The space-time 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 force-free 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 space-time 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 Yang-Mills equations for the weak and strong fields remain unchanged
if the space-time 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 space-time
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 space-time points coincide;
it means that the two space-time 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
space-time is written as R(3, 1) = {(X,
Y,
Z,
T)
| X, Y,
Z,
T∈R},
which forms a 4-dimensional 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
|
(X1, Y1, Z1,
T1)
⋅
(X2,
Y2,
Z2,
T2)
= X1X2 + Y1Y2
+ Z1Z2 - T1T2 |
|
(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((X1, Y1,
Z1,
T1),
(X2, Y2,
Z2,
T2)) |
= |
|(X1, Y1, Z1,
T1)
- (X2, Y2, Z2,
T2)| |
|
= |
|(X1 - X2, Y1
- Y2, Z1 - Z2, T1
- T2)| |
|
= |
((X1 - X2)2 + (Y1
- Y2)2 + (Z1 - Z2)2
- (T1 - T2)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(X1, Y1, Z1,
T1)
⋅
f(X2,
Y2,
Z2,
T2)
= (X1,
Y1,
Z1,
T1)
⋅
(X2,
Y2,
Z2, T2) |
|
(1.7) |
for all ( X1, Y1, Z1,
T1)
and ( X2, Y2,
Z2,
T2)
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 space-time
R(3,
1) that leave the velocity of light c = 1 invariant.
Referring to Lemma 4 [1][3], let S3
= {1, ρ, ρ2,
σ,
σρ,
σρ2}
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 σ2 = 1,
ρ3
= 1 and σρσ-1 = ρ-1.
We identify S3 with transformations of Minkowski space-time
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 |
|
|
|
|
|
|
|
|
ρ2 |
= |
|
|
|
|
|
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 |
|
|
|
|
|
|
|
|
σρ2 |
= |
|
|
|
|
|
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 S3 under matrix multiplication and form
an isomorphic group of transformations S3 of Minkowski
space-time R(3, 1). The transformation group S3
is the permutation group on the space coordinates X, Y, Z
that keeps the time coordinate T fixed. Acting on a right-handed
space coordinate system, the transformations 1, ρ,
ρ 2
give right-handed space coordinate systems whereas the transformations
σ,
σρ,
σρ 2
give left-handed space coordinate systems as shown below in (1.9):
|
|
|
Z
|
|
|
|
|
X |
|
Y |
|
1 |
|
|
|
|
X
|
|
|
|
|
Y |
|
Z |
|
ρ |
|
|
|
|
Y
|
|
|
|
|
Z |
|
X |
|
ρ2 |
|
|
|
|
Z
|
|
|
|
|
Y |
|
X |
|
σ |
|
|
|
|
X
|
|
|
|
|
Z |
|
Y |
|
σρ |
|
|
|
|
Y
|
|
|
|
|
X |
|
Z |
|
σρ2 |
|
|
|
(1.9) |
By (1.8) and (1.9), each transformation in S3 leaves
the expression (1.4) for the Lorentz inner product invariant and hence
satisfies the condition (1.7) for being a Lorentz transformation. Thus,
S3
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(X1, Y1,
Z1,
T1),
f(X2,
Y2,
Z2,
T2))
= η((X1,
Y1,
Z1,
T1),
( X2,
Y2,
Z2, T2)) |
|
(1.10) |
for all ( X1, Y1, Z1,
T1)
and ( X2, Y2,
Z2,
T2)
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 space-time
R(3,
1) that leave the distance between every pair of space-time points
in the metric η invariant. If f is a
Lorentz transformation in O(3, 1), then
|
|
η( f(X1, Y1,
Z1,
T1),
f(X2,
Y2,
Z2,
T2)) |
|
= |
| f(X1, Y1, Z1,
T1)
- f(X2, Y2, Z2,
T2)| |
(by 1.6) |
= |
| f((X1, Y1, Z1,
T1)
- (X2, Y2, Z2,
T2))| |
(by linearity of f ) |
= |
| f(X1 - X2, Y1
- Y2, Z1 - Z2, T1-
T2)| |
(using vector space R(3, 1)) |
= |
( f(X1 - X2, Y1
- Y2, Z1 - Z2, T1
- T2) ⋅ f(X1
- X2, Y1 - Y2, Z1
- Z2, T1 - T2))1/2 |
(by 1.5) |
= |
( (X1 - X2, Y1
- Y2, Z1 - Z2, T1
- T2) ⋅ (X1
- X2, Y1 - Y2, Z1
- Z2, T1 - T2))1/2 |
(by 1.7, since f is in O(3, 1)) |
= |
|(X1 - X2, Y1
- Y2, Z1 - Z2, T1
- T2)| |
(by 1.5) |
= |
| (X1, Y1, Z1,
T1)
- (X2, Y2, Z2,
T2)| |
(using vector space R(3, 1)) |
= |
η((X1, Y1,
Z1,
T1),
(X2,
Y2,
Z2,
T2)) |
(by 1.6) |
|
(1.11) |
for all ( X1, Y1, Z1,
T1)
and ( X2, Y2,
Z2,
T2)
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 space-time 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
space-time. 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 well-defined:
|
|
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 space-time form an abelian group isomorphic to the
additive group of the vector space of Minkowski space-time
R(3,
1).
We shall now show that the only translation that is also a Lorentz transformation
is the identity transformation of Minkowski space-time, 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 non-negative, 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 space-time.
This implies that R'(3, 1) ∩
O(3,
1) = {1}.
We shall now show that the only isometries that fix the origin of Minkowski
space-time are the Lorentz transformations. Suppose f is an isometry
of Minkowski space-time such that f(0, 0, 0, 0) = (0, 0, 0, 0).
Given any (X1, Y1, Z1,
T1)
and (X2, Y2,
Z2,
T2)
in R(3, 1), define
|
(X1*,
Y1*,
Z1*,
T1*) |
= |
f(X1, Y1,
Z1,
T1) |
(X2*, Y2*,
Z2*,
T2*) |
= |
f(X2,
Y2,
Z2,
T2) |
|
(1.18) |
Then
|
|
X1 2 + Y12
+ Z1 2 - T12 |
|
= |
η((X1, Y1,
Z1,
T1),
(0, 0, 0, 0)) 2 |
(by 1.6) |
= |
η(f(X1, Y1,
Z1,
T1),
f(0,
0, 0, 0)) 2 |
(since f is an isometry) |
= |
η((X1*, Y1*,
Z1*,
T1*),
(0, 0, 0, 0)) 2 |
(by 1.18) |
= |
X1* 2 + Y1*2
+ Z1* 2 - T1*2 |
(by 1.6) |
|
(1.19) |
Similarly
|
|
X2 2 + Y22
+ Z2 2 - T22 |
|
= |
η((X2, Y2,
Z2,
T2),
(0, 0, 0, 0)) 2 |
(by 1.6) |
= |
η(f(X2, Y2,
Z2,
T2),
f(0,
0, 0, 0)) 2 |
(since f is an isometry) |
= |
η((X2*, Y2*,
Z2*,
T2*),
(0, 0, 0, 0)) 2 |
(by 1.18) |
= |
X2* 2 + Y2*2
+ Z2* 2 - T2*2 |
(by 1.6) |
|
(1.20) |
Now, since f is an isometry
|
|
η(f(X1,
Y1,
Z1,
T1),
f(X2,
Y2,
Z2,
T2))2
|
= |
η((X1, Y1,
Z1,
T1),
(X2, Y2,
Z2,
T2))2 |
(by 1.10) |
⇒ |
η((X1*,
Y1*,
Z1*,
T1*),
(X2*, Y2*,
Z2*,
T2*))2
|
= |
η((X1, Y1,
Z1,
T1),
(X2, Y2,
Z2,
T2))2 |
(by 1.18) |
⇒ |
(X1* - X2*)2
+ (Y1* - Y2*)2
+ (Z1* - Z2*)2
- (T1* - T2*)2 |
= |
(X1 - X2)2 + (Y1
- Y2)2 + (Z1 - Z2)2
- (T1 - T2)2 |
(by 1.6) |
|
(1.21) |
Expanding both sides of the last equation in (1.21) we obtain
where
|
A* |
= |
X1* 2 + Y1*2
+ Z1* 2 - T1*2 |
B* |
= |
X2* 2 + Y2*2
+ Z2* 2 - T2*2 |
C* |
= |
-2 X1*X2*
-2 Y1*Y2* -2
Z1*Z2*
+2 T1*T2* |
|
|
A |
= |
X1 2 + Y12
+ Z1 2 - T12 |
B |
= |
X2 2 + Y22
+ Z2 2 - T22 |
C |
= |
-2 X1X2 -2 Y1Y2
-2
Z1Z2 +2 T1T2 |
|
|
|
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 X1*X2*
-2 Y1*Y2* -2
Z1*Z2*
+2 T1*T2*
|
= |
-2 X1X2 -2 Y1Y2
-2 Z1Z2 +2 T1T2 |
|
⇒ |
X1*X2*
+ Y1*Y2* +
Z1*Z2*
- T1*T2*
|
= |
X1X2 + Y1Y2
+
Z1Z2 - T1T2 |
(dividing by -2) |
⇒ |
(X1*, Y1*,
Z1*,
T1*)
⋅
(X2*,
Y2*,
Z2*,
T2*)
|
= |
(X1, Y1, Z1,
T1)
⋅
(X2,
Y2,
Z2, T2) |
(by 1.4) |
⇒ |
f(X1, Y1, Z1,
T1)
⋅
f(X2,
Y2,
Z2,
T2)
|
= |
(X1, Y1, Z1,
T1)
⋅
(X2,
Y2,
Z2, T2) |
(by 1.18) |
|
(1.23) |
Hence, if f is an isometry of Minkowski space-time 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
space-time can be uniquely written as the product of a translation and
a Lorentz transformation. Suppose f is an isometry of Minkowski
space-time. 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'')-1f(X',
Y',
Z',
T')
|
|
= |
|
g'g-1 |
|
(1.25) |
But f(X'', Y'', Z'', T'')-1f(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'')-1f(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 space-time. The abelian group (module) of the vector
space
R'(3, 1) is isomorphic to the abelian group (module)
R(3,
1) of Minkowski space-time 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(g1g2)
|
= |
(g1g2)-1 f (g1g2) |
|
= |
g2-1(g1-1
f g1)g2 |
|
= |
g2-1( f ξ(g1))g2 |
|
= |
(f ξ(g1))ξ(g2) |
|
= |
f ξ(g1)ξ(g2) |
|
(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 space-time.
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 space-time 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:
|
( f1, g1)( f2,
g2)
|
= |
( f1( f2 ξ(g1)),
g1g2) |
|
= |
( f1(g1-1f2g1),
g1g2) |
|
(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).
|