Relativity and the Lorentz Transform
Relativity only occurs and the Lorentz transform becomes only relevant when observers obtain information delivered from the past via vibrations or deformations of the field in which they are embedded. This field transports information with a constant velocity. We call that velocity the light speed.
A complete model offers an overall view that offers access to all dynamic geometric data whether they belong to the past, to the present status quo, or to the future. I call that view the creator’s view, but you can also call it the storage view. Besides of the creator’s view the complete model must also offer an observer’s view. Observers are discrete objects that travel with the current static status quo.
If the observer moves with uniform speed relative to the observed event, then for an observer that travels with light speed holds:
x= c t ⟹ x ²= c² t²
For an observer that travels with speed v holds:
x² − c² t ²= x’²− c² t’²
Here x an t are coordinates of the observer and x’ and t’ are the coordinates of the observed event.
This corresponds with a hyperbolic transformation:
c t’ = c t cosh (ω) −x sinh (ω)
x‘ = x cosh (ω)−c t sinh (ω)
cosh (ω ) = (exp (ω) + exp (-ω ))/2 = c/√(c²-v²)
sinh (ω) = (exp (ω) – exp (−ω))/2 = v/√(c²-v²)
cosh² (ω) − sinh² (ω) = 1
This transformation is the Lorentz equation.
It means that the local metric gets a Minkowski signature:
ds² = dτ² = dt² − dx² − dy² − dz²
dτ is the infinitesimal proper time step
dt is the infinitesimal coordinate time step
The creator’s view sees the Euclidean metric of quaternions that store the dynamic geometric data:
dt² = dτ² + dx² + dy² + dz²
In this metric coordinate time plays the role of quaternionic distance.