Solving Einstein's Equations With Dual Coordinate Frames

TL;DR

This paper presents a dual-coordinate framework for numerically solving Einstein's equations, enabling stable simulations of black-hole spacetimes by combining two coordinate systems and using a feedback control system.

Contribution

It introduces a novel dual-coordinate method that improves stability and flexibility in simulating black-hole mergers compared to traditional single-coordinate approaches.

Findings

Achieved stable evolutions of black-hole spacetimes with dual coordinates.

Successfully evolved binary black-hole systems for multiple orbits.

Implemented a feedback control system to maintain excision boundaries within horizons.

Abstract

A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the way fields are projected onto an orthonormal tetrad basis. These field components are then determined as functions of a second independent coordinate system. The transformation to the second coordinate system can be thought of as a mapping from the original ``inertial'' coordinate system to the computational domain. This dual-coordinate method is used to perform stable numerical evolutions of a black-hole spacetime using the generalized harmonic form of Einstein's equations in coordinates that rotate with respect to the inertial frame at infinity; such evolutions are found to be generically unstable using a single rotating coordinate frame. The dual-coordinate method is also used here to evolve binary black-hole spacetimes for several orbits. The great flexibility of this method allows comoving coordinates to be adjusted with a feedback control system that keeps the excision boundaries of the holes within their respective apparent horizons.