iSoul In the beginning is reality

Complete Lorentz transformation

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 (rvt) and t′ = γ (trv/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 (rvt) = γr (1 – v/c) = γt (cv),

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 (ctrv/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 (c2v2).

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 (c2v2).

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 Galilei transformation and include a factor, γ, in the transformation equation for the positive direction of one axis:

r′ = γ (rc2 t/v) and t′ = γ (tr/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 (rc2 t/v) = γr (1 − c/v) = γt (cc2/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 (c2c4/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 (rc2 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.

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