- s^2 <0, the events are causally related by the propagation of particles of any mass; one may
viewτ, defined byτ^2 =−s^2 /c^2 , as the proper time between the two events.
The coordinates of two inertial framesR(t,x) andR′(t′,x′) are related by an affine transformation
which leaves the Minkowski distances^2 between any two events invariant,
−c^2 (t 1 −t 2 )^2 + (x 1 −x 2 )^2 =−c^2 (t′ 1 −t′ 2 )^2 + (x′ 1 −x′ 2 )^2 (21.2)It is immediate that this construction automatically implies the second postulate that the speed
of light iscin all inertial frames. It is also immediate that space and time translations leave the
Minkowski distance invariant. Amongst the linear transformations, rotations leaves^2 invariant as
well. The remaining transformations are boosts, which act linearly. Using rotation symmetry, any
boost may be rotated to thex-direction, leavingyandz-coordinates untransformed. We may then
parametrize a boost as follows,
ct′ = Act+Bx
x′ = Cct+Dx
y′ = y
z′ = z (21.3)Choosing two events as follows (t 1 ,x 1 ) = (t,x, 0 ,0) and (t 2 ,x 2 ) = (0, 0 , 0 ,0), and requiring invari-
ance of the Minkowski distance between them, gives
−c^2 t^2 +x^2 = −c^2 (t′)^2 + (x′)^2
= −(Act+Bx)^2 + (Cct+Dx)^2 (21.4)Since this relation must hold for all possible events, it must hold for allt,x, which requires
1 = A^2 −C^2
1 = −B^2 +D^2
0 = −AB+CD (21.5)ParametrizingA= chφ,C= shφ,B= shψ, andD= chψsolves the first two equations. The third
equation is solved by havingψ=φ. To gain better physical insight into the transformation, we set
B=C=−βγ A=D=γ≡1
√
1 −β^2(21.6)
whereβis viewed as the parameter taking all possible real values|β|<1. The transformations of
(21.7) take the following form in this parametrization,
ct′ = γ(ct−βx)
x′ = γ(x−βct)
y′ = y
z′ = z (21.7)