6.1 Basic Concepts and Formulae 315
Inverse transformations
x=γ(x′+νt′) (6.12)
y=y′ (6.13)
z=z′ (6.14)t=γ(
t′+
νx′
c^2)
(6.15)
with
γ=1
√
(1−β^2 )=
1
√
(1−ν^2 /c^2 )(6.16)
and
β=ν
c(6.17)
Transformation matrix
The Lorentz transformations (6.8), (6.9), (6.10), and (6.11) can be condensed in the
matrix form
X′=ΛX (6.18)
whereX=⎡
⎢
⎢
⎣
x 1
x 2
x 3
x 4⎤
⎥
⎥
⎦ and X′=
⎡
⎢
⎢
⎣
x 1 ′
x 2 ′
x 3 ′
x 4 ′⎤
⎥
⎥
⎦ (6.19)
are the column vectors with components
x 1 =x,x 2 =y,x 3 =z,x 4 =τ=ict (6.20)
x 1 ′=x′,x 2 ′=y′,x′ 3 =z′,x′ 4 =τ′=ict′ (6.21)withi=
√
−1, andΛis an orthogonal matrixΛ=
⎡
⎢
⎢
⎣
γ 00 iβγ
01 00
00 10
−iβγ 00 γ