Shells around black holes: The effect of freely specifiable quantities in Einstein’s constraint equations

Abstract

We solve Einstein's constraint equations in the conformal thin-sandwich decomposition to model thin shells of non-interacting particles in circular orbit about a non-rotating black hole. We use these simple models to explore the effects of some of the freely specifiable quantities in this decomposition on the physical content of the solutions. Specifically, we adopt either maximal slicing or Kerr-Schild slicing, and make different choices for the value of the lapse on the black hole horizon. For one particular choice of these quantities the resulting equations can be solved analytically; for all others we construct numerical solutions. We find that these different choices have no effect on our solutions when they are expressed in terms of gauge-invariant quantities.

Equilibrium initial data for moving puncture simulations: the stationary 1 + log slicing

“…We include (35) and (36) in figure 2, and find excellent agreement with our numerical results. Before closing this section, we point out that there was no need to assume j = 0 in (28).…”

“…To construct initial data, we solve equations (12), (15) and (16) under the assumption of spherical symmetry withū ij = 0 = ∂ t K, adopting the algorithm and numerical code described in [34]. The boundary conditions at spatial infinity are given by (18a) with Ω j = 0 for nonrotating spacetimes.…”

“…Figure 2. The asymptotic behavior of the shift β and the mean curvature K in spherical symmetry.The solid lines denote the numerical results of section 3; the dashed lines represent the asymptotic scalings(35) and(36).…”