Let φ ≡ arcsin(v/c)

Equivalently: sin(φ) = β ≡ v/c

Then, the Lorentz factor (gamma, a function of velocity) simplifies as

γ = 1/√(1 – v²/c²)

= 1/√(1 – sin²φ)

= 1/cos(φ)

Time dilation and length contraction then simplify to

τ = t∙cos(φ)

ʀ = r∙cos(φ)

Einstein thought that the “rapidity angle” was the 4D angle of rotation corresponding to a Lorentz transformation (boost) in SR, but he was uncharacteristically mistaken. It can be shown that φ is the “4D rotation angle” in a complex, multidimensional (3r+3i) spacetime, because exp(iφ) = cos(φ) + i∙sin(φ), where 0 ≤ φ ≤ π/2 (in special relativity). Dirac’s 6D complex model also elegantly eliminates the problematic concept of “mixing” space and time dimensions at relativistic velocities. RIP!

]]>I agree completely. In 1953, Dirac proposed (in private correspondence) a complex 6D spacetime ansatz, extending Kaluza-Klein’s 5D theorem, thereby attempting to unify QM with GR (a compelling reason!). He very nearly succeeded, but never formally published his framework (because bosonic mass). However, a 3r+3i metric can be shown to be 100% consistent with SR, and with experimental and observational data. I’d post a link to the relevant math (framework, not a “theory”), but Prof. Francis doesn’t permit posting links to such working papers in these comments. ]]>

This is not correct. The relativistic dopper shift can accomadate infinite redshifts under motion that is not faster than the speed of light. in LaTeX:

\frac{f}{f_0} = \sqrt{\frac{c + v}{c – v}}

The only way to come to the conclusion that galaxies are receding faster than c is to use the non-relativistic doppler equation.

So while I agree that the redshifting of galaxies is a general relativistic effect, the argument for why that is the case is not the one the Doctor Francis presented. The reasoning is that the redshift is observed in every direction, relatively uniformly, and that the distance increases monatonically with redshift. Those observations can only brought in line with the Copernican principle by general relativistic effects.

Fun mathematical fact: taking the logarithm of the dopper shift ratio gives the rapidity that corresponds to v. So \ln(1+z) = the rapidity needed to produce the redshift, z. Brief refresher, with \phi = rapidity:

v = c \tanh(\phi)

\gamma = c / sqrt(c^2 – v^2) = \cosh(\phi)

v \gamma = \sinh(\phi)