The standard transformation of reference frames begins with two frames in uniform relative motion along one axis (usually called *x*). Here we take the spatial axis to be the *r*-axis, which parallels the spatial axis of motion. Similarly, the temporal axis is taken to be the *t*-axis, which parallels the temporal axis of motion.

One aspect of the exposition here is that the notation is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames. One can generalize the results here to other directions by rotation.

The two frames are differentiated by primed and unprimed letters. They coincide at time *t* = 0 and their relative speed is *v* or pace *u* = 1/*v*. The main difference between speed and pace is that time (duration) is the independent variable for speed and (travel) length is the independent variable for pace.

We’re assuming the existence of what I’m calling a *characteristic (modal) rate*, which is a speed or pace that is the same for all observers within a context such as physics or a mode of travel. The characteristic speed, *c*, or pace, *b = *1/*c*, may take any positive value, and may represent a maximum, a minimum, or a typical value, depending on the context. In physics, the speed of light traveling in a vacuum is the characteristic speed.

The trajectory of a reference particle (e.g., photon or probe vehicle) that travels at the characteristic speed follows these equations in both frames:

Speed: *r* = *ct* or *r/c = t* and *r′* = *ct′* or *r′/c* = *t′*.

Pace: *r* = *t*/*b* or* br = t* and *r′* *= t′/ b or br′ = t′*.

Consider a point event such as a flash of light that is observed from each reference frame. How are its coordinates in each frame related?

There are two basic relations: *r′* = *r* − *tv* and *t′* = *t – r**u*. However, these relations assume different independent variables so they need to be kept separate.

To find the time and space transformations for each of these requires a characteristic rate, which allows the inter-conversion of space and time. There are thus two transformations, one for speed and one for pace, though we’ll convert pace to speed for convenience:

Speed: *r′* = *r* − *tv* = *r* (1 – *v*/*c*) and

*t′* = *t* (1 – *v*/*c*) = *t* – (*r*/c) (*v*/*c*) = *t – r* (*v*/*c²*).

Pace: *t′* = *t – r**u* = *t – r*/*v* = *t* (1 – *c*/*v*) and

*r′* = *r* (1 – *u*/*b*) = *r* (1 – *c*/*v*) = *r* − c*t* (*c*/*v*) = *r* − *t* (*c²*/*v*).

Note the factors (1 – *v*/*c*) and (1 – *c*/*v*) transform the unprimed to primed coordinates. Note also these limits:

Speed: *t′* = *t – r* (*v*/*c²*) approaches *t* as *c* approaches infinity, as in the Galilei transformation.

Pace: *r′* = *r* − *t* (*c²*/*v*) approaches *r* as *c* approaches zero (it cannot equal zero).

__Lorentz transformation
__

For this we take the previous transformations and include a factor, *γ*, in the transformation equation for the direction of motion:

Speed: *r′* = *γ* (*r* − *vt*) = *γr* (1 – *v/c*),

Pace: *t ′* =

*γ*(

*t – r*

*u*) =

*γ*(

*t*–

*r/v*) =

*γt*(1 –

*c/v*),

with equal values for the other corresponding primed and unprimed coordinates. The inverse transformations are then:

Speed: *r* = *γ* (*r′* + *vt′*) = *γr′* (1 + *v/c*),

Pace: *t ′* =

*γ*(

*t + r*

*u*) =

*γ*(

*t*+

*r/v*) =

*γt*(1 +

*c/v*).

Multiply each corresponding pair together to get:

Speed: *rr′* = *γ²**rr′* (1 – *v²/c²*),

Pace: *tt′* = *γ²**tt′* (1 – *c²/v²*),

Dividing out *rr′* and *tt′* yields:

Speed: 1 = *γ*^{2} (1 – *v*^{2}/*c*^{2}),

Pace: 1 = *γ*^{2} (1 – *c*^{2}/*v*^{2}).

Solving for *γ* leads to:

Speed: *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2}, which applies if |*v*| < |*c*|,

Pace: *γ* = (1 – *c*^{2}/*v*^{2})^{–1/2}, which applies if |*v*| > |*c*|,

and that is the complete Lorentz transformation.