Solving Einstein's Equations With Dual Coordinate Frames

TL;DR

This work tackles instability and boundary-tracking challenges in numerical relativity by proposing a dual-coordinate-frame approach that evolves inertial-frame components $u^{\bar{\alpha}}$ as functions of a comoving frame via a dynamic map $x^a(x^{\bar{a}})$. It demonstrates that single rotating-frame evolutions suffer from constraint growth and angular-harmonic mixing, which can be mitigated by tensor-spherical-harmonic filtering performed in the rotating frame and by adopting horizon-tracking coordinate maps. Applied to Schwarzschild and equal-mass binary black holes, the method yields stable evolutions for multiple orbits and shows promise for guiding mergers with adaptive coordinates. Overall, the dual-coordinate framework provides a flexible, robust platform for long-term numerical relativity simulations and waveform modeling, addressing key stability and boundary-pathologies inherent to traditional single-frame approaches.

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.

Figures (13)

• Figure 1: Constraint violations in evolutions of a Schwarzschild black hole using the generalized harmonic system and coordinates that rotate with angular velocity $\Omega$.
• Figure 2: Constraint violations in dual-coordinate-frame evolutions of Schwarzschild, with comoving coordinates that rotate uniformly with angular velocity $\Omega=0.2/M$. Angular filtering (see text) is performed on the inertial frame components.
• Figure 3: Time-dependence of the inertial frame components of two cross sections (equatorial and polar) of a vector field expressed as functions of rotating coordinates. At $t=0$ the vector field is radial, but the $\partial_{ \bar{x}}$ and $\partial_{ \bar{y}}$ basis vectors rotate with angular velocity $\Omega$. The vectors on the right side of the polar cross section figure at $t=\pi/(2\Omega)$ are pointed into the plane of the figure, while those on the left point out.
• Figure 4: Tensor-spherical-harmonic components of the Schwarzschild spatial three-metric expressed in rotating coordinates. Dashed curves show the components expressed in rotating-frame tensor spherical harmonics (only the $\ell=0$ component is non-zero); solid curves show the inertial-frame tensor-spherical-harmonic components.
• Figure 5: Constraint violations in dual-coordinate-frame evolutions of Schwarzschild, with comoving coordinates that rotate uniformly with angular velocity $\Omega=0.2/M$. These evolutions use the new rotating-frame tensor-spherical-harmonic filtering algorithm. The highest resolution case (dashed curve) uses more angular basis functions ($\ell\leq 13$ instead of $\ell\leq 11$) as a more stringent test of convergence.