Numerical Relativity with the Conformal Field Equations

Abstract

Abstract. I discuss the conformal approach to the numerical simulation of radiating isolated systems in general relativity. The method is based on conformal compactification and a reformulation of the Einstein equations in terms of rescaled variables, the so-called "conformal field equations" developed by Friedrich. These equations allow to include "infinity" on a finite grid, solving regular equations, whose solutions give rise to solutions of the Einstein equations of (vacuum) general relativity. The conformal approach promises certain advantages, in particular with respect to the treatment of radiation extraction and boundary conditions. I will discuss the essential features of the analytical approach to the problem, previous work on the problem -in particular a code for simulations in 3+1 dimensions, some new results, open problems and strategies for future work.

Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations

“…In a realistic problem, such as the binary black hole problem, a global scheme is necessary to provide physically correct outer boundary data, either by using hyperboloidal time slices to extend the Cauchy evolution to infinity (see [22,23] for reviews) or by matching to an exterior characteristic or perturbative solution (see [24] for a review). Harmonic evolution offers important advantages for Cauchy-characteristic matching (CCM) which have led to successful matching in the linearized regime [10].…”

Gravitational wave extraction based on Cauchy–characteristic extraction and characteristic evolution

“…We carry out a series of simulations, varying both the location of the characteristic world-tube and the radius of the Zerilli sphere. We use 80 3 , 120 3 We study the dependence of the signal with the amplitude and conclude that CCE code resolves correctly amplitudes of A 10 −8 . At smaller amplitudes, clean convergence behaviour is contaminated by round-off error.…”

“…The spacetime is evolved using the BSSN code and excision methods described in [69]. The coarsest Cauchy grid has 29 3 points with x i = 0.4M, whilst the characteristic grid has 35 2 × 31 points. The world-tube is at r = 7M.…”

“…One way is to evolve the entire spacetime. For example, a hyperboloidal slicing of the spacetime [1][2][3] allows information to propagate to I + in finite time. Conformal compactifications, such as those suggested by the conformal field equations [4,5] or employed in [6], allow I + to be located at a finite computational coordinate in a regular way.…”

The Art and Science of Black Hole Mergers

“…One approach, devised originally by H Friedrich, [12] is being investigated intensively by a group of scientists at the AEI and their collaborators at other institutions. [13] In this idea, the field equations are expressed in a conformally compactified way, which means that the spacetime is mapped onto a computational domain in which the outgoing light rays reach "infinity" in a finite coordinate distance. There is a natural boundary condition there, and the result is a simulation of the whole spacetime with automatically varying refinement.…”