A method of obtaining the rate of perihelion precession in a system of two charged masses

Abstract

The motion of a charged test body around a massive charged source can be described by the Klein-Gordon equation with gravitational and electromagnetic fields. When this equation is transformed into a coordinate system rotating with angular velocity Ω, it is found that for each state withψ=R(u)/uYj,k(θ, φ) expiωt, there is a corresponding rate of rotationΩn,j,k at which the general reiativistic Klein-Gordon equation approximately simplifies into a Schrodinger-Kepler equation, whose characteristic energies differ from those of the equation for a hydrogenlike system because the coupling factorsGmM andqQ must be multiplied by (1+4E/mc2) and (1+2E/mc2)1/2, respectively, in the general reiativistic case. The energy levels in the fixed frame are found to be-ħωfixed=ħωrtating-ħkΩn,j,k, wheren,j, k are the principal, angular momentum, and azimuthal quantum numbers. Applied to the case of a pair of rotating charged bosons of zero spin, the resulting fine structure agrees with the known fine-structure levels of this problem. Applied to the motion of Mercury around the Sun, the first-order calculation gives a rate of perihelion precession of 42.98 sec arc/century. If the Sun and Mercury had electrical chargesQ andq, then to the first order the rate of precession would be approximately 42.98 (1−3y) sec arc/century. (Herey is the ratio of the electric to gravitational force.) Consideration of upper limits on the field strength on the surfaces of the Sun and Mercury, indicated by Stark shifts and molecular binding energy, show that the electric part of the rate of precession, (−129y), is far below 0.1 sec arc/century, and so need not be considered in the test of general relativity based on Mercury's perihelion precession.