Feynman's Near Field

[Jack Sarfatti]

$ #Coulomb Force
The late Richard Feynman derived an original formula for the retarded
classical electromagnetic fields produced by an electric charge moving
in an arbitrary timelike way. The electric field vector (actually a set
of tensor components) at a given point event in spacetime consists of
three terms. The first near-field term is the intuitive "retarded"
Coulomb force which says that the electric field we find now at some
point in space depends only on the position and motion of the source
point-like charge at the earlier time by which it would take light
travelling at speed c, to travel the distance r' from the charge to the
field point. This term by itself, says that the frame-invariant
spacetime interval between a source event and its field event is exactly
zero in all frames of reference. However, that is not the end of the
story which is rather subtle and is not taught precisely in most
university courses in classical electrodynamics. Feynman's second term
is a near-field "correction" which is the rate of change of the first
retarded Coulomb field multiplied by r'/c which is the retardation delay
time. This counter-intuitive rigorous acausal correction term works the
opposite way tending to cancel the causal retardation in the first term!

The correction is a linear extrapolation into the future made by Nature
not by man. Feynman says explicitly on p. 21-2 of Vol II of his Cal
Tech Lectures on Physics (Addison-Wesley): "If the field is changing
slowly, the effect of the retardation is almost completely removed by
the correction term, and the two terms together give us an electric
field that is the 'instantaneous Coulomb field' ... to a very good
approximation. The importance of Feynman's little known classical
computation cannot be exaggerated. It means, that the classical near-
field effectively acts outside the lightcone with a spacelike influence.

This matches up with Feynman's demonstration, in quantum
electrodynamics, that the near field is due to the combined effect of
virtual timelike and longitudinal polarized photons off the mass shell
which exactly cancel each other out leaving only transversely polarized
real photons to make up the classical far-field or radiation field that
propagates energy to infinity. The transverse polarized field is the
third term in Feynman's equation (21.1). In the limit of charge speed v
<< c and for large distances r' this transverse field is proportional to
1/r' multiplied by the component of vector acceleration of the point-
like source charge which is at right angles to the spatial line r' from
the field point at time t to the retarded source position at time t -
r'/c.