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2.2 Strahlkörper parameterizations

Most apparent horizon finders, and some event-horizon finders, assume that each connected component of the apparent (event) horizon has S2 topology. With the exception of toroidal event horizons (discussed in Section 4), this is generally a reasonable assumption.

To parameterize an 2 S surface’s shape, it is common to further assume that we are given (or can compute) some “local coordinate origin” point inside the surface such that the surface’s 3-coordinate shape relative to that point is a “Strahlkörper” (literally “ray body”, or more commonly “star-shaped region”), defined by Minkowski [138, Page 108] as

a region in n-D Euclidean space containing the origin and whose surface, as seen from the origin, exhibits only one point in any direction.

The Strahlkörper assumption is a significant restriction on the horizon’s coordinate shape (and the choice of the local coordinate origin). For example, it rules out the coordinate shape and local coordinate origin illustrated in Figure 1View Image: a horizon with such a coordinate shape about the local coordinate origin could not be found by any horizon finder which assumes a Strahlkörper surface parameterization.

View Image

Figure 1: This figure shows a cross-section of a coordinate shape (the thick curve) which is not a Strahlkörper about the local coordinate origin shown (the large dot). The dashed line shows a ray from the local coordinate origin, which intersects the surface in more than one point.

For event-horizon finding, algorithms and codes are now available which allow an arbitrary horizon topology with no Strahlkörper assumption (see the discussion in Section 5.3.3 for details). For apparent horizon finding, the flow algorithms discussed in Section 8.7 theoretically allow any surface shape, although many implementations still make the Strahlkörper assumption. Removing this assumption for other apparent horizon finding algorithms might be a fruitful area for further research.

Given the Strahlkörper assumption, the surface can be explicitly parameterized as

r = h (θ, φ), (4 )
where r is the Euclidean distance from the local coordinate origin to a surface point, (θ,φ) are generic angular coordinates on the horizon surface (or equivalently on S2), and the “horizon shape function” h : S2 → ℜ+ is a positive real-valued function on the domain of angular coordinates defining the surface shape. Given the choice of local coordinate origin, there is clearly a one-to-one mapping between Strahlkörper 2-surfaces and horizon shape functions.

There are two common ways to discretize a horizon shape function:

Spectral representation
 
Here we expand the horizon shape function h in an infinite series in some (typically orthonormal) set of basis functions such as spherical harmonics Yℓm or symmetric trace-free tensors2,
h(θ,φ) = ∑ a Y (θ,φ). (5) ℓm ℓm ℓ,m

This series can then be truncated at some finite order ℓmax, and the Ncoeff = ℓmax(ℓmax+1 ) coefficients {aℓm } used to represent (discretely approximate) the horizon shape. For reasonable accuracy, ℓmax is typically on the order of 8 to 12.

Finite difference representation
 
Here we choose some finite grid of angular coordinates {(θk,φk )}, k = 1, 2, 3, …, Nang on 2 S (or equivalently on the surface)3, and represent (discretely approximate) the surface shape by the Nang values
{h(θk,φk )} k = 1,2,3,...,Nang. (6)
For reasonable accuracy, Nang is typically on the order of a few thousand.

It is sometimes useful to explicitly construct a level-set function describing a given Strahlkörper. A common choice here is

F ≡ r − h(θ,φ). (7 )

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