Elimination of Retarded Functions from Two-Body Classical Relativistic Electrodynamics

Interaction problems in classical relativistic electro-dynamics are frustratingly difficult to solve, since the differential equation of motion involves functions of both the present and retarded times. In this paper the author eliminates the retarded functions from the equation of motion for the two-body problem in which two equal masses with opposite charges that are equal in magnitude are released from rest and allowed to fall into one another. In the final form of the equation of motion only the present time, the present position, and derivatives of the present position with respect to the present time appear—in other words, it is an equation with one unknown and the present time only. To accomplish this simplification of the equation of motion, the author employs the Lorentz condition, $∇·A+δΦ/cδt=0$⁠, and the expression $E=−∇Φ−δA/cδt$ to obtain a partial differential equation in the retarded separation distance R that can be solved. When the resulting relationships are substituted into the Lorentz force equation it contains only functions of the present time.