RCCZ4: A Reference Metric Approach to Z4

TL;DR

This paper introduces a modified covariant Z4 formulation with a reference metric to improve the handling of constraint violations in numerical relativity, demonstrating robustness in black hole and scalar field collapse simulations.

Contribution

It proposes a new FCCZ4-based approach coupling constraint violations to a reference metric, offering an alternative to existing hyperbolic formulations in numerical relativity.

Findings

Works with black holes and scalar field collapse

Achieves strong hyperbolicity via pseudodifferential reduction

Comparable performance to FCCZ4 with potential for further control

Abstract

The hyperbolic formulations of numerical relativity due to Baumgarte, Shapiro, Shibata & Nakamura (BSSN) and Nagy Ortiz & Reula (NOR), among others, achieve stability through the effective embedding of general relativity within the larger Z4 system. In doing so, various elliptic constraints are promoted to dynamical degrees of freedom, permitting the advection of constraint violating modes. Here we demonstrate that it is possible to achieve equivalent performance through a modification of fully covariant and conformal Z4 (FCCZ4) wherein constraint violations are coupled to a reference metric completely independently of the physical metric. We show that this approach works in the presence of black holes and holds up robustly in a variety of spherically symmetric simulations including the critical collapse of a scalar field. We then demonstrate that our formulation is strongly hyperbolic through the use of a pseudodifferential first order reduction and compare its hyperbolicity properties to those of FCCZ4 and generalized BSSN (GBSSN). Our present approach makes use of a static Lorentzian reference metric and does not appear to provide significant advantages over FCCZ4. However, we speculate that dynamical specification of the reference metric may provide a means of exerting greater control over constraint violations than what is provided by current BSSN-type formulations.