Mathematical Principles of Theoretical Physics

2.2 Lorentz Invariance

With this principle, the special theory of relativity was developed in two directions: a) the
introduction of relativistic mechanics, replacing the classical mechanics, and b) the develop-
ment of relativistic quantum mechanics.

x^1

x^2

x^3

̃x^1

̃x^2

x ̃^3

v

Figure 2.1

We now introduce typical Lorentz transformations^1. Let(x,t)and( ̃x, ̃t)be two inertial
systems which are in relativistic motion with a constant velocityv, as shown in Figure2.1,
wherevis in thex 1 -axis direction. Then the Lorentz transformation for the two inertial
systems is given by

(2.2.3) ( ̃x^1 , ̃x^2 , ̃x^3 ) =

(

x^1 −vt

√

1 −v^2 /c^2

,x^2 ,x^3

)

, ̃t=

t−x^1 v/c^2

√

1 −v^2 /c^2

,

wherecis the speed of light.
The vacuum speed of light is invariant under the above Lorentz transformation (2.2.3).
To see this, notice that the speed of light is ̃c=d ̃x^1 /d ̃tmeasured in the ̃x-coordinate system,
and isc=dx^1 /dtmeasured in thex-coordinate system. Consequently,

̃c=

dx ̃^1

d ̃t

=

dx^1 −vdt

√

1 −v^2 /c^2

/dt−dx (^1) v/c 2
√
1 −v^2 /c^2

=

dx^1 /dt−v

1 −(dx^1 /dt)(v/c^2 )

=

c−v

1 −v/c

=c.

2.2.2 Minkowski space and Lorentz tensors

We recall that each symmetry is characterized by three ingredients: space/manifold, trans-
formation group, and tensors. For the Lorentz invariance, the transformation group is the
Lorentz group, consisting of Lorentz transformations suchas those given by (2.2.3), and the
corresponding space is the Minkowski space introduced below.

Minkowski Space

We know that the Newtonian mechanics is defined in the Euclidean spaceR^3 , and the
timetis regarded as a parameter. From the Lorentz transformation(2.2.3), we see that in
relativistic physics, there is no absolute space and time, and the timetcannot be separated
from the spaceR^3. Namely the Minkowski spaceM^4 is a 4-dimensional space-time manifold
defined by

(2.2.4) M^4 ={(x^0 ,x^1 ,x^2 ,x^3 )|x^0 =ct,(x^1 ,x^2 ,x^3 )∈R^3 },

(^1) General Lorentz transformations are linear transformations that preserves the Minkowski metric (2.2.6), and the
transformation (2.2.3) is called a boost.