displaystyle v^{2}/c^{2}} , where v {\displaystyle v} is a typical velocity and c {\displaystyle c} is the speed of light. This expansion is known as the post-Coulombic expansion. Within this expansion, ϕ EM {\displaystyle \phi ^{\text{EM}}} contributes to the two-body potential already at 0th order, while A → EM {\displaystyle {\vec {A}}^{\text{EM}}} contributes only from the 1st order and onward, since it couples to electric currents and hence the associated potential is proportional to v 2 / c 2 {\displaystyle v^{2}/c^{2}} .
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