are reminiscent of the way in which the components of a vector on the plane transform
under a rotation of the corresponding coordinate axes. It is then natural to describe
the effect of a change in the inertial frame as a kind of “rotation” of the space and
time axes
x, t
. Considering also the other two coordinates
y, z
, which do not transform
if the two inertial frames are in the
standard configuration
, one may regard a Lorentz
transformationasakindof“rotation”inafourdimensionalspace,thefourthdirection
being spanned by the time variable.
A more precise definition of this kind of rotation will be given in
Chap. 4
; for the
time being we call this fourdimensional space of points
space–time
or
Minkowski
space
. Every point in space–time defines an
event
which occurs at a point in space
of coordinates
x, y, z
, at a time
t
, and is labeled by the four coordinates
t, x, y, z
.
In threedimensional Euclidean space
R
3
a rotation of the coordinate axes im
ply a transformation in the components
x
,
y
,
z
of the relative position vec
tor between two points, which however does not affect their squared distance

x

2
=
x
2
+
y
2
+
z
2
. In analogy to ordinary rotations in Euclidean space, a
Lorentz transformation preserves a generalized “squared distance” between events
in
Minkowski space
which generalizes the notion of distance between two points in
space. To show this let us recall that, in determining the Lorentz transformations, we
required the equality:
x
2
+
y
2
+
z
2
−
c
2
t
2
=
x
2
+
y
2
+
z
2
−
c
2
t
2
.
(1.72)
the left and righthand sides of this equation being separately zero, in accordance
to the constancy of the speed of light in every inertial frame. The two events in
A
and
B
, in that case, were the emission of a spherical light wave in
O
=
O
at
the time
t
=
t
=
0 and the passage of the spherical wavefront through a generic
point of coordinates
x, y, z, t
and
x
,
y
,
z
,
t
, respectively. Consider now a light wave
which is emitted in a generic point, instead of the origin, at a generic time, and let
the spherical wave propagate for a time
t
in
S
. Equation
1.72
can be written as:

x

2
−
c
2
t
2
≡
x
2
+
y
2
+
z
2
−
c
2
t
2
=
x
2
+
y
2
+
z
2
−
c
2
t
2
= 
x

2
−
c
2
t
2
.
(1.73)
It is now a simple exercise to verify that equality (
1.73
) holds even if the two events do
not refer to the propagation of a light ray. It is sufficient to express the primed quanti
ties on the righthand side in terms of the unprimed ones by using the Lorentz trans
formations (
1.57
)–(
1.60
). One then finds that (
1.73
) is identically satisfied. Defining