A "Spanish physicist" has reviewed Setterfield's paper at
http://www.setterfield.org/criticalreview.htm. I shall not go through all six of his criticisms here, nor do I agree with all six. HIs last criticism, however, together with Setterfield's response, calls for further comment.
The Spanish physicist writes:
According to the author, epsilon varies like h and therefore alpha varies like c.
Setterfield responds:
Incredibly, I have never said that alpha varies like c. What I have said is that all atomic constants which vary do so in such a way as to give rise to the redshift. Alpha is one of these ‘constants.’ Therefore the maximum change in alpha could only be 1.7 and not “many magnitudes.”
Who is right? In fact, neither. As we shall see below the Reviewer, in determining that alpha varied as c in Setterfield's model, assumed that in Setterfield's model electrical charge was conserved. It is easy to show that it is not. As I shall show below in Setterfield's model electrical charge varies directly with c, so alpha varies inversely with c. Both the reviewer and Setterfield were wrong.
The fine structure constant alpha = e^2/(2*pi*epsilon*h*c), where h is Planck's constant, c the speed of light, -e the charge of the electron and epsilon the permitivity of free space. Since Setterfield has written that the product h*c is constant in his model (although h and c each vary with time) it follows that the time-dependence of alpha is carried in its e^2/epsilon term.
Since Setterfield has written that epsilon varies inversely with c, it would follow that alpha is proportional to c if the electronic charge, -e, were constant. The Spanish physicist Reviewer evidently thought that e was constant in Setterfield's model. That would not be a bad physical assumption except that it is contradicted by the version of Maxwell's Equations that Setterfield has chosen to use.
Let us consider conservation of electrical charge using Setterfield's version of Maxwell's Equations. Our treatment will closely parallel that used in standard (constant c) electrodynamics to derive the equation of continuity for electrical currents and charges.
The two relevant Maxwell Equations are, according to Setterfield in Section 3.4 of his paper:
epsilon * div E = rho (M1)
curl H = J + epsilon (d E/d t) (M2).
These are different from what Setterfield has written because he wrote the equations for charge and current-free vacuum. Remember that Setterfield allows epsilon to be time (but not space) dependent. Here rho is the charge density and J the current density. Since I cannot type a partial derivative sign here I have used "d" to denote partial differentiation. Please do not confuse this with its standard meaning of total differentiation.
Now we proceed in the standard way. We take the divergence of (M2). Since the divergence of a curl of a vector is identically zero we have:
0 = div J + epsilon div (d E/d t) (MK1).
But because div and d/dt are just plain differential operators, it follows that they commute, so we have:
0 = div J + epsilon (d(div E)/d t) (MK2).
Substituting Equation (M1) into (MK2) we obtain:
0 = div J + epsilon (d(rho/epsilon)/d t) (MK3).
This is the equation of continuity for charge and current density in Setterfield's model. Compare it to its counterpart from standard electrodynamics (constant epsilon) where one finds
0 = div J + d(rho)/dt, which is what Equation (MK3) reduces to if epsilon is time-independent.
Equation (MK3) is called the equation of continuity because of the following consideration:
Integrate Equation (MK3) over some finite (fixed) volume of space.
0 = Integral_V(dV div J/epsilon) Integral_V(dV (d((rho/epsilon)/d t)) (MK4).
The second integral, that of rho/epsilon over V, is just the total charge Q in volume V divided by epsilon.
The first integral may be transformed by means of the Divergence Theorem to an integral over the surface of V of the outward normal component of J/epsilon. The latter, physically, is just the total rate of net current passing outward through the surface of volume V divided by epsilon per unit time.
0 = Integral_S(dS.J/epsilon) + d(Q/epsilon)/dt MK5).
Thus we see that the rate of increase of (charge (Q) divided by epsilon) within a fixed volume is equal to the total current flowing into that volume through its surface divided by epsilon.
If there is no current flowing through the surface of the volume, Equation (MK5), implies that Q/epsilon is constant within that volume. In other words, absent current, Q is proportional to epsilon.
This holds for any charge Q (absent currents) in Setterfield's theory, including the electronic charge -e. We can imagine a volume containing just one electron with no electrical current going in or out of that volume. Then we have that e/epsilon is constant, or e is proportional to epsilon in Setterfield's theory.
Since epsilon is proportional to 1/c in Setterfield's theory, e is also inversely proportional to c. Therefore e^2/epsilon is also proportional to 1/c, and therefore alpha varies as 1/c.
This means that when ca was 10^10 times its present value sometime during early creation week, alpha was the same factor smaller than its present value of ~1/137. Obviously this is not in agreement with experiment, which shows alpha changing by at most only a vew parts in 10^5 back to a billion or so atomic years ago.
Setterfield has said that the change in alpha for his model is not more than that allowed by experiment back to z = 1.7 (a few parts in 10^5), but we have derived a different relationship, that alpha varies inveresly as c, from Setterfield's own model.
Which should you believe? Should you believe what Setterfield says about Setterfield's model or what may be derived from Setterfield's model.
All scientists immediately know the answer. What counts is what may be derived from Setterfield's model. What Setterfield has to say is of no special significance. Indeed, were there an infinite number of Setterfield clones, each swearing in an infinite number of courtrooms an infinite number of times on an infinite stack of Bibles that his model has alpha varying only slightly, all of that testimony would be easily outweighed by a single, simple derivation from his model of what his model actually predicts.
Setterfield's model predicts that alpha varies inversely with c. The Spanish physicist Reviewer was mistaken when he said that alpha varied with c, but he had the right idea. Setterfield did not, and does not.
