Begin with a particle on some prescribed trajectory, and with a field point which is on the light cone at some point in the history of the particle. The vector R points from the field point to the particle. This direction for the vector R is preferred for differentiation in time, since there are no sign inversions. By definition, the vector R is always on the light cone.
The time at the tail of the vector is t. The time at the head of the vector is T, with T = t-(R dot R)1/2/c = t-R/c. R can be differentiated with respect to either t or T, with dR/dT being the retarded velocity, and dR/dt being the velocity of the retarded intersection.
With V representing the retarded velocity, and Ru being a unit vector parallel to R, two auxiliary equations are easily derived.
dR/dt = V/(1 + 1/c V dot Ru)
V = (dR/dt)/(1 - 1/c dR/dt)
d2R/dt2 is not zero for straight-line unaccelerated motion. Using this notation, the Liénard-Wiechert [3, 4] retardation equations are
Ψ = q (1 - 1/c dR/dt)/R
A = q (dR/dt)/(R c)
q is the charge. Ψ is the scalar potential. A is the vector potential. The equations give the retarded potentials for an accelerated charge. The solutions are in the Lorentz gauge. The fields are computed from the potentials by the equations B = curl A, E = -grad Ψ - 1/c ∂A/∂t. The solutions are always solutions to the Maxwell equations. The retardation equations were first derived in the year 1898.
The LW equations can be derived in more than one way . One way consists of transforming to the frame of reference of the charged particle, then assuming that the advanced potential is Ψ = q/Ra in that frame of reference, where Ra is the distance to the field point at the advanced time. The advanced potential is then transformed back to the frame of reference of the field point, where it becomes the retarded potential.
The derivation of the mass equation will proceed the same way, but there is a complication. The potentials are always arbitrary to within a constant of integration, so a constant scalar and a constant vector can be added to the potential solution in either frame of reference. The Lorentz transform transforms these constants to a different pair of constants in the other frame of reference, and the constants are of no consequence in either frame of reference. However, the problem at hand is to derive the retardation equations, and it turns out that these constants of integration do matter. They matter in the case of charge also, but for charge the correct choice is zero for both constants. For the mass solution there is a particular choice for which the solutions for an accelerated mass do not contain an electromagnetic wave that does not decay with distance. The conservation of energy requires that this choice be taken. The charge solution can be obtained without transforming to the second frame of reference , and the mass solution can also, in which case the complication of the constants of integration does not occur.
After choosing the constant of integration, the solution in Section 1 becomes
A = G m Rua/c2
Ψ = G m/c2,
where Rua is the unit vector parallel to the position vector pointing from the source to the field point at the advanced time. The time at the field point is the source time + Ra/c.
The calculations for both charge and mass are carried through in a computer generated file. The charge solution was given above. The mass solution is
A = -G m/c2 Ru (1 - 1/c dR/dt)
Ψ = -G m/c3 dR/dt
In Section 1 the position vector was taken as pointing from the source to the field point. In order to avoid sign inversions when differentiating in time the retarded vector R has been taken as pointing from the field point to the particle, which inverted the sign of the retarded solution. Further, the signs of both the scalar and vector equations could be inverted if the sign of the acceleration equation was also inverted also, and it would make no difference. The solution can be parameterized by the retarded velocity instead of dR/dt, and that solution was shown as an intermediate step in the computer file.
In the second frame of reference the mass used in the derivation was γ m, where γ = 1/(1 - v2/c2)1/2, so the mass in the equation is the rest mass in the frame of reference of the field point. The mass transforms conventionally, but γ was absorbed into the other terms in the inverse transformation.
The equations are linear in field strength, so each mass in a system can be
independently retarded, and the solutions added. If the mass is accelerated
then at least two masses are required in order to conserve momentum.