Cauchy-characteristic extraction in numerical relativity

Abstract

We treat the calculation of gravitational radiation using the mixed timelike-null initial value formulation of general relativity. The determination of an exterior radiative solution is based on boundary values on a timelike world tube ⌫ and on characteristic data on an outgoing null cone emanating from an initial cross section of ⌫. We present the details of a three-dimensional computational algorithm which evolves this initial data on a numerical grid, which is compactified to include future null infinity as finite grid points. A code implementing this algorithm is calibrated in the quasispherical regime. We consider the application of this procedure to the extraction of waveforms at infinity from an interior Cauchy evolution, which provides the boundary data on ⌫. This is a first step towards Cauchy-characteristic matching in which the data flow at the boundary ⌫ is two-way, with the Cauchy and characteristic computations providing exact boundary values for each other. We describe strategies for implementing matching and show that for small target error it is much more computationally efficient than alternative methods.

Characteristic extraction in numerical relativity: binary black hole merger waveforms at null infinity

“…The formalism for the numerical evolution of Einstein's equations, in null cone coordinates, is well known [30,33,[37][38][39][40]. For the sake of completeness, we give a summary of those aspects of the formalism that will be used here.…”

“…If the data is consistently passed back and forth between the evolution schemes at Γ during the evolution, the method is known as Cauchy-characteristic matching (CCM). Given astrophysical initial data, such a method has only discretisation error [33], and a complete mathematical specification has been developed [34]. However, efforts to implement CCM encountered stability problems at the interface, and in the general case have not been successful.…”

Black hole evolution by spectral methods

“…There are two aspects to this problem: First, one must formulate a procedure for handling the boundary, such as imposing an analytic condition (e.g., a Sommerfeld condition) on the fundamental variables, matching to a wave perturbation described by the Zerilli equation [28], or matching to a characteristic evolution code that propagates the solution out to null infinity [29][30][31]. Second, one must construct a FD approximation of either the analytic boundary condition or the matching condition.…”

General relativistic null-cone evolutions with a high-order scheme

“…A prominent result was the first stable dynamical evolution of a black hole spacetime in three spatial dimensions achieved by the Pittsburgh group [11,12]. Since then, the Pittsburgh null code (or PITTNullCode) has become the main building block for current implementations of characteristic extraction used in numerical relativity simulations [1,2,3,4,5], and is now part of the publicly available Einstein Toolkit [13].…”

“…Over the years, there have been several improvements to the original algorithm as implemented in [11,12]. Stereographic coordinates were replaced by more uniform angular grids [15,16].…”