The Lorentz transformation for velocities in all directions is known from the special relativity theory. This may be generalized to lenticities in all directions. Begin with the definition of the vector β from a previous post:
β =
- v/c = k/v if |v| < |c|, with velocity v and speed of light c,
- v/k = c/v if |v| > |c|, with lenticity v and pace of light k,
- 0 if |v| = |c|,
with γ = 1 / √(1 – β²). Recall that pace is the inverse of speed, with space (length) as the independent variable in the denominator instead of time. Lenticity is the vector version of pace, that is, pace with direction or three pace components. Note that a/b = (1/a)/(1/b) = b/a.
The Lorentz transformation for velocity, that is, all speeds in all directions may be derived as follows. Note that the transformation applies only to movement in the parallel direction. Given the vectors of movement, v and v, separate the space and time vectors into parallel and perpendicular components: r = r∥ + r⊥ and t = t∥ + t⊥.
The subluminal Lorentz transformation uses time as the independent variable and may be expressed with |t| = t as
r´ = r⊥ + γ(r∥ – vt).
The subscripts may be eliminated by substituting r⊥ = r – r∥ and using
r∥ = r∥ v/v = ((r · v)/v) (v/v)
since (v/v) is a dimensionless unit vector in the same direction as r∥ if |v| < |c|, and r∥ = (r · v)/v is the projection of r onto the direction of v.
Substituting for r∥ and factoring v gives
r´ = r + ((γ – 1)/v²) (r · v – γt) v.
Similarly, the superluminal Lorentz transformation uses space (length) as the independent variable and may be expressed with |r| = r as
t´ = t⊥ + γ(t∥ – vr/c²).
The subscripts may be eliminated by substituting t⊥ = t – t∥ and using
t∥ = t∥ v/v = ((t · v)/v) (v/v)
since (v/v) is a dimensionless unit vector in the same direction as t∥, if |v| > |c|, and t∥ = (t · v)/v is the projection of t onto the direction of v.
Substituting for t∥ and factoring v gives
t´ = t + ((γ – 1)/v²) (t · v – γr/c²) v.