Coordinate singularities in harmonically sliced cosmologies

Abstract

Harmonic slicing has in recent years become a standard way of prescribing the lapse function in numerical simulations of general relativity. However, as was first noticed by Alcubierre [Phys. Rev. D 55, 5981 (1997)], numerical solutions generated using this slicing condition can show pathological behaviour. In this paper, analytic and numerical methods are used to examine harmonic slicings of Kasner and Gowdy cosmological spacetimes. It is shown that in general the slicings are prevented from covering the whole of the spacetimes by the appearance of coordinate singularities. As well as limiting the maximum running times of numerical simulations, the coordinate singularities can lead to features being produced in numerically evolved solutions which must be distinguished from genuine physical effects. 04.25.Dm, 98.80.Hw

Binary Black Hole Coalescence

“…In such a decomposition the wave-like character of the field equations in not manifest, and the primary reason quoted for using harmonic gauge (in particular harmonic time slicing) was for its geometric "singularity avoiding" properties. However even within ADM evolutions harmonic coordinates were seldom used due to the notion that they would generically lead to the formation of "coordinate shocks" [235,236,237,238,239]. An in-principle solution to this problem noted by Garfinke [226] (and see an earlier discussion of this by Hern [237]) was to use generalized harmonic coordinates (GHC), first introduced by Friedrich [240].…”

Computational Cosmology: From the Early Universe to the Large Scale Structure

“…This choice is computationally cheap, simple to implement, and certain choices of F 2 (i.e., 1 + ln γ ) can mimic maximal slicing in its singularity avoidance properties [8]. On the other hand, numerical solutions derived from harmonically-sliced foliations can exhibit pathological gauge behavior in the form of coordinate “shocks” or singularities which will affect the accuracy, convergence and stability of solutions [5, 86]. Also, evolutions in which the lapse function is fixed by some analytically prescribed method (either geodesic or near-geodesic) can be unstable, especially for sub-horizon scale perturbations [7].…”

“…where b is the bias factor chosen to match present observations of rms density fluctuations in a spherical window of radius R h = 8 h −1 Mpc. Also, P (k) is the Fourier transform of the square of the density fluctuations in equation (72), and W (k) = 3 (kR h ) 3 (sin(kR h ) − (kR h ) cos(kR h )) (74) is the Fourier transform of a spherical window of radius R h . Overdensity peaks can be filtered on specified spatial or mass scales by Gaussian smoothing the random density field [25] Bertschinger [40] has provided a useful and publicly available package of programs called COSMICS for computing transfer functions, CMB anisotropies, and gaussian random initial conditions for numerical structure formation calculations.…”

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

“…In particular, there has been a large amount of work on casting the evolution equations into first-order symmetric [2, 182, 195, 3, 21, 155, 248, 443, 22, 74, 234, 254, 383, 377, 18, 285, 285, 86] and strongly hyperbolic [62, 63, 12, 59, 60, 13, 64, 367, 222, 78, 58, 82] form; see [182, 352, 188, 353] for reviews. For systems involving wave equations for the extrinsic curvature; see [128, 2]; see also [424] and [20, 75, 374, 379, 436] for applications to perturbation theory and the linear stability of solitons and hairy black holes.…”