Optimal alignment of Lorentz orientation and generalization to matrix Lie groups

2025/06/17 by Sha, Congzhou M
#Computational Physics (physics.comp-ph) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2506.14994

Abstract

There exist elegant methods of aligning point clouds in \mathbb R³. 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 Λ such that Λv_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.

Citations

• Quaternions and rotations: applications to Minkowski's four-vectors, electromagnetic waves and polarization optics
• Simple exponential acceleration of the power iteration algorithm
• Array programming with NumPy
• SciPy 1.0: fundamental algorithms for scientific computing in Python

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