Relativity may be derived as an algebraic relation among differentials. Consider motion in the x spatial dimension, with a differential displacement, dx, differential velocity displacement, dv, and arc (elapsed) time t:
dx² = (dx/dt)²dt² = dv²dt² = d(vt)².
Let there be a constant, c:
dx² = d(vt)² = d(cvt)²/c² = d(ct)² (v/c)² = d(ct)² (1 – (1 – v²/c²)).
Let γ² = 1/(1 – v²/c²). Then
dx² = d(vt)² = d(ct)² (1 – 1/γ²) = d(ct)² – d(ct)²/γ² = d(ct)² – d(ct/γ)² = d(ct)² – d(cτ)²,
where τ = t/γ. This may be rearranged as
d(cτ)² = d(ct)² – dx²,
which equals the Lorentz invariant, d(cτ)² = ds².
Alternately, consider motion in the t temporal dimension, with a differential displacement, dt, differential velocity displacement, dv, and arc length x:
dt² = (dt/dx)²dx² = d(1/v)²dx² = d(x/v)².
Let there be a constant, c:
dt² = d(x/v)² = d(x/cv)² c² = d(x/c)² (c/v)² = d(x/c)² (1 – (1 – c²/v²)).
Let λ² = 1/(1 – c²/v²). Then
dt² = d(x/v)² = d(x/c)² (1 – 1/λ²) = d(x/c)² – dx²/(λc)² = d(x/c)² – d(x/λc)² = d(x/c)² – d(σ/c)²,
where σ = x/λ. This may be rearranged as
d(σ/c)² = d(x/c)² – dt²,
which equals the Lorentz invariant, d(σ/c)² = dw².
Equivalently, consider motion in the ξ temporal dimension, with a differential displacement, dξ, differential lenticity displacement, du, and elapsed arc time ξ:
dξ² = (dξ/dx)²dx² = du²dx² = d(ux)².
Let there be a constant, k = 1/c:
dξ² = d(ux)² = d(kux)²/k² = d(kx)² (u/k)² = d(kx)² (1 – (1 – u²/k²)).
Let λ² = 1/(1 – u²/k²). Then
dξ² = d(ux)² = d(kx)² (1 – 1/λ²) = d(kx)² – d(kx)²/λ² = d(kx)² – d(kx/λ)² = d(kx)² – d(kσ)²,
where σ = x/λ. This may be rearranged as
d(kσ)² = d(kx)² – dξ²,
which equals the Lorentz invariant, d(kσ)² = dw².