Excision scheme for black holes in constrained evolution formulations: Spherically symmetric case

Excision techniques are used in order to deal with black holes in numerical simulations of Einstein’s equations and consist in removing a topological sphere containing the physical singularity from the numerical domain, applying instead appropriate boundary conditions at the excised surface. In this work we present recent developments of this technique in the case of constrained formulations of Einstein’s equations and for spherically symmetric spacetimes. We present a new set of boundary conditions to apply to the elliptic system in the fully constrained formalism of Bonazzola et al. [Phys. Rev. D 70, 104007 (2004)], at an arbitrary coordinate sphere inside the apparent horizon. Analytical properties of this system of boundary conditions are studied and, under some assumptions, an exponential convergence toward the stationary solution is exhibited for the vacuum spacetime. This is verified in numerical examples, together with the applicability in the case of the accretion of a scalar field onto a Schwarzschild black hole. We also present the successful use of the excision technique in the collapse of a neutron star to a black hole, when excision is switched on during the simulation, after the formation of the apparent horizon. This allows the accretion of matter remaining outside the excision surface and for the stable long-term evolution of the newly formed black hole.

Figure 1

Evolution in terms of coordinate time $t$ of the lapse $N$, minus its time asymptotic value $N_{\lim }\simeq 0.549$ (left top panel, logarithmic scale); of $b$, the projection of the shift ${\beta }^i$ onto the normal to the excision surface [Eq. (3.2)], minus the lapse $N$ (right top panel); of the conformal factor $\psi$ (left bottom panel), at the excision surface ($r=0.916M_{\text{ADM}}$); and of the AH coordinate radius $r_{*}$ (right bottom panel), for a spherically symmetric vacuum (i.e. Schwarzschild) BH spacetime, using excision boundary conditions described in Sec. 3b.

Figure 2

Evolution in terms of coordinate time $t$ of the outward expansion ${\theta }^{(l)}$ [Eq. (3.3)], for a spherically symmetric BH spacetime, as in Fig. 1.

Figure 3

Evolution in terms of coordinate time $t$ of the variation of the AH irreducible mass (4.2) with respect to the initial ADM mass (4.1), for a spherically symmetric BH spacetime, using excision boundary conditions described in Sec. 3b.

Figure 4

Evolution in terms of the coordinate time $t$ of the absolute value of the accreted scalar field, $|\varphi |$, at the excision surface, $\varphi (r=1,t)$.

Figure 5

Evolution of the ratio between the AH irreducible mass (4.2) and the ADM one (4.1), as a function of the coordinate time $t$, for the case of the accretion of a massless scalar field.

Figure 6

Evolution in terms of the coordinate time $t$ of the relative variation of the ADM mass, defined as in Eq. (4.1), with respect to its initial value, for the case of the accretion of a massless scalar field.

Figure 7

Radial profiles of the rest-mass density, for different moments around the start of the excision in the simulation of the collapse of a neutron star to a BH. Excision starts at $t=0.43788\mathrm{ms}$.

Figure 8

Evolution in terms of coordinate time $t$ of the AH irreducible mass (4.2) for the collapse of a neutron star to a BH. The vertical dotted line indicates the moment when excision is turned on.

Figure 9

Evolution in terms of coordinate time $t$ of the relative variation, with respect to its initial value at $t=0$, of the ADM mass (4.1) for the spacetime of a neutron star collapsing to a BH. The vertical dotted line indicates the moment when excision is turned on.