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

We solve Einstein’s constraint equations in the conformal thin-sandwich decomposition to model thin shells of noninteracting particles in circular orbit about a nonrotating 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.

Figure 1

The average error over all collocation points $E^N$ as a function of the number of collocation points $N$ for two analytic representations of the Schwarzschild geometry in isotropic coordinates. The label $AP$ denotes the analytic puncture solution presented in 21, while the label $\mathrm{KS}$ denotes the Kerr-Schild solution in 11. The errors drop off exponentially as a function of the number of collocation points $N$, until a “floor” of specified tolerance has been reached.

Figure 2

The conformal factor $\psi$ and the lapse $\alpha$ as a function of areal radius for the analytic shell solution ($K=0$ and ${\alpha }_{\mathrm{EH}}=0$) for a Lorentz factor of $W=1.20$ and a mass ratio of $M_{\mathrm{BH}}/M_{\mathrm{SH}}=1$, corresponding to an areal shell radius of about $8.013M_{\mathrm{BH}}$. The field variables are continuous across the shell, but, according to the jump conditions (21) their derivatives are not. Our numerical solutions agree with the analytical ones to within better than ${10}^{-10}$, making the error far smaller than the line width in the graph.

Figure 3

The ADM binding energy (29) $E_{\mathrm{B}}$ as a function of the shell’s areal radius $R$ for an extreme-mass-ratio sequence with $M_{\mathrm{BH}}/M_{\mathrm{SH}}=1000$. Values of the horizon lapse ${\alpha }_{\mathrm{AH}}$ range from zero to unity in increments of 0.1. The graph includes 11 solid lines [representing an evaluation of the binding energy (29) in maximal slicing] and 11 dashed lines (for Kerr-Schild slicing). All 22 lines coincide within our numerical error. We computed these sequences using $N=26$ collocation points in each domain.

Figure 4

The ADM binding energy $E_{\mathrm{B}}$ as a function of the shell’s areal radius $R$ for an equal-mass sequence, $M_{\mathrm{BH}}/M_{\mathrm{SH}}=1$. As in Fig. 3, values of ${\alpha }_{\mathrm{AH}}$ again range from zero to unity in increments of 0.1, and the graph again contains 11 lines each for Maximal and Kerr-Schild slicing. All 22 lines again coincide within our numerical error. We obtained these results with $N=20$ collocation points in each domain.