
437Theory of Special Relativity
where
γββ
=− =11
2
/,/
(13.16)
is the Lorentz factor.
If β ≪ 1, γ ≅ 1, then Equation 13.15 reduces to the Galilean transformations. Thus, the Galilean
transformation is a rst approximation to the Lorentz transformations for β ≪ 1.
When the velocity,
u , of S′ relative to S is in some arbitrary direction, Equation 13.15 can be
given a more general form in terms of the components of
and
perpendicular and parallel to
u:
′
=− =
′
+
′
=
′
=
′
⊥⊥ ⊥⊥
rrut rrut
rr rr
() ()
ttturctturc=−⋅=
′
+⋅
()
. (13.17)
The Lorentz transformations are valid for all types of physical phenomena at all speed ...