The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell'

The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell's Equations

Dimensional analysis of Maxwell's equations in a planar electromagnetic wave form implies wave propagation at a speed of c, defined as (e_om_o)^-1/2. But such analysis does not specify anything at all about the specific values of e_o or m_o. Thus Maxwell's equations in and of themselves say nothing about the specific velocity of propagation of an electromagnetic wave, nor of the detectable velocity or range of velocities in any particular observer's frame of reference. The generally accepted frame-invariance of c, and hence e_o and m_o, constitutes an assumption. The Lorentz transformations allow the preservation of the form of Maxwell's equations in any inertial frame of reference (IFR) under this assumption, an assumption which Einstein raised to the status of a postulate.

But really it may be only the experimental means by which we measure the speed of light c, or e_o and m_o, that produces the observed frame invariance.

The principle of equivalence tells us that in a uniformly moving reference frame, any experiment performed in that frame should produce the same results as if performed in a "stationary" frame. Clearly, therefore, unless one is willing to assume the existence of an aether (which implies a preferred reference frame), all experiments will result in a measured "velocity" of c in any uniformly moving frame of reference, regardless of the actual behavior of the light itself. For example, we may determine a value for e_o by measuring the capacitance between two plates. Observers in different IFRs performing this experiment in their own reference frames will each obtain the same result. Thus each of several observers in different IFRs will measure or observe the velocity of light from a distant source to be intersecting their apparatus at a velocity of c.

In light of the above arguments, the second postulate can be more precisely worded to state: "The observed velocity of light is constant from all inertial frames of reference." In order to understand the distinction, we must develop a model which obeys the modified second postulate (with the word observed), but violates the original. To do this, we state what will henceforth be referred to as the Radiation Continuum Model (RCM) of light, and then look at some of the properties of this model:

Electromagnetic radiation propagates at all velocities from zero to some undetermined upper value C. As demonstrated by laboratory measurements, only that component of this radiation which passes a physical observer at a relative velocity of c in the observer's frame of reference can produce any physical interaction and hence be detected. All other velocity components of this radiation are undetectable by that observer, or by any other electro-mechanical device which is stationary in that frame of reference. Any observer in motion relative to the first observer will, in general, detect a different component of the radiation, that component being the one which has a relative velocity of c in its frame of reference.

While the above description may at first seem strange, due to our being accustomed to thinking of light as having a single velocity, it must be realized that this model is no more unusual than SRT itself. Under SRT, the distance to or time since an event takes on an infinitude of values, being dependent on the velocity of any particular observer. Under RCM theory, distances and time maintain their intuitive Galilean properties, while the velocity of light with respect to a given source takes on an infinitude of values. This leads to several testable differences between RCM and SRT, which will be outlined in the Summary.

If one assumes the truth of Einstein's second postulate as stated, and also wishes to preserve the truth and invariance of Maxwell's equations, then one must adopt the Lorentz transformations. If one attempts to use Galilean transformations and adopt Einstein's second postulate, then Maxwell's equations will not be invarinat. This does not mean that Maxwell's equations are inherently only Lorentz invariant. One must deal with three quantities, length, time and the speed of electromagnetic (EM) propagation in transforming Maxwell's equations between reference frames. When one chooses, as did Einstein, to assume that the velocity of EM propagation is always strictly c, then one must use Lorentz transformations to keep the form of Maxwell's equations consistent. This is done at the expense of standard concepts of length and time, both of which become distorted to accommodate the constancy of EM velocity.

The radiation continuum model simply adds one word to Einstein's second postulate, which we will call the modified second postulate, stated as follows: "The observed velocity of light is constant for all inertial frames of reference, and is independent of the motion of the source." Through the use of the modified second postulate, making the observed velocity of light dependent on the observer, it is a simple matter to show Galilean invariance of Maxwell's equations.

Consider the situation depicted in figure 1, where the observers in the stationary (K) and moving (K') frames are at the origins S and S' respectively. We will let the origins be initially coincident at the time of an event at A, a distance x from S. In the RCM, where 0<c<C, the component velocity of light in the non-moving K frame is c. In the moving K' frame (moving with a constant velocity v), c' = c, but in the K frame c' is c+v. Also, we will assume absolute time, such that t' = t, rather than arbitrarily assuming that time itself is different for various IFRs. Without any loss in generalization, we will restrict motion of the K' frame along the x axis. In this case, the Galilean transformations are:

t = t t' = t

x = x x' = x + vt

y = y y' = y

z = z z' = z (1)

(1)

From a simple treatment of wave mechanics, where we have chosen a wave propagating in the x direction with Ez = 0, we can write: [3]

(2)

where c represents the velocity of propagation. If we let a flash of light occur at some distance x from the origin of K, then we can let x be represented by c times the time it takes the light to reach an observer at the origin of K. Thus, we can derive the following relations:

(3)

(4)

(5)

(6)

where we can pull a out of the partial as a constant. We can also derive two very useful realtions from (3), whereby we express

in terms of

and c:

(7)

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

(8)

(9)

(10)

Equations (9) and (10) demonstrate that the wave will remain form invariant in going from the K frame to K', provided that the velocity observed in K', as measured in K, is c + v. Such a wave will have a velocity as observed or measured 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 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 will not place any constraints on the upper bound of C.

We could, of course, assume that rather than the K' and K observers being susceptible only to that component of light which has a velocity of c in their respective reference frames, that c' and c are actually physically the same. There is no obvious reason for doing so, since the only thing we know about EM propagation is that we only observe components with a velocity of c in our own reference frame. However, if we do make the assumption that c' = c, then we must abandon the Galilean transform of equation (1). Now, the Galilean transform is actually a special case of Lorentz-type transformations. This type of transformation simply picks some arbitrary velocity V which transforms into itself. In the Galilean case, V is infinite. In the case of special relativity, V = c, and we know the implications of invoking that particular ceiling.

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

(11)

However, this is no profound statement. Since c' was forced to transform into c, the left side of equation (10) could simply begin with c'² and, of course, such a statement holds under a Galilean transformation as well. Consider again the simple case of motion in the x direction only. We now have, using equations (6) and (10):

(12)

Now we will consider the transformation of Maxwell's equations themselves. Our goal here is simply to ensure that the assumed wave equation of (8) is actually valid. We are still considering a wave traveling in the x direction, with Ez = 0, so we can make the following statements: [1]

(13)

Substituting equation (13) into these equations, we can write them explicitly in three dimensions as:

At this point we need to develop explicitly the transformation for J. 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:

All that remains is to show that primed equations (18) - (25) in K' remain form invariant under the Galilean transformations of (1). We will express the equations one at a time in terms of the primed quantities and show how they transform under equations (1), (6), (7), (10) and (27).

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: [6]

We will consider the source stationary in the K system, with the wave traveling in the same direction in which the K' system is moving. From the perspective of the source, all velocity components of the EM wave leaving the source must equate to the same frequency, thus the wavelength varies with the component velocity. We will therefore subscript the notation l with the velocity of the component with respect to the source whenever we consider a wavelength. This allows us to derive a simple relation between component velocity with respect to the source and the wavelength of that component, compared with the wavelength of the component leaving at c, as would be perceived by an observer stationary in K. If we let c' = ac as before, this relation becomes:

(38)

Note that it is not practical to obtain a clear visual image of how such a wave appears. Since the effective wavelength of a given photon takes on physical significance (as far as our experiments are concerned) only upon being absorbed by an observer, we can not actually visualize this wave as, say, a standing wave or a pulse traveling down a string. The wavelength is simply a mathematical expression of the ratio between the particular velocity component of interest and the overall frequency (or energy) contained in the photon as a whole. The visual image of a wavelength may or may not have significance either while the photon is in flight or upon absorption.

From (37) and (38) we can express the wave equation for any velocity component c', as it appears with respect to the source, stationary in K, as follows:

(39)

Now, the wavelength in (39), lc', will be the same under any transformation between inertial reference frames. The wavelength is simply a measure of length, such as a millimeter, and a millimeter in K' is the same as a millimeter in K. While distances, such as the distance of the origin of each system to an event, transform under equation (1), the definition of a given length, such as a ruler carried by each observer, will remain the same in each system under the Galilean transforms. Thus we can see how the wave of equation (39) will look to an observer in K' by replacing t with t (since t' = t), x' with x, c' with c, and expressing lc' as al:

(40)

Notice how the wave number, k, the number of waves in 2p units of length, transforms to 2p/l_c’, as it should for an observed wavelength of l_c’. Also, the frequency of the wave for the K' observer has been shifted to the red:

(41)

Stated simply, the observed frequency in K' is equal to the frequency of the source times the ratio of c (the observed velocity) to the component velocity as emitted in K, the reference frame of the source. Note that we can also express the red-shifted frequency as c over the wavelength observed in K', or simply c/l_c’. This formula (41) for the Doppler shift is basically the formula attributable to Newton. Experimental evidence obtained from radar ranging to Venus illustrates a preference for (41) over the special relativistic formula [5].

We will 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 will be 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 this observer is sensitive, c', illustrated in figure 2, is given by:

(42)

(43)

(44)

To obtain the frequency as observed in K', we simply replace c' with c, as we did in (40), and obtain:

(45)

Once again, we see that the observed frequency in K' is equal to the frequency of the source times the ratio of c (the observed velocity) to the initial velocity, c', where, from (42), c/c’=g.

Returning to figure 2, we see that the observer in K' must look along the line given by the resultant of his velocity and the c' component traveling along the x axis. From the figure, this angle is easily determined to be given by the following:

(46)

Thus, an observer moving in a direction transverse to the line joining the earth to a star would have to tilt his telescope at the angle given by (46). This predicted angle is consistent with observation. [2]

We will allow light to fall apparently perpendicular to an observer in a reference frame, K', moving with a velocity of vy with respect to the source in K, as in figure 3. In order for light to appear to be perpendicularly incident, it must have a velocity made up of c along the x axis, and of v in the y direction. Just as when one leads a moving target with a snowball, this light will track the observer in the y direction, and appear to have a velocity of c with respect to the moving observer in the x direction. The actual velocity in K of this component of light, c', is then given by:

(47)

In a manner analogous to the previous section, it is clear that the frequency as observed in K' will be given by:

(48)

In figure 4, the observer is traveling to the left with a constant velocity. A burst of light is leaving the source, so that it will strike the observer at the angle indicated. It is interesting at this point to again break the motion of the observer into two orthogonal components, the first being along the line of sight to the source, the second being perpendicular to this line.

While one could calculate the effect due to one motion, and then shift the resulting frequency again to account for the other motion, it is easier in practice to simply determine the initial velocity required to ultimately strike the observer with a speed of c at the angle indicated. Once this velocity is determined, the Doppler shift is simply given by the ratio of c to the initial velocity as before. It is clear from the figure that the initial velocity component of light leaving the source is given by:

(49)

From this, we get the observed frequency of light striking an observer at an arbitrary angle of incidence, by multiplying the source frequency by the ratio of c to the velocity of the component as measured in K. This is done below, where the right hand term has been derived by dividing through by c and expanding the denominator, noting that

, and

(50)

From equation (50), it is clear that the Doppler shift due to radial motion, proportional to v/c will swamp the effect due to transverse motion, proportional to v²/c², unless cosq approaches 0. This is why experimental measurements of the transverse Doppler effect are limited to cases in which all motion is transverse, such as rotor experiments utilizing the Mossbauer effect, where even the radial motion of the atoms themselves is well controlled. Such measurements confirm equation (45), which is identical to the special relativistic formula. When any radial motion is involved, especially for v<<c, equation (41) provides very accurate results, while equation (50) would prove more accurate than the limits of experimental error for most studies.

The analysis of the force on a moving charged particle above a neutral, current carrying wire is an interesting one. As we view the experiment from different reference frames, using the covariant special relativistic approach, the force is caused by different electromagnetic properties, first a magnetic force, then an electric one. First a neutral wire, then a charged one [3]. We would like to be able to solve this problem without resorting to length contraction, since such effects obviously imply preference for Lorentz transformations, which we have been very careful to eliminate from our treatment of Maxwell's equations.

We begin with the formula for the force on a moving particle outside an infinitely long current carrying wire:

We also note that 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 [3].

We now define v_o as being equal to the velocity of the charged particle with respect to the mass of the wire or medium 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 the moving negative charges in the frame of the wire as v. Thus, 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 frame D of figure 5. If we wish to consider velocities of particles or wires in a direction other than the direction of current flow, we must project the velocities onto this direction and define a unit velocity vector in the direction of current flow. For these discussions, we will, with no loss due to the generalization, consider the case of movements only along the direction of this vector.

Generally, if we have a current I in a wire stationary with respect to our reference frame, we attribute the current to the motion of some of the negative electrons, the conduction electrons, while the positive nuclear charges stay relatively fixed with respect to the wire. We will define the density of the conduction electrons to be r_-, and the density of the charges at rest then becomes r₊. Further, r₊ is the negative of r_-, since we are considering an uncharged wire. Now, depending on our frame of reference, we have two currents, one due to the flow of the negative charge density with respect to our IFR, the other due to the flow of the positive charge density with respect to our IFR. Thus, we have the following expression for the total current in the wire, where v_- and v₊ are the velocities of the negative and positive charge densities, respectively, with respect to our reference frame, and A is the cross-sectional area of the wire [4]:

(52)

We can rewrite equation (51) as follows, where we replace B with the equation for the field at a distance r due to a current I [3]:

(53)

Substituting (52) into (53) yields the total expression for the force on a charged particle moving with respect to a current carrying wire. This expression remains applicable when viewed from any inertial frame of reference, as several examples will show:

(54)

Consider first the situation depicted in frame A of figure 5. In this figure, we are stationary in the reference frame of the wire. Thus, the entire current is due to the motion of the negative charge density, or I = r_-v_-A. Further, the velocity of the charged particle with respect to the current carrying wire, v_o, is v/2. Thus, the force on this particle becomes:

(55)

This is the expected answer for the force on a moving charged particle due to a stationary current carrying wire.

Next, consider the situation depicted in frame B of figure 5. In this example, we are moving with respect to the reference frame of the wire to the right at a constant velocity of v. Thus, in our reference frame, the velocity of the negative charge density is zero, but the velocity of the positive charge density is -v. Additionally, the velocity of the charged particle with respect to the current carrying wire, v_o, is still v/2, the same value used in the previous example. Also, since r₊ = -r_- , and v₊ = -v_-, the force on the charged particle will be given by:

(56)

Now, consider the case depicted in frame C of figure 5. Here we are moving at the same velocity with respect to the wire as is the charged particle, v/2. In our reference frame, there are two currents in the wire, and the charged particle is still moving with respect to the wire at a velocity of v/2. In this example, the force on the particle is given by:

(57)

Finally, we analyze the case depicted in frame D of figure 5. In frame D we are moving at a velocity with respect to the wire of 2v. In this frame, the total current is given by:

(58)

As in each of the previous cases, the total current is exactly the current we observe when stationary with respect to the wire. Once again, the velocity of the charged particle with respect to the current carrying wire is v/2. Thus the force on the charged particle is given by:

(59)

From the four examples provided above, we see that the force on a moving charged particle due to a current carrying wire is the same regardless of the reference frame of the observer. Even more importantly, the force is a magnetic force in all frames of reference. Such is not the case in the covariant, relativistic view of things. In special relativity, an observer moving with the charged particle will not see a magnetic force at all. Instead, the moving wire will produce an electric field. This requires that the previously uncharged wire suddenly becomes charged when viewed from another reference frame. This is accomplished by 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. This length contraction causes an increased charge density of the positive charges--same density, shorter length--and thus the wire becomes charged. This is unsettling at best. As we move from one reference frame to another, we see what was once a magnetic field effect vanish, we see an electric field arise where there was none, and a previously neutral wire acquires an excess positive charge. The most confusing point is that the velocity of the particle, v_o, is measured with respect to our arbitrary frame of reference, rather than the frame of reference of the current carrying wire. True, the math works out in special relativity such that the correct answer is achieved, but the solution appears contrived and counterintuitive. We have seen how we obtain a Galilean form invariance for Maxwell's equations utilizing RCM theory. Here we see how much simpler the analysis of classic problems becomes when we apply this intuitive interpretation of Maxwell's equations.

We can use equation (51) to determine the force on a charged particle in a magnetic field of some fixed value B. However, the geometry of this equation as viewed from the laboratory frame is equivalent to that as 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. It is well known that the photon is the carrier of the electromagnetic force, and also that such forces must interact at a velocity of c with respect to the observer. We can therefore 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 back along the hypotenuse of the triangle of figure 6. In the figure, the value of the cross product in equation (51) would be given by:

(60)

In a particle accelerator, we wish to provide a constant acceleration to a particle of mass m, to keep it, say, confined to motion in a circle of defined radius. Referring to Newton:

(61)

(62)

Thus we see that, for a given B field, the effective acceleration of a fast moving particle, a’, is given by the slow velocity acceleration, a, divided by g. In SRT, no allowance is made for the reduction in force obtained by (60), and the following expression for the mass of the moving particle is obtained:

(63)

where m_o represents the so-called rest mass of the particle. Utilizing the obvious force reduction of (60), RCM produces the following expression for the mass of the moving particle:

(64)

It is clear from this presentation that the preferred interpretation is that of a decrease in the effective force on the particle, and that the concept of mass increase with velocity is unnecessary.

The original basis for the Lorentz transformations, and indeed, all of special relativity, was the assumption that the observed or measured velocity of interaction of light with matter also represented an actual unique velocity of the electromagnetic wave itself. This is an arbitrary decision, not born out by Maxwell's theories or by any actual 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 value of c is based on experimental measurement rather than an analysis of first principles. The modified second postulate of RCM theory, however, says nothing about the actual propagation of EM energy (although RCM theory itself does), but only of the relative speed with which it must interact with matter to be detected. Utilizing this modified light principle, we obtain a Galilean form invariance for Maxwell's equations, which is satisfying and intuitive, and accounts for the observed Doppler shift in both the radial and transverse directions.

There are several distinctions between RCM theory and SRT, which lead to testable differences, some more technologically feasible than others. Among these tests are a slight distinction between RCM and SRT Doppler shift equations at very high velocities. A test of these differences is not likely at the present time. Another test involves the issue of simultaneity. RCM and SRT predict differently the arrival times of light from a distant event at stationary and moving observers. Such a difference may be testable in either a laboratory or in space. Another test, involving topics not covered in this paper, involves different predictions regarding the time unit of clocks constructed in different IFRs. The apparent equal success of each theory in predicting results of experiments to date, combined with the fundamental differences in the underlying mechanics, warrants further analysis of the differences and performance of each theory.

When considering the Doppler shift at an arbitrary angle of incidence as depicted in Figure 4 and equation (50), the similarity between the SRT formula and the RCM formula is quite straightforward. The first point to consider is that the angle used in the RCM calculation is the apparent angle to the source as viewed by the observer. The SRT formula instead uses the actual angle between source and observer. In figure 4, this would be the angle formed by the line joining the source to point O (the observer) and the line representing the velocity of the observer with respect to the source. To a very high degree of accuracy, we can relate these two angles as follows, where q_S is the SRT angle, and q_R is the line of sight angle of RCM:

(65)

Substituting (65) into (50) and expanding, we derive a form of equation (50) dependent on the angle of SRT instead of the line of sight angle of RCM.

The last expression in equation (66) is the relativistic Doppler equation for an arbitrary angle of incidence. In the approximate form I have used to relate the two angles, there should be an additional term in equation (66) on the order of v²cos²q_S/c². However, this term is as much a part of the approximation in relating the two angles as it is to the binomial expansion. In calculating the actual variations between SRT and RCM, without approximations, we see that the worst case variance between predicted Doppler shifts is less than 5E-13 for speeds up to 30 km/sec. Such a level of difference is well below the limits of detectability of any experiment involving current technology. The best hope lies in the measurement of Doppler broadening obtained by reflecting light off the bearings of an ultra-high-speed centrifuge.

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