#Local horizon definition software
In this review I distinguish between a numerical algorithm (an abstract description of a mathematical computation also often known as a “method” or “scheme”), and a computer code (a “horizon finder”, a specific piece of computer software which implements a horizon finding algorithm or algorithms). The scope of this review is limited to the finding of event/apparent horizons and omits any but the briefest mention of the many uses of this information in gaining physical understanding of numerically-computed spacetimes. Baumgarte and Shapiro have also recently reviewed event and apparent horizon finding algorithms. The subject of this review is numerical algorithms and codes for doing this, focusing on calculations done using the 3 + 1 ADM formalism. To gain this information, we must explicitly find the horizons from the numerically-computed spacetime geometry. The usual output of a numerical relativity simulation is some (approximate, discrete) representation of the spacetime geometry (the 4-metric and possibly its derivatives) and any matter fields, but not any explicit information about the existence, precise location, or other properties of any event/apparent horizons. Systems with strong gravitational fields, particularly systems which may contain event horizons and/or apparent horizons, are a major focus of numerical relativity. Minimization methods are slow and relatively inaccurate in the context of a finite differencing simulation, but in a spectral code they can be relatively faster and more robust. Flow algorithms are generally quite slow but can be very robust in their convergence. In many cases, Schnetter’s “pretracking” algorithm can greatly improve an elliptic-PDE algorithm’s robustness. In slices with no continuous symmetries, spectral integral-iteration algorithms and elliptic-PDE algorithms are fast and accurate, but require good initial guesses to converge. In axisymmetry, shooting algorithms work well and are fairly easy to program. There are a large number of apparent horizon finding algorithms, with differing trade-offs between speed, robustness, accuracy, and ease of programming. Most “apparent horizon” finders actually find MOTSs. The MOTS condition is a nonlinear elliptic partial differential equation (PDE) for the surface shape, containing the ADM 3-metric, its spatial derivatives, and the extrinsic curvature as coefficients. An apparent horizon is then defined as a MOTS not contained in any other MOTS. A marginally outer trapped surface (MOTS) in a slice is a smooth closed 2-surface whose future-pointing outgoing null geodesics have zero expansion Θ. In contrast to an event horizon, an apparent horizon is defined locally in time in a spacelike slice and depends only on data in that slice, so it can be (and usually is) found during the numerical computation of a spacetime. The last of these is generally the most efficient and accurate. There are three basic algorithms for finding event horizons, based on integrating null geodesics forwards in time, integrating null geodesics backwards in time, and integrating null surfaces backwards in time.
The event horizon is defined nonlocally in time: it is a global property of the entire spacetime and must be found in a separate post-processing phase after all (or at least the nonstationary part) of spacetime has been numerically computed. The event horizon is a (continuous) null surface in spacetime. The event horizon of an asymptotically-flat spacetime is the boundary between those events from which a future-pointing null geodesic can reach future null infinity and those events from which no such geodesic exists.
In this article I review numerical algorithms and codes for finding event and apparent horizons in numerically-computed spacetimes, focusing on calculations done using the 3 + 1 ADM formalism.
Navigate to frontdoor\backend-config.json and define the Horizon back-end config: \templates\secrets.yaml and define the sitecore-hrz-host-name secret with sitecore-secret-provider.Event and apparent horizons are key diagnostics for the presence and properties of black holes.
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