CTTK: A new method to solve the initial data constraints in numerical relativity

TL;DR

This paper introduces a novel scheme in numerical relativity that simplifies initial data constraints by solving an algebraic equation for the mean curvature, enabling rapid solutions for complex matter configurations.

Contribution

The new method replaces the traditional elliptic equation approach with an algebraic solution for mean curvature, avoiding existence and uniqueness issues in initial data construction.

Findings

Rapid convergence of the method for complex initial conditions

Effective in inhomogeneous spacetimes with scalar field perturbations

Applicable to black hole spacetimes with scalar clouds

Abstract

In numerical relativity simulations with non-trivial matter configurations, one must solve the Hamiltonian and momentum constraints of the ADM formulation for the metric variables in the initial data. We introduce a new scheme based on the standard Conformal Transverse-Traceless (CTT) decomposition, in which instead of solving the Hamiltonian constraint as a 2nd order elliptic equation for a choice of mean curvature $K$, we solve an algebraic equation for $K$ for a choice of conformal factor. By doing so, we evade the existence and uniqueness problem of solutions of the Hamiltonian constraint without using the usual conformal rescaling of the source terms. This is particularly important when the sources are fundamental fields, as reconstructing the fields' configurations from the rescaled quantities is potentially problematic. Using an iterative multigrid solver, we show that this method provides rapid convergent solutions for several initial conditions that have not yet been studied in numerical relativity; namely (i) periodic inhomogeneous spacetimes with large random Gaussian scalar field perturbations and (ii) asymptotically flat black hole spacetimes with rotating scalar clouds.