Optimal alignment of Lorentz orientation and generalization to matrix Lie groups

TL;DR

This paper introduces two methods for optimally aligning 4-vectors via Lorentz transformations, addressing the failure of Euclidean methods in Minkowski space and extending to general matrix Lie groups.

Contribution

It presents a novel least squares approach and a Lie algebra-based method for Lorentz alignment, generalizable to other matrix Lie groups.

Findings

The first method uses direct optimization in Lorentz group parametrization.

The second method leverages Lorentz algebra for simplicity and computational efficiency.

The second method can be extended to other matrix Lie groups.

Abstract

There exist elegant methods of aligning point clouds in $\mathbb R^3$. Unfortunately, these methods fail to generalize to the case of Minkowski space, as we will show. Instead, we propose two solutions to the following problem: given inertial reference frames $A$ and $B$, and given (possibly noisy) measurements of a set of 4-vectors $\{v_i\}$ made in those reference frames with components $\{v_{A,i}\}$ and $\{v_{B,i}\}$, find the optimal Lorentz transformation $\Lambda$ such that $\Lambda v_{A,i}=v_{B,i}$. The first method is direct least squares optimization through a parametrization of $SO(3,1)_+$ in terms of the familiar boost and rotation vectors. The second method takes a detour through the Lorentz algebra; in addition to being conceptually simple and possessing a computational advantage over the first method, it can easily be generalized to the alignment of vector representations in other matrix Lie groups.