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The advance of the pericenter of the orbit of a test body around a massivebody in general relativity can be calculated in a number of ways. In onemethod, one studies the geodesic equation in the exact Schwarzschild geometryand finds the angle between pericenters as an integral of a certain radialfunction between turning points of the orbit. In another method, one describesthe orbit using osculating orbit elements, and analyzes the "Lagrange planetaryequations" that give the evolution of the elements under the perturbing effectsof post-Newtonian (PN) corrections to the motion. After separating theperturbations into periodic and secular effects, one obtains an equation forthe secular rate of change of the pericenter angle. While the different methodsagree on the leading post-Newtonian contribution to the advance, they do notagree on the higher-order PN corrections. We show that this disagreement isillusory. When the orbital variables in each case are expressed in terms of theinvariant energy and angular momentum of the orbit and when account is taken ofa subtle difference in the meaning of "pericenter advance" between the twomethods, we show to the third post-Newtonian order that the different methodsactually agree perfectly.

Pericenter advance in general relativity: Comparison of approaches at high post-Newtonian orders