Maximal slicing for puncture evolutions of Schwarzschild and Reissner-Nordström black holes

Abstract

We prove by explicit construction that there exists a maximal slicing of the Schwarzschild spacetime such that the lapse has zero gradient at the puncture. This boundary condition has been observed to hold in numerical evolutions, but in the past it was not clear whether the numerically obtained maximal slices exist analytically. We show that our analytical result agrees with numerical simulation. Given the analytical form for the lapse, we can derive that at late times the value of the lapse at the event horizon approaches the value 3 16 ͱ3Ϸ0.3248, justifying the numerical estimate of 0.3 that has been used for black hole excision in numerical simulations. We present our results for the nonextremal Reissner-Nordström metric, generalizing previous constructions of maximal slices.

Collisions of charged black holes

“…The space part of the Reissner-Nordström solution in isotropic coordinates is given by Eq. (2.10) with a conformal factor [66,67]…”

Slice stretching effects for maximal slicing of a Schwarzschild black hole

“…Following up on earlier work done together with Brügmann, [4][5][6], in the present paper one particular singularity avoiding slicing is looked at, namely maximal slicing corresponding to the condition that the mean extrinsic curvature of the slices vanishes at all times [7]. This geometrically motivated choice of the lapse function has been used frequently in numerical relativity (for simulations of a single Schwarzschild black hole see e.g.…”

Well-Posed Initial-Boundary Value Problem for a Constrained Evolution System and Radiation-Controlling Constraint-Preserving Boundary Conditions

“…Finally, by repeating the constructions of Theorems 2 and 1, with the only difference being that ∂ i ϕ is now the forcing term in the second equation of (23), we establish the existence result and the estimate (26). The term ϕ(t) L2(Ω) is added to (26) by considering the inequality…”

“…Several authors (e.g., [16,26]) had experimented with boundary conditions motivated by physics rather than the equation analysis. Surprisingly, the stability of the calculations improved when physical data was employed.…”

“…We consider a model problem of the vector wave equation The choice of the model problem is motivated by re-formulations of Einstein's field equations (e.g., [3,34,4,31,24,11,12,26]) in which second order in space equations are coupled to first order differential constraints (if necessary, (1) can be reduced to first order in time by introducing π i = ∂ t u i ; all results can be repeated for the pair (u i , π i ) without much complication). First order symmetric hyperbolic reductions of (1) has been used as a source of insight in numerical relativity in the past [6,9,21,27] and methods for constructing constraint-preserving boundary conditions were investigated using the same model problem.…”