Optimal alignment of Lorentz orientation and generalization to 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.