iSoul In the beginning is reality

# Algebraic relativity

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()²,

where τ = t/γ. This may be rearranged as

d()² = d(ct)² – dx²,

which equals the Lorentz invariant, d()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/cvc² = 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, , differential legerity displacement, du, and elapsed arc time ξ:

² = (/dx)²dx² = du²dx² =  d(ux)².

Let there be a constant, k = 1/c:

² = 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

² = 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()²,

where σ = x/λ. This may be rearranged as

d()² = d(kx)² – ²,

which equals the Lorentz invariant, d()² = dw².