Expanding Glome to Special Relativity

In Big Fat Coincidence and Problems that Don’t (Seem to) Exist, I laid out how the FUGE model works.  What I didn’t do, because I didn’t think about it at the time, was explain how one reaches an explanation of (Special) Relativity from an expanding glome.  So here goes …

The equation for a glome is:

Δx² + Δy² + Δx² + (cΔτ)² = r² = (cΔt)²

where x, y and z are spatial units, τ is a temporal unit and r is the radius, which is given by the change in time, t, times c, which is a constant required to mediate the units.

We can use this equation to consider a change in spatial location on the glome in the period Δt:

v² = Δx²/Δt² + Δy²/Δt² + Δz²/Δt²

so:

v².Δt² + (cΔτ)² = (cΔt)²

and then, rearranging:

c²Δτ² = c²Δt² - v².Δt²

Δτ² = Δt² - (v²/c²).Δt²

Δτ = √(1 - (v²/c²)).Δt

Which is the equation for temporal dilation where Δt is by convention expressed as t' and Δτ as t.  Note that “(a)fter compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame”.  If we are counting ticks, we are actually measuring a frequency (at a rate of one tick per second) and this is why the time dilation equation will usually appear as something like this:

t' = t / √(1 - (v²/c²))

To get length contraction, one simply multiplies through by c:

Δτ.c = √(1 - (v²/c²)).Δt.c

ΔL = √(1 - (v²/c²)).ΔL_o

Alternatively, given that the surface volume of the expanding glome is flat in the FUGE model, one could merely use the approach described in Galilean to Special in One Page.

As for mass-energy, the total energy of a mass is given by:

E_total = m.v_spacetime² = m.v² + m.c²Δτ² = mc² ≈ m_o.c² + ½m_o.v²

See On Time where I explain why m.v² + m.c²Δτ² ≈ m_o.c² + ½m_o.v².