The transformation of a time warp

The following calculations derive the Liénard-Wiechert retardation equations from the assumption that the scalar potential represents a displacement in time, and that the vector potential represents a displacement in space. In view of the mathematical form of the Lorentz transform, the demonstration is mathematically trivial, but it is nevertheless interesting in a physical sense. The calculation demonstrates that it would be difficult to show in the laboratory that the 4-potential does not represent displacements in space and time.

The calculation begins in the frame of reference of the field point. The coordinates are then transformed to the frame of reference of moving point charge. In that frame of reference the vector potential is zero, and the displacement in time is proportional to the scalar potential, which is q/(4 π ε0 Ra), where Ra is the distance to the field point at the future light cone position. Ra is parameterized by the transformed coordinates, then the displacement in time is added to the transformed time. The composite is transformed back to the first frame of reference. The original coordinates are then subtracted from the deformed system, leaving the LW solution as the residual.

The solutions for the displacements in space and time (MKS units) at the field point are

Δ r = ξ q vr/[4 π ε0 Rr (1 + Rur dot vr/c)]
Δ t = ξ q/[4 π ε0 Rr (1 + Rur dot vr/c)].

vr = retarded velocity
Rur = unit vector pointing from the field point to the particle at the retarded time
Rr = magnitude of the retarded position vector
ε0 = vacuum permittivity
q = charge
ξ = scaling constant, about 10-27 sec/volt, developed in the pulsar section.

Except for the scaling relationships, these are the Liénard-Wiechert retardation equations. Acceleration terms appear after the solutions are differentated to obtain the fields, and the equations are known to work well for accelerated charges. The solutions are always solutions to the Maxwell equations.


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