Except as noted below, I generally follow the sign and notation conventions of Wald [160]. I assume that all
spacetimes are globally hyperbolic, and for event-horizon finding I further assume asymptotic flatness; in
this latter context
is future null infinity. I use the Penrose abstract-index notation, with summation
over all repeated indices. 4-indices
range over all spacetime coordinates
, and 3-indices
range over the spatial coordinates
in a spacelike slice
. The spacetime coordinates are
thus
.
Indices range over generic angular coordinates
on
or on a horizon surface. Note that
these coordinates are conceptually distinct from the 3-dimensional spatial coordinates
. Depending on
the context,
may or may not have the usual polar-spherical topology. Indices
label angular
grid points on
or on a horizon surface. These are 2-dimensional indices: a single such index uniquely
specifies an angular grid point.
is the Kronecker delta on the space of these indices or, equivalently, on
surface grid points.
For any indices and
,
and
are the coordinate partial derivatives
and
respectively; for any coordinates
and
,
and
are the coordinate partial
derivatives
and
respectively.
is the flat-space angular Laplacian operator on
,
while
refers to a finite-difference grid spacing in some variable
.
is the spacetime 4-metric, and
the inverse spacetime 4-metric; these are used to raise and
lower 4-indices.
are the 4-Christoffel symbols.
is the Lie derivative along the 4-vector field
.
I use the 3 + 1 “ADM” formalism first introduced by Arnowitt, Deser, and Misner [14]; York [163] gives
a general overview of this formalism as it is used in numerical relativity.
is the 3-metric defined in a
slice, and
is the inverse 3-metric; these are used to raise and lower 3-indices.
is the
associated 3-covariant derivative operator, and
are the 3-Christoffel symbols.
and
are the 3 + 1 lapse function and shift vector respectively, so the spacetime line element is
I often write a differential operator as , where the “
” notation
means that
is a (generally nonlinear) algebraic function of the variable
and its 1st and 2nd angular
derivatives, and that
also depends on the coefficients
,
, and
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 . Then given some norm
on
, the algorithm’s radius of convergence about a
solution
is defined as the smallest
such that the algorithm will converge to
the correct solution
for any initial guess
with
. We only rarely know the
exact radius of convergence of an algorithm, but practical experience often provides a rough
estimate1.
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