But what of the "(vt)" term in the "Lorentz/Fitzgerald" contraction?
If this equation is no longer Galilean, but the above term is in the "()" "dimension", it can be interpreted as "polarization" of a photon, or "spin" of an electron. This means that photons can interact via their polarizations; like polarizations repel, unlike polarizations attract.
So we have ((ct'),(VT),}), where ct' is the new "rest mass" for the boson defined by c,v, and VT is the spin perturbation that accounts for the fermion.. The resultant "spin mass" can be thought of as the cross product between the radius ct' and the tangent to the radius at the circumference, which corresponds to the "v x B" term in the Lorentz Force equation....
, where the first term correpnds to inline force (between charges "permittivity") and the second corresponds to "spin") So, (very) roughly speaking, the quantity dependent on E corresponds to the "boson" part of the characterization, and the quantity depending on B corresponds to the "fermion" part of the equation.
m0 = 1 and v=0 gives the classical expression.
Note that V can be either positive or negative, corresponding to two "directions" of spin. For V = 0, the boson equation obtains. Also, permittivity and permittivity have not been explicitly included -see Coulomb's and Gauss's law for reference).
(It is interesting that the trace of the four dimensional EM field tensor is 0, subject to symmetry conditions on A . If anyone is at all interested, I'll attempt to clarify)
(Note: this equation requires a bit more discussion (in particular, one has to get much more persnickety about signs, and Dirac applies both positive and negative values to m0, etc.), but shows the relation of spin to mass in the Lorentz force equation in relation to the vector potential. Also, F is a vector sum. I'll write more on it later on.)
Dividing the r.h.s. by m gives the vector (the vector potential) as the Lorentz force per unit "rest" mass (EM. STR) (ct), which now includes "spin" Note that this is also subject to a "perturbation", which requires further modification of the equation (which can be interpreted as "gravity" (inertial) perturbation/interaction in GTR).
(In classical atomic (spin-orbit) quantum mechanics, one ignores the 0 component in favor of the magnetic moment (0) , and considers the and vectors. Modifying these to include relativity results in the hyperfine structures of energy levels of the atom...)
(if the Lorentz forces are equal and opposite, the parts of the equation are squared independently, similar to the "time dilation" equation to derive the energy)
The term then adds mass to the "time dilation" equation via the prescription [tex]\pm[/tex(VT), where V and T are now different from the v,t of , and define an added energy to that given by the rest mass definition. These are not subtracted, since the energy equation is ultimately squared, but rather contributes to the two particles split by the B field in the Stern-Garlach experiment; if there is no B field, the particles do not interact - Physics interprets this as the particles having an intrinsic spin, so that fermions are defined differently from bosons. (If the positive mass is equal to the negative mass, the VT term disappears, but because of the squaring, the "rest" mass ct' changes (increases) yet again i.e., the pair has recombined to add vectorially to the "bare" mass ct' i.e., the spins are equal and opposite).
("spin/polarization mass" is actually a cross product between a change in radius and circumference)..
The Pauli equation only interprets the VT term, where as the Dirac equation interprets the whole equation, which includes the (vt',ct) - defined "rest" mass as well as spin/ploarization.
(Actually there is a direct "head-on" interaction in addition to the tilted "spin" interaction, but this would correspond to a "red shift" - much harder to detect. - For gravity B probe, the spin would correspond to the "geodetic" effect, and the "head-on" effect would correspond to "frame-dragging". Frame- dragging is much more difficult to detect, since it lies along the geodesic, rather than being the "shear" effect of the geodetic..)
There is much more to be said about this, but the bottom line is that polarization converts the boson model of photons to fermions, and allows interaction (Note that Newton's law of gravity does not include spin, and I think the was Einstein's motivation for interpreting the non-linear (coordinate dependent) version of such an interaction as gravity. Since such a transformation depends on BOTH space and time, the transformation is contra-variant, and thus changes the effective speed of light in space-time (actually momentum/energy) diagram)...
So the photons do interact, and are slowed down in one direction by the "headwind" created by the other since they interact like fermions via polarization; the angular conic section is necessary so that plane waves are not involved (which would have no interaction).... and if there is no "headwind", the photons travel at c, since they are now non-interacting bosons...
(One can also say the "headwind" is an increased "density" in the vacuum created by the opposing beam. This will, of course, change the E and B fields to effective E' and B' fields - i.e. D and H fields - within the beam path)
Well, that's quite exhausting for me this morning, but is my basic interpretation of what is going on (IMO, at least for the next three minutes) - the angle produces the interacting local E,B that allows the photons to interact, which slows them down compared to the case where only one "lens" is used....
Cosmologically, this "headwind" is responsible for red shift (photons toward a distance source - e.g. "from behind" us, interacting with incoming photons from the source, lowering their energy, and thus the locally observed red shift, interpreted relative to a local blackbody)
Photon "spin" interactions are responsible for "Einstein" rings.... (For Einstein, this is independent of photon spin (an EM effect), but comes from a "curvature" due to an un-specified mass - unspecified because it would involve action at a distance, which Einstein is desperate to avoid...)