This is a continuation of a series of posts that began with *Lorentz for space and time*.

The standard Lorentz transformation applies only if |*v*| < |*c*|. The complete transformation for all real values of *v* is presented here based on both the relative space, absolute time (R-A) Galilei transformation as well as the complementary absolute space, relative time (A-R) Galilei transformation. The absolute is associated with one dimension and the relative with three dimensions but there is no necessary connection.

__R-A Lorentz transformation
__

Consider the relative space and absolute time Galilei transformation and include a factor, *γ*, in the transformation equation for the positive direction of one axis:

*r′* = *γ* (*r* − *vt*) and *t′* = *γ* (*t* – *rv/c²*)

where *r* is one spatial coordinate (the others are unchanged) and *t* is the temporal coordinate. The inverse R-A transformations are then:

*r* = *γ* (*r′* + *vt′*) and *t* = *γ* (*t′* + *r′v/c²*).

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

*r* = *ct* and *r′* = *ct′*.

Combined with the Galilei transformation for space this leads to

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

*ct* = *r* = *γ* (*r′* + *vt′*) = *γr′* (1 + *v/c*) = *γt′* (*c* + *v*).

Alternately, the Galilei transformation for time combined with the standard speed leads to

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

*r = ct = γ* (*ct′ + r′v/c*) = *γr′* (1 + *v/c*) = *γt′* (*c* + *v*).

Multiplying these pairs together for space and dividing out *rr′* yields:

1 = *γ*^{2} (1 – *v*^{2}/*c*^{2}).

Or multiplying these pairs together for space and dividing out *tt′* yields:

*c*^{2} = *γ*^{2} (*c*^{2} – *v*^{2}).

Multiplying these pairs together for time and dividing out *rr′* leads to:

*γ*^{2} (1 – *v*^{2}/*c*^{2}) = 1.

Or multiplying these pairs together for time and dividing out *tt*′ leads to:

*c*^{2} = *γ*^{2} (*c*^{2} – *v*^{2}).

Whichever way is done yields

*γ* = (1 – *v*^{2}/*c*^{2})^{–1/2},

which is the standard Lorentz transformation and applies only if |v| < |c|.

__A-R Lorentz transformation
__

Consider the absolute space and relative time Galilei transformation and include a factor, *γ*, in the transformation equation for the positive direction of one axis:

*r′* = *γ* (*r* − *c*^{2} *t/v*) and *t′* = *γ* (*t* − *r/v*).

where *r* is the spatial coordinate and *t* is one temporal coordinate (the others are unchanged). The inverse A-R transformations are then:

*r* = *γ* (*r′* + *c*^{2} *t′/v*) and *t* = *γ* (*t′ + r′/v*).

Again the standard speed is

*r = ct* and *r′ = ct′*.

Combine these together to get for space

*ct′ = r′* = *γ* (*r* − *c*^{2} *t/v*) = *γr* (1 − c/v) = *γt* (*c* − *c*^{2}/*v*),

*ct = r* = *γ* (*r′* + *c*^{2} *t′/v*) = *γr′* (1 + *c/v*) = *γt′* (*c* + *c*^{2}/*v*).

and for time

*r′/c = t′* = *γ* (*t − r/v*) = *γr* (1/*c* − 1/*v*) = *γt* (1 − *c/v*),

*r/c = t* = *γ* (*t′ + r′/v*) = *γr′* (1/*c* + 1/*v*) = *γt′* (1 + *c/v*).

Multiplying these pairs together for space and dividing out *rr′ yields:*

1 = *γ*^{2} (1 – *c*^{2}/*v*^{2}).

Or multiplying these pairs together for space and dividing out *tt*′ leads to:

*c*^{2} = *γ*^{2} (*c*^{2} – *c*^{4}/*v*^{2}).

Multiplying these pairs together for time and dividing out *rr′* yields:

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

Or multiplying these pairs together for time and dividing out *tt*′ leads to:

1 = *γ*^{2} (1 – *c*^{2}/v^{2}).

Whichever is done, this yields

*γ* = (1 − *c*^{2}/*v*^{2})^{–1/2},

which is the complementary Lorentz transformation that applies only if |*v*| > |*c*|.

__Complete Lorentz transformation
__

The complete Lorentz transformation is then

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

which applies only if |*v*| < |*c*|, and

*r′* = *γ* (*r* − *c*^{2} *t/v*), *t′* = *γ* (*t − r/v*), and *γ* = (1 − *c*^{2}/*v*^{2})^{–1/2},

which applies only if |*v*| > |*c*|. If |*v*| = |*c*|, then *r′ = r* and *t′ = t*.