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 KleinGordon 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)/uY _{ j,k }(θ, φ) expiωt, there is a corresponding rate of rotationΩ _{ n,j,k } at which the general reiativistic KleinGordon equation approximately simplifies into a SchrödingerKepler 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/mc ^{ 2 }) and (1+2E/mc ^{ 2 })^{1/2}, respectively, in the general reiativistic case. The energy levels in the fixed frame are found to beħω _{ fixed }=ħω _{ rtating }ħkΩ _{ n,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 finestructure levels of this problem. Applied to the motion of Mercury around the Sun, the firstorder 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 (13y) 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.
 Publication:

General Relativity and Gravitation
 Pub Date:
 June 1978
 DOI:
 10.1007/BF00759542
 Bibcode:
 1978GReGr...9..469S
 Keywords:

 Charged Particles;
 Field Theory (Physics);
 KleinGordon Equation;
 Perihelions;
 Precession;
 Relativistic Effects;
 Bosons;
 Coordinate Transformations;
 Electromagnetic Fields;
 Energy Levels;
 Gravitational Fields;
 Mercury (Planet);
 Quantum Mechanics;
 Schroedinger Equation;
 Astrophysics;
 Neutron Stars;
 Relativistic Astrophysics;
 Background Radiation;
 Gravitation Theory