### 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
ε_{0} 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 **v**_{r}/[4 π ε_{0} R_{r}
(1 + **Ru**_{r} dot **v**_{r}/c)]

Δ t = ξ q/[4 π ε_{0} R_{r}
(1 + **Ru**_{r} dot **v**_{r}/c)].

**v**_{r} = retarded velocity

**Ru**_{r} = unit vector pointing from the field point to
the particle at the retarded time

R_{r} = magnitude of the retarded position vector

ε_{0} = 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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