2.2. LORENTZ INVARIANCE 49
Theorem 2.18.The Minkowski metric (2.2.5) or (2.2.6) is invariant under the coordinate
transformation (2.2.7). Namely the metric(g ̃μ ν)in{ ̃xμ}is the same as that in{xμ}:
(2.2.11) ( ̃gμ ν) = (gμ ν).
In other words, ds^2 in{x ̃μ}is also expressed as
ds^2 =−c^2 d ̃t^2 +d ̃x^2 , x ̃= (x ̃^1 ,x ̃^2 , ̃x^3 ).
The proof of Theorem2.18needs to use the properties of tensors. In fact, the Minkowski
metric (2.2.5) is a second-order covariant Lorentz tensor, i.e. under thetransformation (2.2.8),
gμ νtransforms as
(2.2.12) (g ̃μ ν) = (lαβ)(gμ ν)(lβα)T,
where(lμμ)is the inverse of(Lνμ)given by (2.2.10). Then, a direct computation we can get
(2.2.11) from (2.2.12).
Lorentz Transformation Group
Each transformation (2.2.8) corresponds to a Lorentz matrix(Lνμ)given by (2.2.9). These
matrices constitute a group in the multiplication as
√^1
1 −β^2
−√β
1 −β^2
0 0
−√β
1 −β^2
√^1
1 −β^2
0 0
0 0 1 0
0 0 0 1
√^1
1 −γ^2
−√γ
1 −γ^2
0 0
−√γ
1 −γ^2
√^1
1 −γ^2
0 0
0 0 1 0
0 0 0 1
=
√^1
1 −α^2
−√α
1 −α^2
0 0
−√α
1 −α^2
√^1
1 −α^2
0 0
0 0 1 0
0 0 0 1
,
where
β=
v
c
, γ=
u
c
, α=
w
c
, w=
u+v
1 +uv/c^2
,
wherewis the velocity composed byuandvby the theory of special relativity.
Thanks to Theorem2.18, we can define Lorentz transformation group as all linear trans-
formations of the Minkowski spaceM^4 , that preserve the Minkowski metric; see (2.2.12):
(2.2.13) LG={(Lμν):M^4 →M^4 |(gμ ν) = (Lαβ)(gμ ν)(Lαβ)T,det(Lμν) = 1 }.
Elements ofLGare also called Lorentz matrices, and relativistic physicsis referred to the
invariance of action under the Lorentz group. The invariance of the Minkowski metric implies
the Lorentz invariance of physical laws.
In this definition, we require det(Lμν) =1, and such transformations are often called
proper Lorentz transformation in physics literatures. Theparity transformation and time re-
versal are two special linear transformations of the Minkowski space, which are not elements
inLGas defined in (2.2.13) and are often dealt with separately: