Lorentz with 3D time

Just as three dimensions of space are combined with one dimension of time, so we can combine three dimensions of time with one dimension of space. The place to start is the Lorentz transformation. Let’s take a common approach, that of spherical wavefronts of light but instead of taking three length coordinates and converting time into length via the speed of light, let’s take three duration coordinates and convert space into duration via the speed of light.

Here’s a revision of the Wikipedia text, using Greek letters for time and Latin letters for space:

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with a velocity v with respect to O in the positive ξ-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 distance r and r′ and a duration τ and τ′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light, c, is the same in both frames, so for the radial coordinates to the point P:

τ = r / c and τ’ = r’ / c.

The equation of a sphere of time (duration) in frame O is given by

ξ2 + η2 + ζ2 = τ2.

For a spherical wavefront that becomes

ξ2 + η2 + ζ2 = r2 / c2.

Similarly, the equation of a sphere in frame O’ is given by

ξ’2 + η’2 + ζ’2 = τ’2.

so the spherical wavefront satisfies

ξ’2 + η’2 + ζ’2 = r’2 / c2.

The origin O’ is moving along the ξ-axis. Therefore,

η’ = η and ζ’ = ζ.

Other than these equations the derivation follows as before. A slightly different Lorentz factor is found (call it g):

g2 = v2 / (v2 – c2).

Compare that with the usual Lorentz factor, gamma:

γ2 = c2 / (c2 – v2).

Note that γ2 + g2 = 1. Also note that γ is real only if v < c and g is real only if v > c.

The 3D time Lorentz transformation is then

r’ = g (r – ξ c2 / v)

ξ’ = g (ξ – r/ v)

η’ = η and ζ’ = ζ.

In this Lorentz transformation length is the independent variable, whereas in the usual Lorentz transformation time is the independent variable.