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 rely on the positive definite property of the Euclidean metric, and do not easily extend to the indefinite Minkowski metric. In this paper, 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 method we outline is conceptually simple and easily extends to alignment problems in other matrix Lie groups.