Precession-tracking coordinates for simulations of compact-object-binaries

TL;DR

This paper introduces two coordinate transformation methods, based on Euler angles and quaternions, for simulating precessing binary black hole systems, highlighting the quaternion method's robustness for strong precession.

Contribution

It presents and compares two coordinate transformations for precessing binary systems, demonstrating the quaternion approach's effectiveness over Euler angles in strong precession scenarios.

Findings

Euler-angle method fails at 90-degree precession due to gimbal lock.

Quaternion method handles any precession angle without singularities.

Both methods work well for moderate precession.

Abstract

Binary black hole simulations with black hole excision using spectral methods require a coordinate transformation into a co-rotating coordinate system where the black holes are essentially at rest. This paper presents and discusses two coordinate transformations that are applicable to precessing binary systems, one based on Euler angles, the other on quaternions. Both approaches are found to work well for binaries with moderate precession, i.e. for cases where the orientation of the orbital plane changes by much less than 90 degrees. For strong precession, performance of the Euler-angle parameterization deteriorates, eventually failing for a 90 degree change in orientation because of singularities in the parameterization ("gimbal lock"). In contrast, the quaternion representation is invariant under an overall rotation, and handles any orientation of the orbital plane as well as the Euler-angle technique handles non-precessing binaries.