I’ve written several related posts, such as one on the *Complete Lorentz transformation*. This post extends the previous post on the Galilean transformation to the Lorentz transformation, and what I’m now calling the dual Lorentz transformation, in order to show their similarities and differences.

There are many expositions of a Lorentz transformation, such as *here*. It is standard to describe them in terms of two reference frames and their coordinate systems in uniform relative motion along the *x*-axis. Here we take the spatial axis to be the *r*-axis, which is parallel to the spatial axis of motion. Similarly, the temporal axis is taken to be the *t*-axis, which is parallel to 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.

The two frames are differentiated by primed and unprimed letters. Their relative speed is *v*, and their relative pace is *u* = 1/*v*. The key difference between speed and pace is their independent unit of measure: speed is measured per unit of time (duration), whereas pace is measured per unit of space (length).

A Lorentz transformation requires what I’m calling a *characteristic (modal) rate*, in units of speed or pace, which is the same for all observers within a context such as physics or a mode of travel. The characteristic speed, *c*, or pace, *ç* (c-cedilla), may take any positive value, and may represent a maximum or a minimum, depending on the context. In the context of physics the characteristic rate is that of light traveling in a vacuum.

Note that the trajectory of a reference particle (or probe vehicle) that travels at the characteristic rate follows these equations in the two frames:

speed: *r* = *ct* or* r/c = t* and *r′* = *ct′*, or *r′/c* = *t′*,

pace: * çr* =

*t*or

*r*=

*t/ç*and

*çr′*=

*t′*or

*r′*=

*t′/ç*.

__Lorentz transformation
__

This starts with the Galilean transformation and includes a factor, *γ*, in the transformation equation for the direction of motion, along with the characteristic rate:

speed (–): *r′* = *γ* (*r* − *vt*) = *γr* (1 – *v/c*) = *ct′* = *γ* (*ct* – *rv/c*) = *γt* (*c* − *v*),

pace (–): *r ′* =

*γ*(

*r*–

*t/u*) =

*γr*(1 –

*) =*

*ç*/u*t*=

*′*/*ç**γ*(

*t/*–

*ç**r*) =

*ç*/u*γt*(1/

*ç*– 1/

*u*),

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*) = *ct* = *γ* (*ct′* + *r′v/c*) = *γt′* (*c* + *v*),

pace (+): *r* = *γ* (*r ′* +

*t*) =

*′*/u*γr′*(1 +

*) =*

*ç*/u*t/b*=

*γ*(

*t*+

*′*/*ç**r*) =

*′**ç*/u*γt′*(1/

*ç*+ 1/

*u*).

Multiply each corresponding pair together to get:

speed: *rr′* = *γ²**rr′* (1 – *v²/c²*) = *c²tt′* = *γ²tt′* (*c²* − *v²*),

pace: *rr´* = *γ²rr′* (1 – *ç²/u²*) = *tt ′/ç² *=

*γ²tt′*(1/

*ç²*– 1/

*u²*).

Dividing out *rr′* yields:

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

pace: * ç²* =

*γ*

^{2}(1 –

*ç*

^{2}/

*u*

^{2}).

Or dividing out *tt′* yields:

speed: *c*^{2} = *γ*^{2} (*c*^{2} – *v*^{2}),

pace: 1/*ç*² = *γ*^{2} (1/*ç²* – 1/*u²*).

Either way, solving for *γ* leads to:

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

pace: *γ* = (1 – *ç*^{2}/*u*^{2})^{–1/2}.

which is the standard Lorentz transformation and applies only if |*v*| < |*c*| or |*u*| > |*ç*|.

__Dual Lorentz transformation
__

Now start with the dual Galilean transformation and include a factor, *λ*, in the transformation equation for the direction of motion, along with the characteristic rate:

speed (–): *t ′* =

*λ*(

*t*–

*r/v*) =

*(1 –*

*λ*t*c/v*) =

*r′/c*=

*λ*(

*r/c*–

*tc/v*) =

*(1/*

*λ*r*c*– 1/

*v*),

pace (–): *t′* = *λ* (*t* – *ur*) = * λt* (1 –

*u/*) =

*ç**çr′*=

*λ*(

*–*

*ç*r*tu/*) =

*ç**(*

*λ*r*ç*–

*u*),

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

speed (+): *t* = *λ* (*t ′* +

*r*) =

*′*/v*(1 +*

*λ*t′*c/v*) =

*r/c*=

*λ*(

*r*+

*′*/c*t*) =

*′*c/v*(1/*

*λ*r′*c*+ 1/

*v*),

pace (+): *t* = *λ* (*t′* + *ur′*) = * λt′* (1 +

*u/*) =

*ç**çr*=

*λ*(

*+*

*ç*r*′**t*) =

*′*u/*ç**(*

*λ*r′*ç*+

*u*).

Multiply each pair together to get:

speed: *tt ′* =

*λ*²*tt*(1 – c²/v²) =

*′**rr′*/c² =

*λ*²*rr*(1/

*′**c²*– 1/

*v²*),

pace: *tt ′* =

*(1 –*

*λ*² tt′*u²/ç²*) =

*=*

*ç*²rr*′**(*

*λ*² rr′*ç²*–

*u²*).

Dividing out *tt′* yields:

speed: 1 = * λ²* (1 –

*c²*/

*v²*),

pace: 1 = *λ*^{2} (1 – *u²/ç²*).

Or dividing out *rr′* yields:

speed: 1/*c²* = * λ²* (1/

*c²*– 1/

*v²*),

pace: * ç²* =

*λ*

^{2}(

*ç²*–

*u²*).

Either way, solving for *λ* leads to:

speed: *λ* = (1 − *c*^{2}/*v*^{2})^{–1/2},

pace: *λ* = (1 – *u*^{2}/*ç*^{2})^{–1/2},

which is the dual Lorentz transformation and applies only if |*v*| > |*c*| or |*u*| < |*ç*|.

__Complete Lorentz transformations
__

The Lorentz transformation is then

speed: *r′* = *γ* (*r − vt*) and *t′* = *γ* (*t* – *rv/c²*), with *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2}, or

pace: *r ′* =

*γ*(

*r*–

*t/u*) and

*t*=

*′**γ*(

*t*–

*r*), with

*ç*²/u*γ*= (1 –

*ç*

^{2}/

*u*

^{2})

^{–1/2}.

which applies only if |*v*| < |*c*| or |*u*| > |*ç*|.

The dual Lorentz transformation is then

speed: *t′* = *λ* (*t − r/v*) and *r′* = *λ* (*r* − *c*^{2} *t/v*), with *λ* = (1 − *c*^{2}/*v*^{2})^{–1/2}, or

pace: *t′* = *λ* (*t* – *ur*) and *r′ *= *λ* (*r* – *tu/ ç²*), with

*λ*= (1 –

*u*

^{2}/

*ç*

^{2})

^{–1/2},

which applies only if |*v*| > |*c*| or |*u*| < |*ç*|.

If |*v*| = |*c*|, then *r′ = r* and *t′ = t*.

Note that in each case *γ* is an even function of *v* or *u*, as it needs to be (see *here*).