Lorentz Groups In Relativistic Space-Time Johar M. Ashfaque
Abstract The aim of this paper is to explore the role of SO(4), played by the Lorentz groups in relativistic space-time.
Contents 1 Introduction 1.1 Discussion: SL(2, C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Special Relativity And The Lorentz Group 2.1 The Basic Postulates Of Special Relativity . . . . . . . . . . . . . . 2.2 The Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Generalised Lorentz Group: O(1, 3) . . . . . . . . . . . . . . . 2.4 The Mutually Disjoint Subsets Of The Generalised Lorentz Group:
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4 Conclusion 4.1 Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Special Examples To The Disjoint Subsets Of O(1, 3) A.0.1 The Special Examples . . . . . . . . . . . . . . A.0.2 Identity: An Example Of L↑+ . . . . . . . . . . A.0.3 P: An Example Of L↑− . . . . . . . . . . . . . . A.0.4 T: An Example Of L↓− . . . . . . . . . . . . . . A.0.5 PT: An Example Of L↓+ . . . . . . . . . . . . . A.1 Discussion: The Special Examples . . . . . . . . . . .
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3 The Generators Of The Restricted Lorentz 3.1 Time-Preserving Transformations . . . . . 3.1.1 The Three Simple Spatial Rotations 3.2 Lorentz Boosts And Rapidity . . . . . . . . 3.2.1 Boost Along The x1 -axis . . . . . . .
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SO+ (1, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
Introduction
In this paper, we present a study of the Lorentz groups which arise in Einstein’s theory of special relativity, and seek to explore the role of SO(4) played by these Lorentz groups in relativistic space-time. Consider the double cover SL(2, C) → Λ, as a basis of our study, where SL(2, C) is regarded as a 6-dimensional real Lie group, and Λ is the component of the Lorentz group containing the identity element Λ = {A ∈ M4 (R)|AT SA = S, det(A) = 1, a11 > 0}, where S = diag(−1, 1, 1, 1) is the diagonal matrix.
1.1
Discussion: SL(2, C)
We begin by introducing the special linear group SL(2, C) of all the complex, 2 × 2 matrices with A such that det A = 1. We consider the matrices A such that AT = A, which, as [5] suggests, are known as the Hermitian matrices. There is a natural interaction between the relativistic space-time and the space of 2 × 2 complex Hermitian matrices, given by ( ) x0 − x3 x1 + ix2 A= , x1 − ix2 x0 + x3 for xi ∈ R4 for i = 0, 1, 2, 3. The group SL(2, C) is the group of linear transformations as ( ) ( )( ′ ) ( ) x a b x ax′ + by ′ = = , y c d y′ cx′ + dy ′ where the parameters a, b, c, and d are complex numbers. Consider ( ) a b det = ad − bc = 1, c d as SL(2, C) is the group of complex unimodular matrices. We also know that there are only six independent real parameters, as d = 1+cb a , which define a transformation within the group. For more information, I recommend [1, 5, 6, 11, 12].
2
Special Relativity And The Lorentz Group
In this section, we will begin by stating the basic postulates of special relativity before introducing the properties and the underlying concepts of the Lorentz group.
2
2.1
The Basic Postulates Of Special Relativity
The basic postulates of Einstein’s theory of special relativity are 1. Invariance of uniform motion: If a particle has constant velocity in a given inertial frame, then its velocity is constant in every inertial frame. 2. Invariance of the speed of light: The speed of light is invariant across all inertial frames. 3. Principle of special relativity: The laws of physics are the same in all inertial frames. For further reference, see [6, 7, 14].
2.2
The Lorentz Group
The space-time of special relativity is the defined by R4 , where the coordinates x0 , x1 , x2 , and x3 determine an event. The time t is related to the coordinate x0 by x0 = ct, where c is the speed of light and x1 , x2 , x3 are the spatial coordinates which define the position vector x. The space R4 , as [6] suggests, can be expressed as R1,3 = R1 × R3 . The vectors X = (x0 , x1 , x2 , x3 ) are called the four-vectors. Definition 2.1 A Lorentz transformation is a linear map Λ : R4 → R4 that preserves the Minkowski metric tensor, that is to say Λ′ GΛ = G, where Λ′ is the transpose of the matrix Λ and −1 0 0 0 −1 0 0 0 0 1 0 0 0 . G = (gij ) = 0 0 1 0 = 0 δij 0 0 0 1 0 The set of all Lorentz transformations is a group under composition known as the Lorentz group. Remark 2.2 Taking determinants on both sides of Λ′ GΛ = G, we obtain det Λ = ±1, as the determinant is multiplicative and det Λ = det Λ′ , where Λ′ is the transpose of the matrix Λ. Note. We have that G−1 = G. Definition 2.3 Every linear transformation which leaves the quadratic form φ(X) = −(x0 )2 + (x1 )2 + (x2 )2 + (x3 )2 .
(1)
invariant is called a general Lorentz transformation. Definition 2.4 A Lorentz transformation is a general Lorentz transformation satisfying a00 ≥ 1. The proper Lorentz transformations are linear transformations which define the displacement, due to a uniform motion, of the time from the spatial coordinates in relativistic space-time. The proper Lorentz transformations leave invariant the quadratic form, see Equation (1).
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Definition 2.5 A proper Lorentz transformation is a Lorentz transformation satisfying det Λ = 1. The set of all proper Lorentz transformations forms a group, called the proper Lorentz group, which is a subgroup of the generalised Lorentz group. Remark 2.6 The inverse of a Lorentz transformation and the product of two Lorentz transformations are again Lorentz transformations. In fact, any proper Lorentz transformation can be expressed as a product of a boost and a rotation. Consider the transformation X → X′ = ΛX. Then we have that φ(ΛX) = φ(X). The set of matrices Λ that satisfy the relation Λ′ GΛ = G forms the generalised Lorentz group, which is a real Lie group and is denoted O(1, 3). Remark 2.7 The group O(1, 3) is the orthogonal group, defined as O(1, 3) = {Λ|Λ′ GΛ = G}.
2.3
The Generalised Lorentz Group: O(1, 3)
The group O(1, 3) is known as the generalised Lorentz group, and the elements of this group are the Lorentz transformations. It has four connected components. The two components that preserve the time coordinates, form the subgroup O+ (1, 3) known as the orthochronous Lorentz group. The quotient group O(1, 3)/O+ (1, 3) contains two elements, namely the matrices 1 0 0 0 −1 0 0 0 0 1 0 0 0 , −I = 0 −1 0 . I= 0 0 1 0 0 0 −1 0 0 0 0 1 0 0 0 −1 The two components that preserve the orientation of space form the subgroup SO(1, 3) which are the elements with determinant 1. The identity component of the Lorentz group, is the set of all Lorentz transformations preserving both the orientation and the direction of time, which form the subgroup SO+ (1, 3) known as the proper orthochronous Lorentz group, or the restricted Lorentz group. This subgroup is the group of proper Lorentz transformations and is a normal subgroup of the Lorentz group. There are six types of transformations that generate SO+ (1, 3). Three of the generators are simple spatial rotations, and the other three are space-time operators known as boosts. Remark 2.8 Lorentz transformations which preserve the direction of time are called orthochronous. Those which preserve orientation are called proper.
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2.4
The Mutually Disjoint Subsets Of The Generalised Lorentz Group: O(1, 3)
The orthochronous Lorentz transformations are those for which a00 ≥ 1 and the anti-orthochronous Lorentz transformations are those for which a00 ≤ −1. There are four types of homogeneous Lorentz transformations, which are denoted by L↑+ , L↑− , L↓+ and L↓− . The indices + and − correspond to the characteristic det Λ = ±1. The arrow ↑ corresponds to the characteristic a00 ≥ 1, whilst the arrow ↓ corresponds to the characteristic a00 ≤ −1. Combining the characteristics: det Λ = ±1 and a00 ≤ −1 or a00 ≥ 1, we have Sheet Type The orthochronous rotations The orthochronous reflections The anti-orthochronous rotations The anti-orthochronous reflections
Set L↑+ L↑− L↓+ L↓−
Characteristics det Λ = 1, a00 ≥ 1 det Λ = −1, a00 ≥ 1 det Λ = 1, a00 ≤ −1 det Λ = −1, a00 ≤ −1
Table 1: The disjoint subsets of the the generalised Lorentz group: O(1, 3). See Appendix A for examples for each of the subsets of O(1, 3). Amongst the disjoint subsets, only L↑+ forms a subgroup SO+ (1, 3), of orthochronous rotations, known as the proper Lorentz group. The four sets are mutually disjoint. As [8] suggests, we have L+ = L↑+ ∪ L↓+ and
L↑ = L↑+ ∪ L↑− .
It turns out that the Lorentz group L↑+ , is doubly covered by SL(2, C). Remark 2.9 The quotient group O(1, 3)/O+ (1, 3) = O(1, 3)/L↑ = {I, −I}. Remark 2.10 The anti-orthochronous transformations form a set denoted L↓ , which does not form a subgroup since it does not contain the unit matrix.
3
The Generators Of The Restricted Lorentz Group: SO+ (1, 3)
There are two special types of Lorentz transformations, as [6, 7, 14] suggest, the time-preserving transformations and the Lorentz boosts (rotation-free Lorentz transformations).
3.1
Time-Preserving Transformations
These are Lorentz transformations such that (Λx)0 = x0 for all x.
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Thus, the matrix Λ takes the form
1 0 0 0 0 a11 a12 a13 Λ= 0 a21 a22 a23 0 a31 a32 a33 which can be also be expressed as
1 0 0 0 0 , Λ= 0 [aij ] 0 where A = [aij ] ∈ O(1,3), that is it must preserve (x1 )2 + (x2 )2 + (x3 )2 , which represents the Euclidean distance from the origin in three-dimensional space. The set of all such transformations forms a subgroup of the proper Lorentz group which is isomorphic to the three-dimensional rotation group. 3.1.1
The Three Simple Spatial Rotations
The spatial rotation in the x1 -axis is given by 1 0 0 0 0 1 0 0 Λ= 0 0 cos(θ) − sin(θ) 0 0 sin(θ) cos(θ) The spatial rotation in the x2 -axis is given by 1 0 0 cos(θ) Λ= 0 0 0 sin(θ)
.
0 0 0 − sin(θ) . 1 0 0 cos(θ)
The spatial rotation in the x3 -axis is given by 1 0 0 0 cos(θ) − sin(θ) Λ= 0 sin(θ) cos(θ) 0 0 0
3.2
0 0 . 0 1
Lorentz Boosts And Rapidity
A Lorentz boost along the axis of a given spatial coordinate is a Lorentz transformation that leaves invariant the other two spatial coordinates.
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3.2.1
Boost Along The x1 -axis
The Lorentz boost along the x1 -axis has the form a00 a01 a10 a11 Λ= 0 0 0 0
0 0 1 0
0 0 . 0 1
From Λ′ GΛ = G we obtain a10 2 − a00 2 a10 a11 − a00 a01 a10 a11 − a00 a01 a11 2 − a01 2 0 0 0 0
0 0 1 0
0 −1 0 0 = 0 0 1 0
0 1 0 0
0 0 1 0
0 0 0 1
which yields the relations a00 2 − a10 2 = 1; a11 2 − a01 2 = 1; a10 a11 = a00 a01 . From these relations, we can deduce that a00 = a11 = cosh(θ) and a01 = a10 = sinh(θ), and have that
cosh(θ) sinh(θ) 0 0 sinh(θ) cosh(θ) 0 0 . Λ= 0 0 1 0 0 0 0 1
Thus, as [14] suggests, a Lorentz boost is a hyperbolic rotation. Definition 3.1 The quantity θ is called the rapidity or pseudo-velocity of the transformation. Rapidity, as [14] suggests, determines an isomorphism with the additive group of real numbers. There, also exists a unique number v such that tanh(θ) = − vc . Using v, we write cosh(θ) = γ and
v sinh(θ) = −γ , c where v is the relative velocity between frames along the x1 -axis, c is the speed of light and γ=√
1 1−
is the Lorentz factor. Thus, we have that
v2 c2
,
γ −βγ 0 0 −βγ γ 0 0 , Λ= 0 0 1 0 0 0 0 1 7
where β = vc . Similarly, for boost along the x2 -axis, we have γ 0 Λ= −βγ 0 and for the boost along the x3 -axis
γ 0 Λ= 0 −βγ
0 −βγ 0 1 0 0 , 0 γ 0 0 0 1
0 1 0 0
0 −βγ 0 0 , 1 0 0 γ
which are obtained from the cyclic permutations of x1 , x2 and x3 .
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Conclusion
The aim of this study was to explore the role of SO(4), played by the Lorentz groups in relativistic space-time. We began with a brief discussion on SL(2, C) in Section 1, before shifting our attention to the postulates of special relativity in Section 2. We then considered the generalised Lorentz group O(1, 3), and its mutually disjoint subsets. We saw that the identity component, which preserves both time and space coordinates, is the subgroup SO+ (1, 3) known as the proper Lorentz group and that there are six types of transformations that generate SO+ (1, 3): the three simple spatial rotations, and the three boosts. In Section 3, we looked at the three simple spatial rotations, and the three boosts.
Further Study
4.1
We can extend this project to cover Poincaré group, see [14], as an extension to the full Lorentz group which is generated by the proper orthochronous Lorentz transformations, space reflections, time reversals and translations. Furthermore, we can include an in-depth study of the Dirac equation, and the Dirac spinors in the setting of the relativistic quantum field theory, refer to [2, 3, 4, 6, 11]. But, perhaps the most fascinating direction, as highlighted by [9, 15], would be to explore the areas of spin geometry. Gauge fields and transformations along with Clifford algebras, as [1, 8, 10, 13, 14, 16] suggest, are also interesting areas which can be explored in detail.
A
Special Examples To The Disjoint Subsets Of O(1, 3)
In this section, we consider special examples corresponding to the mutually disjoint subsets of the generalised Lorentz group: O(1, 3). A.0.1
The Special Examples
It should be clear that whilst we can pass from one transformation to another transformation of the same subset, we can not pass from a transformation of one subset to a transformation in another. 8
A.0.2
Identity: An Example Of L↑+
The identity element defined as
1 0 I= 0 0
0 1 0 0
0 0 1 0
0 0 ∈ L↑ . + 0 1
Remark A.1 We have that det I = 1 and a00 = 1. A.0.3
P: An Example Of L↑−
The reflection of the space axes is known as parity, denoted P is defined as 1 0 0 0 0 −1 0 0 ↑ P = 0 0 −1 0 ∈ L− . 0 0 0 −1 Remark A.2 We have that det P = −1 and a00 = 1. A.0.4
T: An Example Of L↓−
The reversal of the time direction is known −1 0 T = 0 0
as time reversal, denoted T is defined as 0 0 0 1 0 0 ∈ L↓ . − 0 1 0 0 0 1
Remark A.3 We have that det T = −1 and a00 = −1. A.0.5
PT: An Example Of L↓+
The product of space reflection and time reversal 1 0 0 0 −1 0 0 0 −1 0 0 0 1 0 PT = 0 0 −1 0 0 0 1 0 0 0 −1 0 0 0
denoted P T is 0 −1 0 0 = 0 0 1 0
defined as
0 0 0 −1 0 0 ∈ L↓ . + 0 −1 0 0 0 −1
Remark A.4 We have that det P T = 1 and a00 = −1.
A.1
Discussion: The Special Examples
The quotient group O(1, 3)/SO+ (1, 3), as [7] hints, is isomorphic to the Klein four-group. Every element in O(1, 3) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group {I, P, T, P T }. The four elements of this isomorphic copy of the Klein four-group label the four connected components of the Lorentz group. 9
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