### Varying Lightspeed, Special Relativity & Maxwell’s Equations

Question: Doesn’t the theory of Special Relativity require the speed of light to be an absolute constant? Further, since Maxwell’s equations can be derived from Relativity, do they not also preclude a variation in lightspeed?

Response: This is a two part question.  First we will deal with relativity and then with Maxwell’s equations.  This first response is technical.  There is a brief layman’s summary at the end.

(i)  Special Relativity
Let us take the standard accepted equation that c2= 1/(em) where c is the speed of light, e is the electric permittivity of free space with m being the magnetic permeability of the vacuum. The objection is that any variation in c and the components of this equation is contrary to the theory of Relativity since the vacuum should have the same properties to any observer in motion. Thus the values of both e and m should be the same for all inertial frames which leads to the conclusion that c should be constant. But the demands of Relativity are still fulfilled if, at any instant, the individual values of e and m are isotropic throughout the cosmos. So, if c is varying, it is doing so on a cosmological scale. This means that, at any instant, c would have the same value in all frames of reference throughout the cosmos.

But we can go further. The formulation which led to the conclusion that the speed of light must be constant contains two equations each describing a velocity [see A.P. French Principles of Modern Physics, pp. 152-155, Wiley 1959]. These two equations must equal each other. The conclusion is then drawn that the two velocities must be equal. Since the velocities involved are the speed of light, c, then c must be the same in all frames of reference and hence a constant. This is one conclusion that can be drawn. But another, equally valid, conclusion can be drawn. Since the velocities in these equations were given by a distance divided by time, and since the distance is required to remain intact, then exactly the same result is achieved by multiplying the velocity, c, by time, t, so that ct is a constant. This maintains the integrity of the equations and, in fact, is the way in which the final formulation appears. Thus French [op. cit.] gives the final equation in this set as being x’2 – c2t’2 = x2 – c2t2 = G. French then states “This expresses the constancy of the velocity of light as measured in different frames of reference. From this equation, all that is needed is that the product c2t’2 = c2t2 or, more generally, that

c12 t12  = c22 t22  =  constant

without saying anything about mutually canceling variations in c and t. This condition is automatically maintained with Zero Point Energy and hence lightspeed variations (as can be seen in the research papers section) where t is atomic time intervals.

Thus, with changing ZPE, the formulations of Relativity are valid if atomic time, t, is used for any measurements. The same is true for the Schwarzschild metric which requires the invariance of the expression x2 + y2 + z2 – c2t2. A comment by Van Flandern is pertinent here. He had measured the atomic clock as slowing compared with orbital time. As a consequence, he stated “Assumptions such as the constancy of the velocity of light … may be true only in one set of units (atomic or [orbital]), but not the other” [NBS Special Publication 617]. This is the kernel of what has already been said above. Since the run-rate of the atomic clock is proportional to c, atomic time intervals, t, will be inversely proportional to c. It then becomes apparent that c will always be a constant in the atomic frame of reference, while ct will always be constant in the orbital frame of reference.

Some have explored other alternatives. Thus Magueijo pointed out recently that “…the urge to reconcile VSL [variable speed of light] to relativity is motivating much ongoing work… It now appears that the constancy of c is not so essential to relativity after all; the theory can be based on other postulates. [J. Magueijo, “Plan B for the Cosmos,” Scientific American, January 2001, p.47].

(ii)  Maxwell’s equations
Those who were taught to derive Maxwell’s equations from Relativity may object that the basic equation we started with is obtained on the assumption that c is a constant.  But, as shown in Bleany & Bleany Electricity and Magnetism, pp. 236-239, Oxford, 1962, this equation can be readily derived without any initial assumptions about the behavior of c, e, or m, just as Maxwell himself did.  This is done by obtaining a set of four simultaneous partial differential equations based on (a) Gauss’s theorem applied to electrostatics; (b) Gauss’s theorem applied to magnetic fields; (c) Faraday’s and Lenz’s law of electromagnetic induction; and (d) Ampere’s law for magnetomotive force. The results are discussed by Bleaney and Bleaney, as referenced above.

The outcome is that the equations are valid provided that the changes in the speed of light are slow compared with the rate of ticking of atomic clocks or the speed of atomic processes. But this condition is always maintained with a varying ZPE since a ZPE change manifests as a change in e, m, and c, as well as proportional changes in atomic clock rates which govern atomic processes. Thus the required condition is always met by the very basis of the formulation.

A final point can be made. Another derivation of Maxwell’s equations shows an almost infinite variation in e, m, and c is permitted [K. Wanser, personal communication]. The analysis here therefore indicates that both Maxwell’s equations and Einstein’s formulations can be satisfied with a varying ZPE, e, m, and c.

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Layman’s summary: There are two basic properties of space:  the electric property (permittivity)  and the magnetic property (permeability).  These govern the speed of light.  It is considered in physics today that it is a basic requirement of relativity that the speed of light be constant.  If this is correct, then the properties of space must remain the same throughout time as well, since they govern the speed of light.

However, all that relativity really requires is that all the properties of space should be the same at any given time, anywhere in space, no matter in what direction or how fast any observer is moving.  Therefore, the properties of space ARE allowed to change, as long as that, at any given moment, they are the same throughout the universe.   This would also apply to the speed of light.  It is not forbidden by relativity that it should change, but rather that any change be universe-wide.  Research has shown that the speed of light is not a constant.  Therefore it needs to be recognized that the properties of space have also changed with time.  The data indicating this is the basis for much of the work I have been doing for the last 27 years.

There are those who would dispute the idea of change in light speed being possible because of the way the relativity equations were put together.   The standard equation for velocity is distance divided by time (v=d/t).  This is something we learned in high school.  When we multiply both sides by ‘t’, we get vt=d.   Distance, here, is how far the light is traveling.  Now, if you travel a specific distance at a faster speed, it will take you less time.  The distance for the slow traveler and the fast traveler is the same, but if the speed goes up the time goes down.  This is an inverse relationship.  This applies to the speed of light.  The distance remains the same, but if it is going faster then the amount of time the travel takes is less.  Thus, all distances in the different velocity equations remain equal to each other – the same – while it is not necessary for either time or speed to remain the same.  It is just that the two of them are in inverse relation to each other.

There are basic equations which are, in essence, v=d/t.  As long, therefore, as the distance remains the same, there is no requirement for either time or velocity to remain constant.  Thus, there is no requirement in relativity for the speed of light to remain constant.   The assumption that the speed of light is constant is not a necessary assumption.  It is only distance that must remain constant – which means that velocity times time (vt) is a constant, since, as explained above, the faster light goes the shorter a time it will take to arrive at the destination.

August 16, 2006

response from Dr. Mark Kluge and Setterfield's response to Kluge