Post-Newtonian

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 E__//ffect 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.