Consider again the now-classic scenario in which observer *K* is at rest and observer *K′* is moving in the positive direction of the *x* axis with constant velocity, *v*. This time there is a standard constant speed, *c*. The basic problem is that if they both observe a point event *E*, how should one convert the coordinates of *E* from one reference frame to the other?

We return first to the *spatial Galilei transformation* and include a factor, *γ*, in the transformation equation for the positive direction of the *x* axis:

*r _{x}′ = γ* (

*r*)

_{x}− vt_{x}where *r _{x} * is the spatial coordinate and

*t*is the temporal coordinate. Only the coordinates of the

_{x}*x*axis are affected; the other coordinates do not change.

The inverse spatial transformation is then:

*r _{x} = γ* (

*r*

_{x}′ + vt_{x}*′*).

The trajectory of a reference particle or probe vehicle that travels at the standard speed in the positive direction of the *x* axis will follow the equations:

*r _{x} = ct_{x}* and

*r*

_{x}′ = ct_{x}′.With the spatial transformations we conclude that

*r _{x}′ = ct_{x}′ = γ* (

*r*) =

_{x}− vt_{x}*γ*

*t*(

_{x}*c − v*),

*r*(

_{x}= ct_{x}= γ*r*) =

_{x}′ + vt_{x}′*γ*

*t*(c + v).

_{x}′Multiplying these together and cancelling *t _{x} *

*t*leads to:

_{x}′* c^{2} = γ^{2}* (

*c − v*) (

*c + v*) =

*γ*(

^{2}*c*

^{2}*−*v

*)*

^{2}so that

*γ* = (*1 – v ^{2}/c^{2}*)

^{ -1/2}.

which completes the spatial Lorentz transformation.

We return next to the *temporal Galilei transformation* and include a factor, *ρ*, in the transformation equation for the positive direction of the *x* axis:

*t _{x}′ = ρ* (

*t*)

_{x}− r_{x}/vwhere *r _{x} * is the spatial coordinate and

*t*is the temporal coordinate. Only the coordinates of the

_{x}*x*axis are affected; the other coordinates do not change.

The inverse temporal transformation is then:

*t _{x} = ρ* (

*t*).

_{x}′ + r_{x}′/vThe trajectory of a reference particle or probe vehicle that travels at the standard speed from the origin will follow the equations:

*r _{x}/c = t_{x}* and

*r*

_{x}′/c = t_{x}′.With the temporal transformations we conclude that

*t _{x}′ = r_{x}′/c = ρ* (

*t*) =

_{x}− r_{x}/v

*ρ**r*(1/

_{x}*c −*1/

*v*),

*t*

_{x}=*r*(

_{x}/c =*ρ**t*) =

_{x}′ + r_{x}′/v

*ρ**r*(1/

_{x}′*c +*1/

*v*).

Multiplying these together and cancelling *r _{x} *

*r*leads to:

_{x}′*1/c ^{2} = ρ^{2}* (1/

*c −*1/

*v*) (1/

*c +*1/

*v*) =

*ρ*(1/

^{2}*c*1/

^{2}−*v*)

^{2}so that

*ρ* = (*1 – c ^{2}/v^{2}*)

^{-1/2}.

which completes the temporal Lorentz transformation.