Lorentz Transformations

In these notes we study Lorentz transformations and focus on the group of proper, orthochronous Lorentz transformations, donated by L+. (These Lorentz transformations have determinant one and preserve the direction of time.) The 2×2 matrices with determinant one, denoted SL(2, C), play a key role, as there is a map from SL(2,C) onto L+, that is 2-to-1 and a homomorphism. In this correspondence, the subgroup SU(2) ⊂ SL(2, C) of unitary matrices with determinant one, maps to pure rotations (real, orthogonal Lorentz transformations). On the other hand hermitian matrices in SL(2, C) map to Lorentz transformations that are pure boosts (symmetric Lorentz transformation matrices). The unique polar decomposition of a matrix A ∈ SL(2, C) into a unitary matrix U followed by a hermitian matrix H, namely A = HU , yields the unique decomposition Λ = BR of an arbitrary Lorentz transformation Λ ∈ L+ into a pure R rotation followed by a pure boost B. We give some explicit examples. I Lorentz Transformations I.1 Points in R and Lorentz Transformations Let us designate a point in R by the 4-vector with real coordinates x = (x0, x1, x2, x3) = (ct, ~x). In other words, we use units for which the four components of x all have dimension length, and we say that the four-vector x comprises a time component and a 3-vector spatial part. A homogeneous Lorentz transformation Λ is a 4 × 4 real matrix that preserves the Minkowski length xM of every 4-vector x. Here x′ = Λx , or xμ = 3 ∑