The spacetime of physics is defined by
four real coordinates: the three space coordinates X,
Y,
Z
and the time coordinate
T. The theory of special relativity [4][5]
is concerned with inertial reference frames in which forcefree particles
do not experience any acceleration with respect to the coordinate system.
Inertial reference frames are defined by the group of Lorentz transformations
which are linear transformations of the spacetime coordinates that leave
the velocity of light c invariant. A Lorentz transformation transforms
one inertial reference frame to another that is in uniform motion relative
to the first. One of the main motivations for restricting the theory to
inertial reference frames is that Maxwell's equations for electromagnetism
and the YangMills equations for the weak and strong fields remain unchanged
if the spacetime coordinates are subjected to Lorentz transformations.
Thus, according to the theory of special relativity, light has a constant
velocity of propagation c. If a light signal in a vacuum starts
from a space point ( X,
Y,
Z) at the time T,
it spreads as a spherical wave and reaches a neighboring space point ( X+ dX,
Y+ dY,
Z+ dZ)
at the time T+ dT. Measuring the distance traveled by the
light signal, we must have

(cdT)^{2} = (dX)^{2}+(dY)^{2}+(dZ)^{2} 

(1.1) 

Figure 1.1. A light signal is represented by a sphere of
radius cdT centered at (X, Y, Z) 
The equation (1.1) may be rewritten as

(dX)^{2} + (dY)^{2} + (dZ)^{2}
 (cdT)^{2} = 0 

(1.2) 
Equation (1.2) represents an objective relation between neighboring spacetime
points and it holds for all inertial reference frames provided the transformations
of the coordinates are restricted to those of special relativity, i.e.
Lorentz transformations. By considering the inertial reference frames of
special relativity, it can also be shown that the Lorentz transformations
are precisely the linear transformations that leave the more general quantity

(dS)^{2} = (dX)^{2} + (dY)^{2}
+ (dZ)^{2}  (cdT)^{2} 

(1.3) 
invariant. Note, however, that the vanishing of ( dS) ^{2}
in equation (1.3) does not imply that the two spacetime points coincide;
it means that the two spacetime points can be connected by a light signal.
This is Einstein's physical motivation for the theory of special relativity.
Let us now formulate the theory of special relativity in Minkowski's
notation [6]. Select fundamental Planck
units [7][8] for measuring X, Y,
Z
and T, so that the velocity of light c = 1. Then
Minkowski
spacetime is written as R^{(3, 1)} = {(X,
Y,
Z,
T)
 X, Y,
Z,
T∈R},
which forms a 4dimensional vector space over the real numbers
R
with the usual addition of vectors and scalar multiplication. The Lorentz
inner product in
R^{(3, 1)} is defined as

(X_{1}, Y_{1}, Z_{1},
T_{1})
⋅
(X_{2},
Y_{2},
Z_{2},
T_{2})
= X_{1}X_{2} + Y_{1}Y_{2}
+ Z_{1}Z_{2}  T_{1}T_{2} 

(1.4) 
and the Lorentz norm in R^{(3, 1)} is defined as

(X, Y, Z, T) 
= 
((X, Y, Z, T) ⋅
(X, Y, Z, T))^{1/2} 

= 
(X ^{2} + Y ^{2} + Z ^{2}
 T ^{2})^{1/2} 

(1.5) 
which is a complex number in general. The Lorentz metric in R^{(3,
1)} is given by

η((X_{1}, Y_{1},
Z_{1},
T_{1}),
(X_{2}, Y_{2},
Z_{2},
T_{2})) 
= 
(X_{1}, Y_{1}, Z_{1},
T_{1})
 (X_{2}, Y_{2}, Z_{2},
T_{2}) 

= 
(X_{1}  X_{2}, Y_{1}
 Y_{2}, Z_{1}  Z_{2}, T_{1}
 T_{2}) 

= 
((X_{1}  X_{2})^{2} + (Y_{1}
 Y_{2})^{2} + (Z_{1}  Z_{2})^{2}
 (T_{1}  T_{2})^{2})^{1/2} 

(1.6) 
in agreement with (1.3) which is an expression of the Lorentz metric locally
in terms of infinitesimal differentials. A linear transformation f
: R^{(3, 1)} → R^{(3,
1)} is called a Lorentz transformation if

f(X_{1}, Y_{1}, Z_{1},
T_{1})
⋅
f(X_{2},
Y_{2},
Z_{2},
T_{2})
= (X_{1},
Y_{1},
Z_{1},
T_{1})
⋅
(X_{2},
Y_{2},
Z_{2}, T_{2}) 

(1.7) 
for all ( X_{1}, Y_{1}, Z_{1},
T_{1})
and ( X_{2}, Y_{2},
Z_{2},
T_{2})
in R^{(3, 1)}. The Lorentz transformations form a group
O^{(3,
1)} called the Lorentz group under the binary operation of
function composition. The Lorentz group O^{(3, 1)} consists
precisely of all the transformations of Minkowski spacetime
R^{(3,
1)} that leave the velocity of light c = 1 invariant.
Referring to Lemma 4 [1][3], let S_{3}
= {1, ρ, ρ^{2},
σ,
σρ,
σρ^{2}}
denote the symmetric group on three letters which is abstractly isomorphic
to the dihedral group of order 6 generated by σ,
ρ
subject to the relations σ^{2} = 1,
ρ^{3}
= 1 and σρσ^{1} = ρ^{1}.
We identify S_{3} with transformations of Minkowski spacetime
R^{(3,
1)} via the following correspondence:

1 
= 





1 

0 

0 

0 


0 

1 

0 

0 


0 

0 

1 

0 


0 

0 

0 

1 








ρ 
= 





0 

1 

0 

0 


0 

0 

1 

0 


1 

0 

0 

0 


0 

0 

0 

1 








ρ^{2} 
= 





0 

0 

1 

0 


1 

0 

0 

0 


0 

1 

0 

0 


0 

0 

0 

1 








σ 
= 





0 

1 

0 

0 


1 

0 

0 

0 


0 

0 

1 

0 


0 

0 

0 

1 








σρ 
= 





0 

0 

1 

0 


0 

1 

0 

0 


1 

0 

0 

0 


0 

0 

0 

1 








σρ^{2} 
= 





1 

0 

0 

0 


0 

0 

1 

0 


0 

1 

0 

0 


0 

0 

0 

1 









(1.8) 
It is easy to verify that the six matrices (1.8) satisfy the generating
relations of S_{3} under matrix multiplication and form
an isomorphic group of transformations S_{3} of Minkowski
spacetime R^{(3, 1)}. The transformation group S_{3}
is the permutation group on the space coordinates X, Y, Z
that keeps the time coordinate T fixed. Acting on a righthanded
space coordinate system, the transformations 1, ρ,
ρ ^{2}
give righthanded space coordinate systems whereas the transformations
σ,
σρ,
σρ ^{2}
give lefthanded space coordinate systems as shown below in (1.9):



Z





X 

Y 

1 




X





Y 

Z 

ρ 




Y





Z 

X 

ρ^{2} 




Z





Y 

X 

σ 




X





Z 

Y 

σρ 




Y





X 

Z 

σρ^{2} 



(1.9) 
By (1.8) and (1.9), each transformation in S_{3} leaves
the expression (1.4) for the Lorentz inner product invariant and hence
satisfies the condition (1.7) for being a Lorentz transformation. Thus,
S_{3}
is a subgroup of the Lorentz group
O^{(3, 1)}.
A linear transformation f : R^{(3, 1)} → R^{(3,
1)} is called an isometry if

η( f(X_{1}, Y_{1},
Z_{1},
T_{1}),
f(X_{2},
Y_{2},
Z_{2},
T_{2}))
= η((X_{1},
Y_{1},
Z_{1},
T_{1}),
( X_{2},
Y_{2},
Z_{2}, T_{2})) 

(1.10) 
for all ( X_{1}, Y_{1}, Z_{1},
T_{1})
and ( X_{2}, Y_{2},
Z_{2},
T_{2})
in R^{(3, 1)}. The isometries form a group
I^{(3,
1)} called the Poincaré group under the binary operation
of function composition. The Poincaré group I^{(3, 1)}
consists precisely of all the transformations of Minkowski spacetime
R^{(3,
1)} that leave the distance between every pair of spacetime points
in the metric η invariant. If f is a
Lorentz transformation in O^{(3, 1)}, then


η( f(X_{1}, Y_{1},
Z_{1},
T_{1}),
f(X_{2},
Y_{2},
Z_{2},
T_{2})) 

= 
 f(X_{1}, Y_{1}, Z_{1},
T_{1})
 f(X_{2}, Y_{2}, Z_{2},
T_{2}) 
(by 1.6) 
= 
 f((X_{1}, Y_{1}, Z_{1},
T_{1})
 (X_{2}, Y_{2}, Z_{2},
T_{2})) 
(by linearity of f ) 
= 
 f(X_{1}  X_{2}, Y_{1}
 Y_{2}, Z_{1}  Z_{2}, T_{1}
T_{2}) 
(using vector space R^{(3, 1)}) 
= 
( f(X_{1}  X_{2}, Y_{1}
 Y_{2}, Z_{1}  Z_{2}, T_{1}
 T_{2}) ⋅ f(X_{1}
 X_{2}, Y_{1}  Y_{2}, Z_{1}
 Z_{2}, T_{1}  T_{2}))^{1/2} 
(by 1.5) 
= 
( (X_{1}  X_{2}, Y_{1}
 Y_{2}, Z_{1}  Z_{2}, T_{1}
 T_{2}) ⋅ (X_{1}
 X_{2}, Y_{1}  Y_{2}, Z_{1}
 Z_{2}, T_{1}  T_{2}))^{1/2} 
(by 1.7, since f is in O^{(3, 1)}) 
= 
(X_{1}  X_{2}, Y_{1}
 Y_{2}, Z_{1}  Z_{2}, T_{1}
 T_{2}) 
(by 1.5) 
= 
 (X_{1}, Y_{1}, Z_{1},
T_{1})
 (X_{2}, Y_{2}, Z_{2},
T_{2}) 
(using vector space R^{(3, 1)}) 
= 
η((X_{1}, Y_{1},
Z_{1},
T_{1}),
(X_{2},
Y_{2},
Z_{2},
T_{2})) 
(by 1.6) 

(1.11) 
for all ( X_{1}, Y_{1}, Z_{1},
T_{1})
and ( X_{2}, Y_{2},
Z_{2},
T_{2})
in R^{(3, 1)}, showing that the Lorentz group O^{(3,
1)} is a subgroup of the Poincaré group I^{(3, 1)}.
Given a fixed (X', Y', Z', T') in R^{(3,
1)}, a translation of Minkowski spacetime by (X', Y',
Z',
T')
is a linear transformation f_{(X',
Y',
Z',
T')}
: R^{(3, 1)} → R^{(3,
1)} given by

f_{(X', Y', Z', T')}(X,
Y,
Z,T)
= (X + X', Y + Y',
Z + Z',
T
+ T') 

(1.12) 
for all ( X, Y,
Z,
T) in R^{(3, 1)}.
Let R'^{(3, 1)} denote the set of all translations of Minkowski
spacetime. Given two translations f_{(X', Y',
Z',
T')}
and f_{(X'', Y'', Z'',
T'')},
the binary operation of function composition f_{(X',
Y',
Z',
T')
}f_{(X'',
Y'',
Z'',
T'')}
is welldefined:


f_{(X', Y', Z', T')} f_{(X'',
Y'',
Z'',
T'')}(X,
Y,
Z,
T) 
= 
f_{(X', Y', Z', T')}(X
+ X'', Y + Y'',
Z + Z'',
T +
T'') 
= 
(X + X' + X'', Y + Y' + Y'',
Z
+ Z' + Z'',
T + T' + T'') 
= 
f_{(X' + X'', Y' + Y'', Z'
+ Z'', T' + T'')} 

(1.13) 
Under the correspondence f_{(X', Y', Z',
T')} →
( X', Y',
Z', T'), the translations R'^{(3,
1)} of Minkowski spacetime form an abelian group isomorphic to the
additive group of the vector space of Minkowski spacetime
R^{(3,
1)}.
We shall now show that the only translation that is also a Lorentz transformation
is the identity transformation of Minkowski spacetime, i.e. R'^{(3,
1)} ∩ O^{(3, 1)}
= {1}. Suppose f_{(X', Y', Z',
T')}
= g, where f_{(X', Y', Z',
T')}
is a translation in R'^{(3, 1)} and g is a Lorentz
transformation in O^{(3, 1)}. Then, for all (X, Y,
Z,
T)
in R^{(3, 1)}:

g(X, Y,
Z,
T) 
= 
f_{(X', Y', Z', T')}(X,
Y,
Z,
T) 

= 
(X + X', Y + Y',
Z + Z',
T
+
T') 

(1.14) 
In particular,

g(0, 0, 0, 0) 
= 
f_{(X', Y', Z', T')}(0,
0, 0, 0) 

= 
(X', Y',
Z',
T') 

(1.15) 
Then, since g is a Lorentz transformation, for all ( X, Y,
Z,
T)
in R^{(3, 1)}:

0 
= 
X ⋅ 0 + Y ⋅
0 + Z ⋅ 0  T ⋅0 


= 
(X, Y,
Z,
T) ⋅
(0, 0, 0, 0) 
(by 1.4) 

= 
g(X, Y,
Z,
T) ⋅
g(0,
0, 0, 0) 
(by 1.7) 

= 
(X + X', Y + Y',
Z + Z',
T
+
T') ⋅ (X', Y',
Z',
T') 
(by 1.14 and 1.15) 

= 
(X + X')X' + (Y + Y')Y' +
(Z + Z')Z'  (T +
T')T' 
(by 1.4) 

= 
(XX' + YY' + ZZ'  TT') + (X'^{2}
+ Y'^{2} + Z'^{2}  T'^{2}) 


(1.16) 
Put T = 0 in (1.16), then for all X, Y,
Z:

T'^{2} 
= 
(XX' + YY' + ZZ') + (X'^{2} + Y'^{2}
+ Z'^{2}) 


(1.17) 
The LHS of (1.17) is nonnegative, but we can always choose X,
Y,
Z
such that the RHS of (1.17) is negative unless X' = Y' =
Z'
= 0. Thus f_{(X', Y', Z',
T')}
= g must be the identity transformation of Minkowski spacetime.
This implies that R'^{(3, 1)} ∩
O^{(3,
1)} = {1}.
We shall now show that the only isometries that fix the origin of Minkowski
spacetime are the Lorentz transformations. Suppose f is an isometry
of Minkowski spacetime such that f(0, 0, 0, 0) = (0, 0, 0, 0).
Given any (X_{1}, Y_{1}, Z_{1},
T_{1})
and (X_{2}, Y_{2},
Z_{2},
T_{2})
in R^{(3, 1)}, define

(X_{1}^{*},
Y_{1}^{*},
Z_{1}^{*},
T_{1}^{*}) 
= 
f(X_{1}, Y_{1},
Z_{1},
T_{1}) 
(X_{2}^{*}, Y_{2}^{*},
Z_{2}^{*},
T_{2}^{*}) 
= 
f(X_{2},
Y_{2},
Z_{2},
T_{2}) 

(1.18) 
Then


X_{1} ^{2} + Y_{1}^{2}
+ Z_{1} ^{2}  T_{1}^{2} 

= 
η((X_{1}, Y_{1},
Z_{1},
T_{1}),
(0, 0, 0, 0)) ^{2} 
(by 1.6) 
= 
η(f(X_{1}, Y_{1},
Z_{1},
T_{1}),
f(0,
0, 0, 0)) ^{2} 
(since f is an isometry) 
= 
η((X_{1}^{*}, Y_{1}^{*},
Z_{1}^{*},
T_{1}^{*}),
(0, 0, 0, 0)) ^{2} 
(by 1.18) 
= 
X_{1}^{*} ^{2} + Y_{1}^{*2}
+ Z_{1}^{*} ^{2}  T_{1}^{*2} 
(by 1.6) 

(1.19) 
Similarly


X_{2} ^{2} + Y_{2}^{2}
+ Z_{2} ^{2}  T_{2}^{2} 

= 
η((X_{2}, Y_{2},
Z_{2},
T_{2}),
(0, 0, 0, 0)) ^{2} 
(by 1.6) 
= 
η(f(X_{2}, Y_{2},
Z_{2},
T_{2}),
f(0,
0, 0, 0)) ^{2} 
(since f is an isometry) 
= 
η((X_{2}^{*}, Y_{2}^{*},
Z_{2}^{*},
T_{2}^{*}),
(0, 0, 0, 0)) ^{2} 
(by 1.18) 
= 
X_{2}^{*} ^{2} + Y_{2}^{*2}
+ Z_{2}^{*} ^{2}  T_{2}^{*2} 
(by 1.6) 

(1.20) 
Now, since f is an isometry


η(f(X_{1},
Y_{1},
Z_{1},
T_{1}),
f(X_{2},
Y_{2},
Z_{2},
T_{2}))^{2}

= 
η((X_{1}, Y_{1},
Z_{1},
T_{1}),
(X_{2}, Y_{2},
Z_{2},
T_{2}))^{2} 
(by 1.10) 
⇒ 
η((X_{1}^{*},
Y_{1}^{*},
Z_{1}^{*},
T_{1}^{*}),
(X_{2}^{*}, Y_{2}^{*},
Z_{2}^{*},
T_{2}^{*}))^{2}

= 
η((X_{1}, Y_{1},
Z_{1},
T_{1}),
(X_{2}, Y_{2},
Z_{2},
T_{2}))^{2} 
(by 1.18) 
⇒ 
(X_{1}^{*}  X_{2}^{*})^{2}
+ (Y_{1}^{*}  Y_{2}^{*})^{2}
+ (Z_{1}^{*}  Z_{2}^{*})^{2}
 (T_{1}^{*}  T_{2}^{*})^{2} 
= 
(X_{1}  X_{2})^{2} + (Y_{1}
 Y_{2})^{2} + (Z_{1}  Z_{2})^{2}
 (T_{1}  T_{2})^{2} 
(by 1.6) 

(1.21) 
Expanding both sides of the last equation in (1.21) we obtain
where

A* 
= 
X_{1}^{*} ^{2} + Y_{1}^{*2}
+ Z_{1}^{*} ^{2}  T_{1}^{*2} 
B* 
= 
X_{2}^{*} ^{2} + Y_{2}^{*2}
+ Z_{2}^{*} ^{2}  T_{2}^{*2} 
C* 
= 
2 X_{1}^{*}X_{2}^{*}
2 Y_{1}^{*}Y_{2}^{*} 2
Z_{1}^{*}Z_{2}^{*}
+2 T_{1}^{*}T_{2}^{*} 


A 
= 
X_{1} ^{2} + Y_{1}^{2}
+ Z_{1} ^{2}  T_{1}^{2} 
B 
= 
X_{2} ^{2} + Y_{2}^{2}
+ Z_{2} ^{2}  T_{2}^{2} 
C 
= 
2 X_{1}X_{2} 2 Y_{1}Y_{2}
2
Z_{1}Z_{2} +2 T_{1}T_{2} 



In equation (1.22), summands A*, A are equal by (1.19) and
summands
B*,
B are equal by (1.20). Hence summands C*,
C
must be equal:


2 X_{1}^{*}X_{2}^{*}
2 Y_{1}^{*}Y_{2}^{*} 2
Z_{1}^{*}Z_{2}^{*}
+2 T_{1}^{*}T_{2}^{*}

= 
2 X_{1}X_{2} 2 Y_{1}Y_{2}
2 Z_{1}Z_{2} +2 T_{1}T_{2} 

⇒ 
X_{1}^{*}X_{2}^{*}
+ Y_{1}^{*}Y_{2}^{*} +
Z_{1}^{*}Z_{2}^{*}
 T_{1}^{*}T_{2}^{*}

= 
X_{1}X_{2} + Y_{1}Y_{2}
+
Z_{1}Z_{2}  T_{1}T_{2} 
(dividing by 2) 
⇒ 
(X_{1}^{*}, Y_{1}^{*},
Z_{1}^{*},
T_{1}^{*})
⋅
(X_{2}^{*},
Y_{2}^{*},
Z_{2}^{*},
T_{2}^{*})

= 
(X_{1}, Y_{1}, Z_{1},
T_{1})
⋅
(X_{2},
Y_{2},
Z_{2}, T_{2}) 
(by 1.4) 
⇒ 
f(X_{1}, Y_{1}, Z_{1},
T_{1})
⋅
f(X_{2},
Y_{2},
Z_{2},
T_{2})

= 
(X_{1}, Y_{1}, Z_{1},
T_{1})
⋅
(X_{2},
Y_{2},
Z_{2}, T_{2}) 
(by 1.18) 

(1.23) 
Hence, if f is an isometry of Minkowski spacetime such that f(0,
0, 0, 0) = (0, 0, 0, 0) then f must be a Lorentz transformation.
Using this result, we shall now show that every isometry of Minkowski
spacetime can be uniquely written as the product of a translation and
a Lorentz transformation. Suppose f is an isometry of Minkowski
spacetime. To demonstrate the existence, we want to show that f
= f_{(X', Y', Z', T')} g
for at least one translation f_{(X', Y', Z',
T')
}in
R'^{(3,
1)} and at least one Lorentz transformation
g in O^{(3,
1)}. Define (X',
Y', Z',
T') = f(0,
0, 0, 0) and g =
f_{(X',
Y',
Z',
T')}^{1
}f.
Then f_{(X',
Y',
Z',
T')} is
a translation in R'^{(3, 1)} and as a product of two isometries,
g
is certainly an isometry. We must show that g is a Lorentz transformation:

g(0, 0, 0, 0) 
= 
f_{(X', Y', Z', T')}^{1
}f(0,
0, 0, 0) 

= 
f_{(X', Y', Z', T')}^{1}
(X', Y', Z', T') 

= 
(0, 0, 0, 0) 

(1.24) 
Thus, g is a Lorentz transformation by (1.23). To demonstrate the
uniqueness, suppose f = f_{(X'', Y'', Z'',
T'')}g'
for some translation f_{(X'',
Y'', Z'',
T'')}
in R'^{(3, 1)} and some Lorentz transformation
g'
in O^{(3, 1)}:

⇒ 
f_{(X', Y', Z', T')
}g

= 
f 
= 
f_{(X'', Y'', Z'', T'') }g' 
⇒ 
f_{(X'', Y'', Z'',
T'')}^{1}f_{(X',
Y',
Z',
T')}


= 

g'g^{1} 

(1.25) 
But f_{(X'', Y'', Z'', T'')}^{1}f_{(X',
Y',
Z',
T')}
= g'g^{1} belongs to R'^{(3, 1)} ∩
O^{(3,
1)} = {1} by (1.17). Hence f_{(X'',
Y'', Z'',
T'')}^{1}f_{(X',
Y',
Z',
T')}
= g'g^{1} = 1. Thus
f_{(X'',
Y'',
Z'',
T'')}
= f_{(X',
Y',
Z',
T')} and g'
= g, demonstrating the uniqueness of the isometry
f = f_{(X',
Y',
Z',
T')
}g
as the product of a translation and a Lorentz transformation.
We shall now show how the Lorentz group O^{(3, 1)} acts
as a group of automorphisms of the vector space of translations R'^{(3,
1)} of Minkowski spacetime. The abelian group (module) of the vector
space
R'^{(3, 1)} is isomorphic to the abelian group (module)
R^{(3,
1)} of Minkowski spacetime under the correspondence
f_{(X',
Y',
Z',
T')} →
(X',
Y',
Z', T'). Let Aut R'^{(3,
1)}denote the group of automorphisms of the module R'^{(3,
1)}. Define the group representation

ξ:

O^{(3, 1)} 
→ 
Aut R'^{(3, 1)} 

g

→ 
ξ(g) 

(1.26) 
where the right action of ξ( g) on R'^{(3,
1)} is given by

ξ(g):

R'^{(3, 1)} 
→ 
R'^{(3, 1)} 

f

→ 
g^{1} f g 

(1.27) 
Note that we write the action on the right as f ξ( g)
= g^{1} f g for each f in R'^{(3,
1)}. To show that ξ is a group representation,
we must verify that ξ is a homomorphism. For
each
f in R'^{(3, 1)}:

f ξ(g_{1}g_{2})

= 
(g_{1}g_{2})^{1} f (g_{1}g_{2}) 

= 
g_{2}^{1}(g_{1}^{1}
f g_{1})g_{2} 

= 
g_{2}^{1}( f ξ(g_{1}))g_{2} 

= 
(f ξ(g_{1}))ξ(g_{2}) 

= 
f ξ(g_{1})ξ(g_{2}) 

(1.28) 
Hence, ξ is a homomorphism showing how the Lorentz
group O^{(3, 1)} acts as a group of automorphisms of the
vector space of translations R'^{(3, 1)} of Minkowski spacetime.
Following [1] Section III, the abelian
group of the vector space of translations R'^{(3, 1)} is
an Eilenberg module for the Lorentz group O^{(3, 1)} and
we can write the Poincaré group I^{(3, 1)} of isometries
of Minkowski spacetime as the split extension R'^{(3, 1)}] O^{(3,
1)}:

I^{(3, 1)}

= 
R'^{(3, 1)}]O^{(3, 1)} 
= 
{( f, g) f ∈R'^{(3,
1)}, g ∈O^{(3, 1)}} 

(1.29) 
By (1.24) and (1.25), each element fg of the Poincaré group
I^{(3,
1)} is written uniquely as the element ( f,
g) of the
split extension R'^{(3, 1)}] O^{(3, 1)}. The
group operation in the split extension R'^{(3, 1)}] O^{(3,
1)} is given by:

( f_{1}, g_{1})( f_{2},
g_{2})

= 
( f_{1}( f_{2} ξ(g_{1})),
g_{1}g_{2}) 

= 
( f_{1}(g_{1}^{1}f_{2}g_{1}),
g_{1}g_{2}) 

(1.30) 
There is an exact sequence of groups


ι^{(3, 1)} 



π^{(3, 1)} 

R'^{(3, 1)}

→

I^{(3, 1)} 
= 
R'^{(3, 1)}]O^{(3, 1)} 
→

O^{(3, 1)} 

(1.31) 
where the injection function is defined by ι ^{(3,
1)}( f ) = ( f, 1), the projection function is defined
by π ^{(3, 1)}(( f, g))
= g and the sequence is split by the zero function




ο^{(3, 1)} 

I^{(3, 1)} 
= 
R'^{(3, 1)}]O^{(3, 1)} 
←

O^{(3, 1)} 

(1.32) 
defined by ο ^{(3, 1)}( g ) =
( f_{(0, 0, 0, 0)}, g).
