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1 Notation and Terminology

Except as noted below, I generally follow the sign and notation conventions of Wald [160Jump To The Next Citation Point]. I assume that all spacetimes are globally hyperbolic, and for event-horizon finding I further assume asymptotic flatness; in this latter context + 𝒥 is future null infinity. I use the Penrose abstract-index notation, with summation over all repeated indices. 4-indices abc range over all spacetime coordinates a {x }, and 3-indices ijk range over the spatial coordinates {xi} in a spacelike slice t = constant. The spacetime coordinates are thus xa = (t,xi).

Indices uvw range over generic angular coordinates (θ,φ ) on S2 or on a horizon surface. Note that these coordinates are conceptually distinct from the 3-dimensional spatial coordinates i x. Depending on the context, (θ,φ) may or may not have the usual polar-spherical topology. Indices ijk label angular grid points on S2 or on a horizon surface. These are 2-dimensional indices: a single such index uniquely specifies an angular grid point. δij is the Kronecker delta on the space of these indices or, equivalently, on surface grid points.

For any indices p and q, ∂p and ∂pq are the coordinate partial derivatives p ∂∕∂x and ∂2∕∂xp ∂xq respectively; for any coordinates μ and ν, ∂u and ∂ μν are the coordinate partial derivatives ∂ ∕∂μ and ∂2∕∂μ∂ ν respectively. Δ is the flat-space angular Laplacian operator on S2, while Δx refers to a finite-difference grid spacing in some variable x.

gab is the spacetime 4-metric, and ab g the inverse spacetime 4-metric; these are used to raise and lower 4-indices. Γ cab are the 4-Christoffel symbols. ℒv is the Lie derivative along the 4-vector field va.

I use the 3 + 1 “ADM” formalism first introduced by Arnowitt, Deser, and Misner [14]; York [163Jump To The Next Citation Point] gives a general overview of this formalism as it is used in numerical relativity. gij is the 3-metric defined in a slice, and gij is the inverse 3-metric; these are used to raise and lower 3-indices. ∇i is the associated 3-covariant derivative operator, and Γ kij are the 3-Christoffel symbols. α and βi are the 3 + 1 lapse function and shift vector respectively, so the spacetime line element is

2 a b ds = gab dx dx (1 ) = − (α2 − βiβi)dt2 + 2βi dxidt + gij dxidxj. (2 )
As is common in 3 + 1 numerical relativity, I follow the sign convention of Misner, Thorne, and Wheeler [112Jump To The Next Citation Point] and York [163] in defining the extrinsic curvature of the slice as 1 Kij = − 2ℒngij = − ∇inj, where na is the future-pointing unit normal to the slice. (In contrast, Wald [160Jump To The Next Citation Point] omits the minus signs from this definition.) K ≡ Kii is the trace of the extrinsic curvature Kij. mADM is the ADM mass of an asymptotically flat slice.

I often write a differential operator as F = F (y,∂uy,∂uvy;gij,∂kgij,Kij), where the “;” notation means that F is a (generally nonlinear) algebraic function of the variable y and its 1st and 2nd angular derivatives, and that F also depends on the coefficients gij, ∂kgij, and Kij at the apparent horizon position.

There are three common types of spacetimes/slices where numerical event or apparent horizon finding is of interest: spherically-symmetric spacetimes/slices, axisymmetric spacetimes/slices, and spacetimes/slices with no continuous spatial symmetries (no spacelike Killing vectors). I refer to the latter as “fully generic” spacetimes/slices.

In this review I use the abbreviations “ODE” for ordinary differential equation, “PDE” for partial differential equation, “CE surface” for constant-expansion surface, and “MOTS” for marginally outer trapped surface. Names in Small Capitals refer to horizon finders and other computer software.

When discussing iterative numerical algorithms, it is often convenient to use the concept of an algorithm’s “radius of convergence”. Suppose the solution space within which the algorithm is iterating is S. Then given some norm ∥ ⋅ ∥ on S, the algorithm’s radius of convergence about a solution s ∈ S is defined as the smallest r > 0 such that the algorithm will converge to the correct solution s for any initial guess g with ∥g − s∥ ≤ r. We only rarely know the exact radius of convergence of an algorithm, but practical experience often provides a rough estimate1.


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