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) Galilean transformation as well as the complementary absolute space, relative time (A-R) Galilean 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 Galilean 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 Galilean 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 Galilean 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 = γ2 (1 – v2/c2).
Or multiplying these pairs together for space and dividing out tt′ yields:
c2 = γ2 (c2 – v2).
Multiplying these pairs together for time and dividing out rr′ leads to:
γ2 (1 – v2/c2) = 1.
Or multiplying these pairs together for time and dividing out tt′ leads to:
c2 = γ2 (c2 – v2).
Whichever way is done yields
γ = (1 – v2/c2)–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 Galilean transformation and include a factor, γ, in the transformation equation for the positive direction of one axis:
r′ = γ (r − c2 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′ + c2 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 − c2 t/v) = γr (1 − c/v) = γt (c − c2/v),
ct = r = γ (r′ + c2 t′/v) = γr′ (1 + c/v) = γt′ (c + c2/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 = γ2 (1 – c2/v2).
Or multiplying these pairs together for space and dividing out tt′ leads to:
c2 = γ2 (c2 – c4/v2).
Multiplying these pairs together for time and dividing out rr′ yields:
1/c2 = γ2 (1/c2 – 1/v2).
Or multiplying these pairs together for time and dividing out tt′ leads to:
1 = γ2 (1 – c2/v2).
Whichever is done, this yields
γ = (1 − c2/v2)–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 – v2/c2)–1/2,
which applies only if |v| < |c|, and
r′ = γ (r − c2 t/v), t′ = γ (t − r/v), and γ = (1 − c2/v2)–1/2,
which applies only if |v| > |c|. If |v| = |c|, then r′ = r and t′ = t.