Perihelion precession for modified Newtonian gravity

TL;DR

This paper derives a universal formula for perihelion precession in nearly circular orbits without assuming the potential is close to Newtonian, and applies it to modified gravity theories and black hole spacetimes.

Contribution

It introduces a new, general method to calculate perihelion precession applicable to a wide class of potentials and spacetime geometries, extending previous approaches.

Findings

Derived a universal formula for perihelion precession delta.

Applied the formula to modified Newtonian gravity and linearized fourth-order gravity.

Obtained a new formula for stable circular orbits in Schwarzschild spacetime.

Abstract

We calculate the perihelion precession delta for nearly circular orbits in a central potential V(r). Differently from other approaches to this problem, we do not assume that the potential is close to the Newtonian one. The main idea in the deduction is to apply the underlying symmetries of the system to show that delta must be a function of r V''(r)/V'(r), and to use the transformation behaviour of delta in a rotating system of reference. This is equivalent to say, that the effective potential can be written in a one-parameter set of possibilities as sum of centrifugal potential and potential of the central force. We get a universal formula for delta. It has to be read as follows: a circular orbit at this value r exists and is stable if and only if this delta is a well-defined real; and if this is the case, then the angular difference from one perihelion to the next one for nearly circular orbits at this r is exactly 2 pi + delta. Then we show how to apply this result to examples of recent interest like modified Newtonian gravity and linearized fourth-order gravity. In the second part of the paper, we generalize this universal formula to static spherically symmetric space-times. For the Schwarzschild black hole with mass parameter m > 0 it leads to a surprisingly unknown formula. It represents a strict result and is applicable for all values r > 6m and is in good agreement with the fact that stable circular orbits exist for r > 6m only.