Euclidean Special Relativity

This is a summary of definitions and formulae for computing Euclidean special relativity in the potential energy field. Everything is for collinear motion, because that’s easier and much clearer about what’s going on. No hyperbolic functions are necessary.

Warning! High school level math ahead.

The speed of light (c) = 1

v = velocity as a fraction of c (0 ≤ v ≤ 1).

Lorentz alpha (α) = √(1-v²) = cos(arcsin v) = cos θ = relativistic factor (0 < α ≤ 1).

Lorentz gamma (γ) = 1/α = 1/√(1-v²) = relativistic factor (1 ≤ γ).

Rapidity (p°) = γv = v/α = tan(arcsin v) = tan θ = momentum gradient without the rest mass.

Momenergy (e°) = p°c = γvc = kinetic energy gradient without the rest mass.
When c = 1, momenergy is numerically equal to rapidity.

Gradient angle (θ) = arcsin(v) = arctan(p°) = arctan(e°)

Acceleration = p°_a+b = p°_a + p°_b

The relativistic velocity addition formula:
Your stationary perspective of moving body a's perspective of collinear moving body b:
v_a+b = (v_a+v_b)/(1+v_av_b) = sin(arctan((v_a+v_b)/(cos(arcsin v_a)·cos(arcsin v_b))))
θ_a+b = arctan((tan θ_a / cos θ_b) + (tan θ_b / cos θ_a))
p°_a+b = γ_bp°_a + γ_ap°_b = γ_aγ_b(v_a+v_b)

sin θ = v
cos θ = α
tan θ = p° = sin θ / cos θ = γv

Average kinetic energy (KE°) = ½e° = ½p°c = kinetic energy without the rest mass.
Rest energy = (rest mass)·c² = mc²
Average total energy (E) = rest energy + KE°

Photons, having no rest mass, only have momenergy. They have neither rest energy nor average kinetic energy.