48 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
wherecis the speed of light, andM^4 is endowed with the Riemannian metric:
(2.2.5) (gμ ν) =
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
.
The doublet{M^4 ,gμ ν}given by (2.2.4) and (2.2.5) is called the Minkowski space, and the
metric (2.2.5) can be equivalently expressed in the form
(2.2.6) ds^2 =gμ νdxμdxν=−c^2 dt^2 +dx^2 =−c^2 dt^2 + (dx^1 )^2 + (dx^2 )^2 + (dx^3 )^2.
Lorentz transformations are linear transformations of theMinkowski spaceM^4 that pre-
serve the metric (2.2.6). The following coordinate transformation of the Minkowski space is
a Lorentz transformation, called boost, corresponding to Figure2.1:
(2.2.7)
̃x^0
̃x^1
̃x^2
̃x^3
=
√^1
1 −β^2
−√β
1 −β^20 0
−√β
1 −β^2√^1
1 −β^20 0
0 0 1 0
0 0 0 1
x^0
x^1
x^2
x^3
,
whereβ=ν/c. Both transformations (2.2.3) and (2.2.7) are the same in form, but the mathe-
matical implication is changed. Here, (2.2.7) represents the coordinate transformation for the
Minkowski spaceM^4. With Einstein’s summation convention, (2.2.7) is often denoted by
(2.2.8) ̃xμ=Lμνxν,
where
(2.2.9) (Lμν) =
√^1
1 −β^2
−√β
1 −β^20 0
−√β
1 −β^2√^1
1 −β^20 0
0 0 1 0
0 0 0 1
is the Lorentz matrix, and its inverse(lνμ) = (Lμν)−^1 is given by
(2.2.10) (lνμ) =
√^1
1 −β^2√β
1 −β^20 0
√β
1 −β^2√^1
1 −β^20 0
0 0 1 0
0 0 0 1
.
An important property of the Minkowski space is that its Riemannian metric is invariant
under the Lorentz transformation (2.2.7).