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8.1 Zero-finding in spherical symmetry

In a spherically symmetric slice, any apparent horizon must also be spherically symmetric, so the apparent horizon equation (16View Equation) becomes a 1-dimensional nonlinear algebraic equation Θ (h) = 0 for the horizon radius h. For example, adopting the usual (symmetry-adapted) polar-spherical spatial coordinates i x = (r,θ,φ), we have [154Jump To The Next Citation Point, Equation (B7)]
Θ ≡ --∂r√gθθ-− 2K-θθ = 0. (17 ) gθθ grr gθθ
Given the geometry variables grr, g θθ, ∂rgθθ, and K θθ, this equation may be easily and accurately solved using one of the zero-finding algorithms discussed in Appendix A29.

Zero-finding has been used by many researchers, including [141142143144119471399154,  155Jump To The Next Citation Point]30. For example, the apparent horizons shown in Figure 2View Image were obtained using this algorithm. As another example, Figure 9View Image shows Θ (r) and h at various times in a (different) spherically symmetric collapse simulation.

View Image

Figure 9: This figure shows the apparent horizons (actually MOTSs) for a spherically symmetric numerical evolution of a black hole accreting a narrow shell of scalar field, the 800.pqw1 evolution of Thornburg [155]. Part (a) of this figure shows Θ (r ) (here labelled H) for a set of equally-spaced times between t=19 and t=20, while Part (b) shows the corresponding MOTS radius h (t) and the Misner–Sharp [111], [112, Box 23.1] mass m (h) internal to each MOTS. Notice how two new MOTSs appear when the local minimum in Θ(r) touches the Θ=0 line, and two existing MOTS disappear when the local maximum in Θ(r) touches the Θ=0 line.

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