The post Lorentz for space and time derived the standard (spatial) Lorentz transformation and also the temporal Lorentz transformation. It is surprising in this age of relativity how the standard Lorentz transformation is dependent on absolute time. While time is relativized in the sense of Lorentz and Einstein, it remains absolute in the sense of Galileo and Newton. This is why clock synchronization is still so important.
The temporal Lorentz transformation is based on the temporal Galilean transformation in which space is absolute. This is at least a different absolute, and so provides an alternative and comparison.
Consider the temporal coordinate of the Galilean transformation and include a factor, γ, in the transformation equation for the positive direction of the x axis:
ctx′ = γ (rx − vtx)
where rx is the spatial coordinate and tx is the temporal coordinate. Only the coordinates of the x axis are affected; the other coordinates do not change.
The inverse spatial transformation is then:
ctx = γ (rx′ + vtx′).
The trajectory of a reference particle or probe vehicle that travels at the characteristic speed in the positive direction of the x axis will follow the equations:
rx = ctx and rx′ = ctx′.
With the spatial transformations we conclude that
ctx′ = rx′ = γ (rx − vtx) = γ (ctx − vtx)
ctx = rx = γ (rx′ + vtx′) = γ (ctx′ + vtx′).
Multiplying these together and dividing out txtx′ leads to:
γ2 (c − v) (c + v) = c2
so that
γ = (1 − v2/c2)-1/2
which is the spatial Lorentz transformation. So, the temporal coordinate can lead to the spatial transformation as well.
One may similarly consider the spatial coordinate of the temporal Galilean transformation:
ctx′ = rx′ = ρ (rx − c2 tx/v) = ρ (ctx − c2 tx/v)
and its inverse
ctx = rx = ρ (r′x + c2 t′x/v) = ρ (ct′x + c2 t′x/v).
Multiplying these together and dividing out c2txtx′ leads to:
ρ2 (v − c) (v + c) = v2
so that
ρ = (1 − c2/v2)-1/2
which is the temporal Lorentz transformation. So, the spatial coordinate can lead to the temporal transformation as well.