Lorentz-like transformation for time-distance domain

Lorentz-like transformations for time-distance domain

What follows are derivations of the Lorentz transformation for the time domain and a Lorentz-like transformation for the distance domain.

Time Domain

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with velocity v with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a length r_x and r_x′ from the origins of O and O′ respectively.

For round-trip motion, the mean round-trip speed of light, c, is the same in all inertial frames, so for the point P with length vector r = (r_x,0, 0) and

r_x = ct, and r_x′ = ct′.

Consider the standard Galilean transformation with a factor γ, which is to be determined and may depend on β, where β = v/c:

r_x′ = γ(r_x − vt) = γ(r_x − βct) = γ(r_x − βct) = γr_x(1 − β).

The inverse transformation is the same except that the sign of β is reversed:

r_x = γ(r_x′ + vt′) = γ(r_x′ + βct′)= γ(r_x′ + βct′)  = γr_x′(1 + β).

Multiply these two equations together to get

r_xr_x′ = γ² r_xr_x′(1 − β²).

Divide out r_xr_x′ to get

γ² = 1/(1 − β²),

or γ = 1/√(1 − β²).

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Consider what happens if this approach is applied to the independent time t times the constant c.

ct′ = r_x′ = γ(ct − βr_x) = γ(r_x − βr_x) = γr_x(1 − β).

The inverse transformation is the same except that the sign of β is reversed:

ct = r_x = γ(ct′ + βr_x′) = γ(r_x′ + βr_x′) = γr_x′(1 + β).

These are exactly the same equations as before, so the result will be the same.

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Distance Domain

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with rapidity p with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point Q on a spherical wavefront at a duration u_x and u_x′ from the origins of O and O′ respectively.

For round-trip motion, the mean round-trip pace of light, k, is the same in all inertial frames, so for the point Q with duration vector u = (u_x,0, 0) and

u_x = ks, and u_x′ = ks′.

Consider the dual standard Galilean transformation with a factor λ, which is to be determined and may depend on α = p/k (cf. c/v)

u_x′ = λ(u_x − ps) = λ(u_x − αks) = λ(u_x − αu_x) = λu_x(1 − α).

The inverse transformation is the same except that the sign of β is reversed:

u_x = λ(u_x′ + ps′) = λ(u_x′ + αks′) = λ(u_x′ + αu_x′) = λu_x′(1 + α).

Multiply these two equations to get

u_xu_x′ = λ²u_xu_x′(1 − α²).

Divide out u_xu_x′ to get

λ² = 1/(1 − α²), or

λ = 1/√(1 − α²).

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Consider what happens if this approach is applied to the independent distance s times the constant k.

ks′ = u_x′ = λ(ks − αu_x) = λ(u_x − αu_x) = λu_x(1 − α).

The inverse transformation is the same except that the sign of α is reversed:

ks = u_x = λ(ks′ + αu_x′) = λ(u_x′ + αu_x′) = λu_x′(1 + α).

These are exactly the same equations as before, so the result will be the same.

RG 2026-04-04