Discussion on Post-Newtonian Effects Feb. 25

see also References.

For a better physical understanding and motivation I presented here, however, the approach of Kannan & Saha (2009), who consider Starting from the Boyer-Lindkvist form of the Kerr metric around a spinning black hole one can derive geodetic equations of motion for massless test particles in this metric, who just can be obtained analogous to elementary Lagrangian mechanis (using Euler-Lagrange equations in the
metric coordinates). If we consider (v/c)^2 ~ (GM/r c^2) ~ eps and evaluate the metric coefficients as a Taylor series with respect to eps, we get corrections to the Newtonian case, which correspond in their physical meaning at every Post-Newtonian order to those of the full PN treatment mentioned above. For example:

Metric coefficients
Eqs. of Motion
Short
Effect test particle
Analogous Effect in full 2-Body PN
eps^2
eps^0
N
Newtonian
Newtonian
eps^4
eps^2
PN1
1st order perihel shift (conservative)
1st order perihel shift (conservative)
eps^5
eps^3
PN1.5
1st order frame dragging
Lowest Order Spin-Orbit Coupling
eps^6
eps^4
PN2
2nd order perihel shift (conservative)

no analogue
2nd order perihel shift (conservative)

Lowest Order Spin-Spin Coupling
eps^7
eps^5
PN2.5
higher order frame dragging

no analogue
Higher Order Spin-Orbit Coupling
Lowest Order Gravitational Energy Loss (Einstein Quadrupole)
eps^8
eps^6
PN3
3rd order conservative
3rd order conservative
eps^9
eps^7
PN3.5
no analogue
Higher Order Grav. Wave Energy Loss


























Most applications use up to PN2.5 (the lowest order term containing gravitational wave energy losses and their Newtonian feedback on the orbit. Note that a very instructive orbit-averaged derivation from Einstein's Quadrupole Formula for the Two-Body Problem was done by Peters & Mathews (1963) and Peters (1964). Note that the change of semi-major axis and eccentricity due to PN2.5 can be followed up on a few pieces of paper starting from Einstein's Quadrupole Formula. The papers even contain a derivation of gravitational wave spectra, though in a very approximate (orbit-averaged) way.

Note that for the conservative terms generalizations of Kepler's Equations for the two-body problem have been derived, which are dubbed quasi-Keplerian, see Memmesheimer, Gopakumar & Schäfer (2004). Remarkably, due to the breaking of the Keplerian closed orbits in this case there exists three different eccentricities, linked to the oscillatory motion in time (et), radial coordinate (er) and angular motion (ephi). Only in the Keplerian case they all agree and lead thus to a closed orbit.