# The Inconsistency of the Retarded Field Concept for Static Forces

The velocity of light is generally accepted as constituting the ultimate velocity by which 'information' can travel. Severe consequences arise from the uncritical generalization of the invariance properties for the expansion of light signals to other physical phenomena. One of these consists in the circumstance that the static interaction of material bodies, for instance by means of Coulomb-, Lorentz- or gravitational forces, has to be treated as a retarded interaction if large distances or high velocities come into play. However, it has obviously been overlooked that this procedure introduces an asymmetry into the problem which results in different forces in different inertial systems. This contradicts the definition of a force which depends only on relative distances (and in some cases velocities) and should by definition therefore be independent of the coordinate system from which the whole configuration is observed ('principle' of relativity).
The proof that the concept of retarded forces violates the principle of relativity is as follows:

Consider two charged particles moving away from each other with relative velocity v along the x- axis (it is assumed here for simplicity that v=const., which can be achieved by choosing v large compared to the velocity increment due to the interaction of the particles during the considered period of time).

(Fig.1)             v•2 ------------------------------------•1------→x

The force acting on charge q2 at time t is determined by the field (or potential) of charge q1 at the location x2(t) of particle 2. The concept of retardation assumes now that this field is produced by particle 1 not at the same moment but at an earlier instant tret corresponding to the time interval it would take a light signal to traverse the distance between particle 1 and particle 2. Therefore, the potential Φ(t) (one could equally well use the expression for the corresponding force) would have to be written as

(1)       Φ1→2(t) = q1/[x1(tret)- x2(t)]   .

Eq.(1) may now be evaluated in two different coordinate systems:

I)   comoving with particle 1 (i.e. according to Fig.1): in this case x1(t) = x1(0) =const., and therewith

(2)       ΦI1→2(t)= q1/[x1(0)- x2(t)]   =   q1/[x1(0)- x2(0)+v.t]   =   q1/d(t)   ,

with

(3)       d(t)= x1(0)- x2(0)+v.t   ,

i.e. the usual expression for the Coulomb- potential is obtained.

II)   comoving with particle 2: here, x2(t)= x2(0)=const., and therefore

(4)       ΦII1→2(t) = q1/[x1(tret)- x2(0)]   =   q1/[x1(0)+v. tret - x2(0)]   .

With

(5)       tret = t - [x1(t)- x2(0)]/c   =   t - [x1(0)- x2(0) +v.t]/c   ,

it follows that

(6)       ΦII1→2(t) = q1/[x1(0)- x2(0) + v.t -v/c.(x1(0)- x2(0) + v.t )]  =   q1/[d(t) .(1-v/c)]   .

From a comparison of Eqs.(2) and (6), it is evident that the retarded potential (and therefore the force) depends on the inertial system, which contradicts, as explained above, the principle of relativity. Only if instantaneous field quantities are used, does one obtain the same forces in all coordinate systems moving with a constant relative velocity.
This fact re-establishes the strict meaning of Newton's law of action and reaction, indicating the self-evident property of a closed system that the sum of its internal forces vanishes. With the inconsistent assumption of a retarded action of forces, this law would reduce to a purely formal geometrical symmetry expressing the circumstance that a hypothetical field quantum emitted from particle 1 to particle 2 takes the same travelling time as a field quantum sent in the reverse direction. With this picture, there is however no real mutual interaction of particles invariant under a transformation between different inertial system, but only an 'interaction' of particles with field quanta which have been emitted by the other particles but do not stand in a direct relationship to them anymore. The strange possibility would arise that a body experiences a force by a static field originating from a particle which does not exist anymore at that instant (because of decay for instance).
One should be aware of the difference compared to the case of an interaction with an electromagnetic light pulse: the latter can be considered as an independent, self-supporting entity (being emitted by one atomic system and being able to be absorbed by another atomic system), whereas a static field evidently has to be related directly to a material body which, in last instance, can account for the work done on other particles.
The static field concept is thus only consistent if the field is thought of as fixed to the particles, i.e. if unretarded field quantities are used (in the same manner as one end of an ideally rigid rod responds instantaneously to a movement of the other end even if the rod measures lightyears in length (of course in reality a rod can not be strictly rigid because the inertia of atoms in the material leads to a delay of the full response, but by an infinitesimal amount movement is nevertheless propagated instantaneously)). The same argument as above can be applied to the vector potential of a static magnetic field by considering the velocity of the corresponding current system relative to the particle acted upon.
Obviously, a new physical principle has to be formulated by which only the derivatives of charge and current densities with respect to time (which lead to the radiation fields in the theory of electrodynamics) have to be treated with the concept of retardation. The static Coulomb- and Biot-Savart- as well as the gravitational field however only make sense if evaluated unretarded.

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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