Setterfield, in his article "ATOMIC QUANTUM STATES, LIGHT, AND THE REDSHIFT" at
http://www.setterfield.org/quantumredshift.htm#isotropcandrel, actually writes down Maxwell’s Equations using genuine calculus. He discusses their application to his theory of time-varying c,. This is good, since physics is mathematical and physicists are accustomed to communicate amongst themselves with mathematics, and Setterfield’s articles are notorious, if for nothing else, for their lack of substantial mathematics. The inclusion of Maxwell’s Equations, then, comes as a welcome surprises. Unfortunately he blunders in their use. It turns out that the time-dependence he introduces can be removed by a trivial time coordinate transformation, yielding the conventional, constant-c, Maxwell’s Equations
I have previously noted that Setterfield’s introduction of a time-dependent permeability of free space laks physical motivation. The permeability of free space, mu, is DEFINED to have the value of 4*pi*10^-7 MKS units. It enters physics through the proportionality parameter in Ampere’s law, stating that bhe magnetic field produced by a current is proportional to that current. We do not, however, measure magnetic fields except by the forces they exert on currents or moving charges: Those forces are taken to be proportional to the magnetic fields exerting the force. Thus we have two current-carrying wires exerting forces on each other proportional to the product of their currents. That is what is measurable. If, for some geometry of wires, k1 is the proportionality constant between the current in wire 1 and the magnetic field it produces at wire 2, and k2 is the proportionality constant between the force exerted by the magnetic field due to current 1 and the current in wire 2, then all we can measure k1*k2, the ratio of the force between the wires and the product of their currents; but we cannot measure k1 or k2 separately. We cannot separately measure the magnetic field strength. We must adopt some arbitrary convention, and the convention adopted is to choose units of magnetic field strength so that B1 = mu * I1/(2*pi*r), where r is the distance from wire 1, mu is as defined above, I1 is the current in wire 1 and B1 the magnetic field strength due to that current a distance r from the wire 1.
Turning to Setterfield’s version of Maxwell’s Equations, let us see how the time-dependence leaves them.. In Setterfield’s theory he gives Maxwell’s equations in vacuum as:
mu div H = 0
epsilon div E = 0
curl E = - mu (d H/d t)
curl H = epsilon (d E/d t).
(I do not have a “curly dee” partial derivative sign, so I am using just a plain lower-case “d” here. Since I will not be dealing with total time derivatives here (as, for example, in fluid mechanics) this should cause no confusion.
For arbitrary disparate time-dependences of mu and epsilon, these would indeed be mathematically different from the conventional Maxwell equations. However, Setterfield claims that in his theory both mu and epsilon are inversely proportional to c. That is, if mu_n, epsilon_n and c_n are, respectively, the values of mu, epsilon, and c(t) now, then:
mu = mu_n*c_n/c(t)
epsilon = epsilon_n*c_n/c(t).
Now let us let
T = integral from 0 to t with respect to t of (c(t)/c_n.
Then dT/dt = c(t)/c_n.
Then mu*(dH/dt) = mu_n*c_n/c(t)*(dH/dt)
= mu_n*c_n/c(t)*(dH/dT)*(dT/dt)
= mu_n*c_n/c(t)*(dH/dT)*c(t)/c_n
= mu_n(dH/dT).
Similarly
Epsilon(dE/dt) = epsilon_n(dE/dT).
Setterfield’s Maxwell’s Equations become:
mu_n div H = 0
Epsilon_n div E = 0
curl E = - mu_n (d H/dT)
curl H = epsilon_n (d E/dT).
These are just the standard (constant-c) Maxwell Equations in vacuum!. We could, with a trivial transformation of current density and charge density, extend this result to the case with free charges and currents. This we leave as an exercise for the reader.
So now that we have gotten Maxwell’s Equations by a simple time-transformation, it is clear that T-time, rather than t-time, is the suitable time for solving electromagnetic problems. One does not even have to worry about conservation of energy or changing wave amplitudes. (Ironically this is the result that Setterfield tries to obtain via incorrect approximate methods: Energy conservation does depend critically upon whether the time-varying parameters, epsilon and mu, are inside or outside of the partial time derivatives. Setterfield moves them in and out of the time-derivatives; but that is valid only for times short compared to both the frequencies of electromagnetic radiation being considered (as Setterfield correctly points out), but also short compared to c(t)/(dc(t)/dt). Since Setterfield’s theory attempts to deal with cosmological problems the latter condition is not fulfilled, and his argument for energy conservation fails.)
So where is the physics of this so-called c-variation. Clearly not within electrodynamics. It could be that we need to use t-time when we consider the interaction of charged particles with other fields, such as gravitational fields; but Setterfield has not even told us what the modifications are supposed to be in Newton’s Laws of motion under his theory, let alone how electromagnetically and gravitationally “charged” particles (particles with electrical charge and mass) behave. Until he does this most basic thing it is no exaggeration to say that his theory does not predict the time-dependence of anything at all.
