THE LORENTZ TRANSFORMATION GROUP GEORGE GUNNER Contents 1. Introduction 2 2. Theory of Groups 2 2.1. Isomorphisms and Homomorphisms 2 2.2. Definition of a Group and Basic Results 4 2.3. Subgroups 4 2.4. Endomorphisms and Automorphisms 5 3. The Lorentz Group 6 3.1. Minkowski Space 6 3.2. Relativistic Velocity Addition 6 3.3. Dyads 7 3.4. Thomas Gyration 8 3.5. General Representation of the Lorentz Group Using Dyads 9 3.6. Spinor Representation 11 3.7. The Abberation Transformation Subgroup 12 4. Conclusion 13 References 14 1 2 GEORGE GUNNER 1. Introduction The study of special relativity gives rise to the Lorentz transformation, which preserves the inner product in Minkowski space. It is very important within the theory of special relativity where we consider the Minkowski space M(<; <3), but there are more general Minkowski spaces which are of interest. In particular we will follow the work in Prof Ungar's paper entitled "The Abstract Lorentz Transforma- tion Group"[5] and examine M(<;V1) space, where V1 is an arbitrary positive definite inner product space. Using basic results from theory of groups and objects called dyads, we are able to represent any Lorentz transformation by a pure Lorentz transform preceeded by a Thomas gyration, with pure Lorentz transformations can be defined by dyadics. This formulation allows consideration of the Lorentz trans- formation in very general Minkowski spaces. We will then focus on <3, where the Lorentz transformation are typically represented as a 4 × 4 matrix. There is a simpler representation where the Lorentz transformation and Minkowski spacetime coordinates of M(<; <3) are encoded into 2×2 matrices called spinors. We will con- clude this paper by considering a certain subgroup of the Lorentz transformation acting in M(<; <3) space, the abberation subgroup. There exists a representation of the abberation transformation as a linear fractional transformation which has a simple physical interpretation in terms of the well-known "headlight" effect. 2. Theory of Groups 2.1. Isomorphisms and Homomorphisms. We call two sets S and T isomorphic with respect to an algebraic operation (which for our purposes we will call a product and denote by ab = c) if there is a one-to-one correspondence between their elements and the following holds: Given elements a,b 2 S corresponding to x,y 2 T , if (2.1) ab = c; xy = z then c 2 S corresponds to z 2 T . An isomorphism between S and T is denoted by S =∼ T . Two general properties of isomorphisms are (1) Symmetry, that is given S =∼ T , it follows that T =∼ S; (2) Transistivity, that is given S =∼ T and T =∼ R, it follows that S =∼ R. Isomorphic sets generally differ from one another in the nature of their elements and in the operation defined on the two different sets. However, the properties of the operations are indistinquishable. That is, whatever can be proven about an operation in a given set S without reference to the properties of the elements of S is necessarily true for an operation in T which is isomorphic with S. If we omit the requirement that two isomorphic sets S and T have a one-to- one correspondence between their elements, we can obtain a generalization of the concept called a homomorphism. More precisely, a mapping φ : S ! T is called a homomorphism if for any a,b 2 S it follows from (2.2) φa = c φb = d that (2.3) φ(ab) = cd where c,d 2 T , and we then call the set T a homomorphic image of S. If T is a homomorphic image of S, the validity of the associative and commutative laws in S entail the validity of the corresponding laws in T . Additionally, if S has THE LORENTZ TRANSFORMATION GROUP 3 a unit element, eS, then its image must be the unit element in T . The converse statements do not follow.[2] It is possible to obtain all possible homomorphic images of a given set S with one algebraic operation. To accomplish this, we suppose there is a given partition P of S into disjoint subsets. We call call these subsets and denote them by ABC:::. The partition P is called regular if given a1 a2 2 A and b1 b2 2 B, it follows that a1b1 and a2b2 both lie in the same class C. Thus the class C is completely determined by the classes A, B. Moreover, we can call C the product of A and B and we have an algebraic operation defined on the set F of all classes of the regular partition P . We call the set F found in this manner the factor set of S with respect to P . Notice that S can be mapped homomorphically to F , and is called the natural homomorphism. We now come to the following theorem: Theorem 1. If T is an arbitrary homomorphic image of S, and φ a homomorphic mapping of S onto T . then there exists a regular partition P of S into disjoint classes such that T is isomorphic to the factor set F with respect to P . Addition- ally, there exists an isomorphic mapping of T onto F such that performing the mappings and φ in succession is the natural homomorphism of S onto F . Proof. Consider all the elements whose images under the mapping φ coincide and consider them as a single class. In this way, we obtain a regular partition P of S into disjoint classes. That is, if a1 and a2 lie in class A and b1 and b2 lie in class B, we have (2.4) φa1 = φa2 = α; and (2.5) φa1 = φa2 = β: where α β 2 T . By the holomorphism of φ, we have (2.6) φ(a1b1) = φ(a2b2) = αβ which means that a1b1 and a2b2 lie in the same class C. Thus we obtain a factor set F , and we now have a one-to-one correspondence φ between all of the elements of T and all the elements of F . It remains to to associate each element of T with the clas sconsisting of all the originals of this element. Notice that φ is an isomorphism: If the elements α and β are linked with the classes A and B, respecitively, and if the elements a, b are chosen from these classes, then we find that AB is a class which contains ab. But (2.7) φ(ab) = (φa)(φb) = αβ which means that αβ is associated to the class AB under the mapping φ. Now choose some a 2 S and let (2.8) φa = α α = A: Notice that the element a is contained in A. This implies that performing the mappings and φ in succession is the natural homomorphism of S onto F , which is what we wanted to show. 4 GEORGE GUNNER 2.2. Definition of a Group and Basic Results. We call a non-empty set G with one algebraic operation a group if: (1) The operation is associative; (2) The inverse operation can be performed in G. Note that the operation defined on G does not need to be commutative. As we will see in section 3, the Lorentz Group does not have a commutative product. We now quote without proof a few basic results from reference [2]. Lemma 1. Given a group G, there exists a unique element e 2 G called the unit element with the property that ae = ea = a for any a 2 G. Lemma 2. For any a in a group G, there exists a unique element a−1 called the inverse of a such that aa−1 = a−1a = e. Notice that the unit element is its own inverse, and that the inverse of a−1 is a. Lemma 3. If a group G is mapped homomorphically onto a set H with one oper- ation, then H is also a group. Theorem 1 can now be expressed as the following homomorphism theorem for groups: Theorem 2. (The Homomorphism Theorem) If φ is a homomorphic mapping of a group G onto a group H, then there exists a regular partition P of G such that H can be mapped isomoprhically onto the factor group F of G with respect to this partition. Additionally, the isomorphism of H onto F can be chosen such that performing the mappings φ and in succession is the natural homomorphism of G onto F . Notice that if a homomorphism φ of a group G onto a group H carries the element a 2 G into the element α 2 H, that is, (2.9) φa = α; then the image of the inverse of a is the inverse of α.[2] For an element a in a group G, we can define the powers of a, an as a multiplied by itself n times for n positive, a0 = e where e is the unit element of G, and a−n = (an)−1 or a−n = (a−1)n. It is easily verified that the two definitions for negative powers of a are equivalent, that is (an)−1 = (a−1)n.[2] 2.3. Subgroups. We call a non-empty subset H of a group G a subgroup if H itself is a group under the operation defined in G. We can determine that a subset H of G is a subgroup by verifying that[2] (1) Given a,b 2 H, ab 2 H; (2) Given a 2 H, a −12 H. The associative law must hold for the product in H as it holds in G, and similarly it follows from the two requirements that the unit element of G is in H.