One considers three quantities, length, time and the speed of electromagnetic (EM) propagation in transforming Maxwell's equations between reference frames. Einstein assumed the velocity of EM propagation to be strictly c, requiring the Lorentz transformations to keep the form of Maxwell's equations consistent. This was at the expense of standard concepts of length, time, and simultaneity; each becoming distorted to accommodate the constancy of c. RCM simply adds the word observed to the second postulate, and derives the Galilean invariance of Maxwell's equations.

In figure 1, observers in the stationary (K) and moving (K') frames are at the origins S and S' respectively. The origins are initially coincident at the time of a flash at A, a distance x from S. In RCM, where 0 < c < C, the component velocity of light in the non-moving K frame is c. As measured in the K' frame (moving with a constant velocity v), c' = c, but in the K frame c' is c + v. Restricting motion of the K' frame to the x axis, the Galilean transformations become:

x’ = x + vt ; y’ = y ; z’ = z ; t’ = t (1)

Figure 1

From a treatment of wave mechanics, for wave propagation in the x direction with Ez = 0 and c the velocity of propagation, we write:

(2)

If a flash of light occurs some distance x from the origin of K, we can let x be represented by c times the time it takes light to reach an observer at K’s origin. Thus, we derive the following relations:

(4)

(6)

We can also derive two useful relations from (3), whereby we express in terms of and c, where the last expression holds since t' = t:

(7)

Now we wish to examine the wave equation for the same wave in the K' system. We have:

(8)

Substituting (6) into (8) and comparing with (2) yields:

(9)

Equation (9) implies:

(10)

Equations (9) and (10) demonstrate the frame invariance in going from the K frame to K', provided that the velocity observed in K', as measured in K, is c + v. This wave has a velocity as observed in K' of c, as required by experiment. The wave we are considering must have a velocity with respect to the source of c plus the velocity, v, of the K' system. Since v can assume any value, the light must leave the source in a continuum of velocities such that 0 < c < C, where we place no constraints on the upper bound of C.

One interesting consequence of the SR Lorentz group is the invariance of the metric:

(11)

However, since c' is forced to transform into c, the left side of equation (10) could simply begin with c'² and such a statement then holds under a Galilean transformation, where we use the substitution dx = cdt.

(12)

Now we consider the transformation of Maxwell's equations to ensure that the assumed wave equation of (8) is actually valid. Maxwell's equations may be expressed as:

(13)

(14)

J is a vector quantity of current density, equal to the net amount of positive charge crossing a unit area of surface per second. Using the Galilean transformation, the transformation of J' is as follows:

(15)

All that remains is to show that primed equations in K' remain form invariant under the Galilean transformations of (1). We will demonstrate the transform for one quantity. All other equations transform similarly:

(16)

Thus, we see that Maxwell's equations are indeed form invariant under the Galilean transformation we have proposed. Next we will compare the wave as observed in the K' system with that observed in the K system.

We can mathematically express the form of an EM wave propagating in the x direction by the following relations:

(17)

(18)

From (17) and (18) we can express the wave equation for any velocity component c', as it appears with respect to the source, stationary in K, as follows:

(19)

(20)

(21)

We allow light to fall (instantaneously) perpendicular to an observer in a reference frame, K', moving with a velocity of v_y with respect to the source in K. The observer is located at a distance x as measured in K at the time of absorption. If the component velocity of light along the x axis is c, the observer will not be sensitive to it, as the resultant of a velocity v_y and c_x is a value greater than c. For this reason, the velocity of the component of light to which the observer illustrated in figure 2 is sensitive, c', and the corresponding wavelength, are given by:

(22)

The frequency of this light, expressed in K, is given by:

(23)

To obtain the frequency as observed in K', we simply replace c' with c, as we did in (24), and obtain:

(24)

Figure 2

In figure 2, an observer in K' must look along the line given by the resultant of its velocity and the c' component traveling along the x axis. This angle is easily shown to be given by the following:

(25)

An observer moving in a direction transverse to the line joining the earth to a star must tilt its telescope at the angle given by (25), consistent with experience.

In figure 3, the observer travels to the left at velocity, v, while a flash leaves the source, to strike the observer at the angle indicated. We calculate the component velocity with respect to the source that will strike the observer with a speed of c at the angle indicated. The Doppler shift is given by the ratio of c to this initial velocity as before. From the figure the initial velocity component of light leaving the source is given by:

(26)

Multiplying the source frequency times the ratio of c to the velocity of the component as measured in K gives the observed frequency:

(27)

Figure 3

We desire to solve the problem of the force on a moving charged particle above a neutral, current carrying wire without resorting to Lorentz transformations. We begin with the formula for the force on a moving particle outside an infinitely long current carrying wire:

(30)

The magnetic force on a wire is due only to the movement of the charges in it, and thereby depends only on the total current, and not the amount of charge carried by each particle or even its sign. Thus we must be careful, in considering different reference frames, to keep track of both the positive and negative currents in the wire.

We define v_o as the velocity of the charged particle with respect to the mass of the wire carrying the current producing charge density. Thus v_o will not change as we, the observers, change our reference frame. By convention, v_o is positive in the same direction as the flow of a current defined by moving negative charges. We further define the velocity of the current due to moving negative charges in the frame of the wire as v. If we observe a wire moving opposite to the flow of a negative charge current with respect to our reference frame at a velocity of -2v, and a charge above that wire moving the same direction at a velocity of -1.5v with respect to our reference frame, then v_o will be equal to -1.5v-(-2v) = v/2 in the wire's reference frame. This is illustrated in figure 4.

Figure 4

(31)

In (30) we replace B with the equation for the field at a distance r due to a current I:

(32)

Substituting (31) into (32) yields the expression for the force on a charged particle moving above a current carrying wire. An example shows that (33) is valid when viewed from any inertial frame of reference.

(33)

In figure 4, we are moving at a velocity with respect to the wire of 2v. Thus the total current is given by:

(34)

This current is the same current we observe when stationary with respect to the wire. The velocity of the charged particle with respect to the wire is v/2. Thus the force on the charged particle is given by:

(35)

The force on a moving charged particle due to a current carrying wire is the same regardless of the reference frame of the observer. More importantly, the force is a magnetic force in all frames of reference. In SRT the length contraction experienced by one charge density (the one arbitrarily chosen to be in motion by our choice of reference frame), but not the other causes an increased positive charge density and the wire becomes charged. This is unsettling at best. As we move from one reference frame to another, we see what was a magnetic field effect vanish while an electric field arises, and a neutral wire acquires excess positive charge. Equally confusing is that the velocity of the particle, v_o, is measured with respect to our arbitrary frame of reference, rather than the frame of the current carrying wire. The math in special relativity obtains the correct result, but the solution is contrived and counterintuitive.

Equation (30) determines the force on a charged particle in a magnetic field of fixed value B. However, the geometry of this equation as viewed from the laboratory is equivalent to that viewed in the particle’s frame only for small velocities v_o, where the direction of B can be considered normal to the direction of v_o. Since the photon carries the electromagnetic force, such forces must interact at a velocity of c with respect to the observer. Thus we assign to the vector B a velocity component equal to c.

As the magnitude of v_o increases, the lines of the B field no longer appear normal to the direction of v_o, in the reference frame of the particle, but instead are directed along the hypotenuse of the triangle of figure 5. Thus the value of the cross product in equation (30) becomes:

(36)

Figure 5

In a particle accelerator, we provide a constant acceleration to a particle of mass m, to confine its motion to a circle of defined radius. Referring to Newton:

(37)

For a given B field, the effective acceleration of a fast moving particle is given by substituting (37) into (36):

(38)

In SRT, no allowance is made for the reduction in force obtained by (36), and the following expression for the mass of the moving particle is obtained:

(39)

The original basis for the Lorentz transformations, and thus special relativity, was the assumption that the observed velocity of interaction of light with matter represents a unique velocity of the electromagnetic wave. This arbitrary decision is not borne out by Maxwell's theories or by any test that might prove that EM energy actually travels in a continuum of velocities. The second postulate as stated by Einstein does not deserve the status of a postulate, as it is at once overly restrictive and ultimately phenomenological--the nature of c is based on experimental measurement rather than on analysis of first principles. RCM’s modified second postulate, however, says nothing about the actual propagation of EM energy, but only of the relative speed with which it must interact with matter to be detected. Utilizing this modified light principle we obtain an intuitive Galilean form invariance for Maxwell's equations.

RCM places no upper limit on attainable velocities, and allows for the possibility of communications between humans or particles at speeds far in excess of c. This precludes many of the compatibility problems between the highly successful quantum mechanics and relativity theory. We have seen quantitatively in earlier papers how the concepts presented herein account for clock slowing due to motion and gravitational fields [1, 2], the Shapiro time delay [3], Mercury’s anomalous perihelion advance and photon deflection by a gravitational field [4]. Returning to first principles in this important area of electrodynamics should not only resolve problems inherent in relativity theory, but also spur advancement in areas of research that have atrophied since the advent of those theories.

Full mathematical derivations of all concepts presented in this article are available by request from the author, free of charge.