Lorentz transformations in time and two space dimensions

TL;DR

This paper develops matrix and vector formalisms for Lorentz transformations in a spacetime with one time and two space dimensions, exploring their structure, decompositions, generators, and physical interpretations.

Contribution

It introduces a comprehensive formalism for Lorentz transformations in (1+2) dimensions, including matrix decompositions, parameterizations, and generator relations, extending prior (1+1) dimensional analyses.

Findings

Lorentz transformations form the SO(1,2) group with specific matrix decompositions.

Each Lorentz matrix is characterized by boost and rotation parameters.

Explicit formulas relate composite transformation parameters to individual ones.

Abstract

In this article, matrix and vector formalisms for Lorentz transformations in time ($t$) and two space dimensions ($x$ and $y$) are developed and discussed. Lorentz transformations conserve the squared interval $t^2 - x^2 - y^2$. Examples of Lorentz transformations include boosts in arbitrary directions, which mix time ansd space, and rotations in space, which do not. Lorentz transformations can be described by matrices and coordinate vectors. Lorentz matrices comprise the special unitary group SO(1,2). The general form of a Lorentz matrix is derived, in terms of both components and block matrices. Each Lorentz matrix $L$ has the Schmidt decomposition $QDP^t$, where $D$ is a diagonal matrix, and $P$ and $Q$ are orthogonal matrices. It also has the Schmidt-like decomposition $R_2BR_1^t$, where $B$ is a boost matrix, and $R_1$ and $R_2$ are rotation matrices. Hence, a Lorentz matrix is specified by three parameters, namely the boost energy $\gamma$, and the rotation angles $\theta_1$ and $\theta_2$. Each Lorentz matrix has a pair of reciprocal Schmidt coefficients (which are real), and a unit coefficient, which is its own reciprocal. It also has a pair of reciprocal eigenvalues (which are real or complex), and a unit eigenvalue. The physical significances of the input and output Schmidt vectors, and the eigenvectors, are discussed. Every Lorentz matrix can be written as the exponential of a generating matrix. There are three basic generators, which produce boosts along the $x$ and $y$ axes, and a rotation about the $t$ axis (in the $xy$ plane). These generators satisfy certain commutation relations, which show that SO(1,2) is isomorphic to Sp(2) and SU(1,1). Simple formulas are derived for the energy and angle parameters of a composite transformation, in terms of the parameters of its constituent transformations.