One important concept when considering the frame invariance of Maxwell’s equations involves the nature of a spherical wave front. Imagine several inertial reference frames (IFRs), traveling in different directions with different velocities. For an instant, we allow the origins of each of these frames to coincide. At that very instant, we cause there to be a flash of light emitted at the instantaneously coincident origins of all frames. Let us pick one very special reference frame, in which the source is stationary. Clearly, in this IFR, light will expand from the origin of the reference frame in a spherical shell. In other words, assume that within this reference frame, we draw a sphere of radius one light second. We now randomly place light detectors at various points on the sphere. One second after the flash, all detectors on this sphere will simultaneously go off, indicating that light from the flash has reached them. There is no question that, in this frame where source and detectors suffer no relative motion, the light from the source expands in a spherical shell. But what of the other IFRs whose origins only coincided with this special frame at the time of the flash? Will they also experience an expanding spherical shell?

In order to answer this question, we begin with the simplest case. In figure 1 there are two reference frames, K and K’. K will be the reference frame of the source of the flash, occurring at the origin of that frame. K’ is moving at some velocity, v. With no loss due to generalization, we will orient the K’ frame with the x axis pointing along the direction of motion. Also, since we have already verified that the wave front is spherical in K, we will orient this frame so that the axes coincide with those of K’ at the instant the two frames origins coincide. At the instant the two reference frames coincide, we allow a flash of light to be emitted in K. We wish to determine how this wave front will appear to observers in the K’ system. In order to do this, we randomly place light detectors on the surface of a one light-second radius sphere in K’, as we did in the previous example. Our goal is to determine the time of arrival of light at these detectors as measured in K’, as well as the apparent direction of the source as observed in K’. By plotting all such observations, we will obtain a clear picture of the nature and shape of the wave front as observed in K’.

Figure 1.

Let us pick an arbitrary point on the x-y plane of K’. Under RCM theory,⁽¹⁾ the light to which an observer in K’ is susceptible must leave the source in K with the a velocity given below:

(1)

In (1) and in figure 2, q_r represents the aberrated line-of-sight angle to the source, while q_s represents a line drawn from the source at the time of emission to the observer at the time of detection. q_s is the angle used in the calculations of SRT. In order to plot the expansion of this wave front, we require a relation between q_s and q_r. From figure 2 (taken directly from reference [1], we find this relation to be the following:

(2)

Figure 2.

With the relations (1) and (2) we can plot what the wave front seen in K’ will look like as measured in K, realizing, of course, that no observer in K is susceptible to the wave components seen in K’. In table 1, we do this for a K’ velocity of -.1c (in eq. (1) a positive velocity is one directed toward the source in the K system. In this example, K’ is receding from the source in K by definition). The first column provides 16 equally spaced values of q_r. The second column provides the corresponding angle q_s as measured from the origin of K. The column c’ gives the required velocity component with respect to the source to be seen by an observer in K’ at the angle q_r indicated. This combination of q_s and c’ is then broken down into c’_x and c’_y components so that it may be plotted.

Table 1. q_s and c’ computed for sixteen values of q_r

Figure 3 plots the c’_x and c’_y values with respect to the origin of K. Notice how the shape of the wave front is a sphere, centered on the origin of K’.

Figure 3.

We can now take each of the sixteen sample points on the wavefront, and replace q_s with q_r, the aberrated line-of-sight angle experienced by an observer in K’. When this is done in figure 4, we see exactly the form of the wave front as seen in K’. The observers in K’ will all see the light from the flash one second after it has been emitted, and each observer on the sphere will have to point its telescope in the direction of the origin of K’. The observers in K’ will see a spherical wave front, expanding from the origin of K’, with all components of the wave front expanding at a velocity of c as measured in K’.

Figure 4.

Now we can consider alternate orientations of the reference frames. We initially chose the axes of K to coincide with the axes of K’. However, there is nothing special about this orientation. We can choose an arbitrary reference frame, k, with an origin coincident with and stationary in K, and orient the axes in any manner we choose. Clearly, except for adding a small amount of complexity to the derivations, the analysis of all equations as seen from k proceeds exactly as it did for K. Thus, K’ need not be moving along the x axis of K, it can, in fact, be moving in any direction in three space. Also, while we have initially chosen the x axis of K’ to lie along the direction of motion, this is clearly not necessary either. Any reference frame, k’, with an origin coincident with and stationary in K’ will produce the same results. The observed wave will be spherical in k’.

One can easily see that this is the case by observing the equations utilized. All equations use only the terms v, q_{r and} q_s. The velocity v is simply the velocity of the origin of K’ with respect to K, and is quite independent of the orientation of K and K’. This velocity can be expressed in any arbitrary reference frame R, which may or may not coincide with the orientation of K or K’. Further, q_r and q_s are expressed only in terms of the velocity and the origins of K and K’ as well. Thus K’ may have any velocity in any direction with respect to K, and may also have any orientation with respect to K, and the results will still be equivalent. A spherical wave front emitted in K will give rise to a spherical wave front in K’. In figure 5, we illustrate a series of three reference frames. K and K’ are as before, but K" is a reference frame with an arbitrary orientation and arbitrary velocity direction with respect to K. The only restriction on K" for this example is that the origin of K" coincide with K and K’ at the time of the flash. As with the previous example, when the values of c’ are calculated for each value of q_s, the end result is a spherical wave front in K" centered on the origin of that reference frame.

Figure 5.

Finally, we may lift the restriction that the origins of the reference frames coincide at the time of the flash. For any reference frame K", we can define another reference frame k", stationary with respect to and with the same orientation as K", but with the origin shifted in any value of (x, y, z). It is obvious that a spherical wave front in K" will also be spherical in k", but with the center of the wave front shifted from the origin of k" to (-x, -y, -z).

(1) Renshaw, Curtis E., Galilean Electrodynamics, "The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell's Equations," Volume 7, Number 1, January, 1996: If Einstein's second postulate is not invoked, it is a simple matter to show that Maxwell's equations are Galilean Invariant. Doppler, apparent mass increase with velocity, and other effects are explained.

Curt Renshaw
http://renshaw.teleinc.com
email to: crenshaw@teleinc.com
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