Locating Horizons in Dynamical Data

Locating a horizon in dynamical data is an elliptic surface-finding problem embedded in a hyperbolic spacetime evolution. An apparent-horizon finder solves for a marginal surface on one slice; an event-horizon finder reconstructs a global null boundary from many slices. Their numerical outputs answer different physical questions, and both require residual, convergence, and branch checks beyond a successful return code.

Required background. Hypersurface normals and tangential projectors supplies the slice and surface geometry entering the expansion.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon definitions and invariants; Black-Hole Perturbation Theory develops distorted-surface modes; Formulations of General Relativity supplies evolved $3+1$ fields; and Computational Relativity develops discretization, interpolation, and convergence.

The continuum surface equation

On a spacelike slice with induced metric $h_{ij}$, covariant derivative $D_i$, and extrinsic curvature convention $K_{ij}=h_i{}^ah_j{}^b\nabla_a u_b$, let $S$ have outward unit normal $s^i$. With $\ell^a=u^a+s^a$, the apparent-horizon equation is

$$\Theta[S]\equiv D_i s^i+K-K_{ij}s^is^j=0.$$

Represent a star-shaped surface about a chosen center by $r=h(\theta,\phi)$. Then $\Theta[h]=0$ is a nonlinear elliptic equation on the sphere. Star-shapedness is an algorithmic restriction, not a theorem about every MOTS. Spectral methods expand $h$ in spherical harmonics; finite-difference methods discretize angular derivatives; flow methods evolve a trial surface toward $\Theta=0$. Gundlach develops the pseudo-spectral and fast-flow family and tests it on Brill–Lindquist and Kerr data in Gundlach 1998, §§ II–V, pp. 864–873, and Tables II–III. Thornburg derives the $3+1$ equation and compares the broader algorithm family in Thornburg 2007, §§3.1–3.2, pp. 27–57, PDF.

An event horizon instead follows null generators or evolves a null level-set backward from a late cross-section near the settled horizon. Backward evolution is stable because outgoing generators that diverge to the future converge onto the event horizon to the past. It requires stored spacetime data, interpolation in space and time, a sufficiently late seed surface, and treatment of generators joining at caustics or crossovers; see Thornburg 2007, §2, pp. 11–24, PDF.

Inputs, outputs, and branch finding

For a MOTS calculation, record the continuum $h_{ij},K_{ij}$ fields, their sign conventions, slice time, asymptotic end, desired topology, surface center, angular representation, interpolation order, initial guess, nonlinear solver, residual norm, and stopping tolerance. Search parameters matter: a Newton solve can converge to an inner branch or fail when the initial guess lies outside its basin. Pretracking or continuation in time and in a modified expansion can reveal branches before or around pair creation.

Reader-facing outputs should be geometric: the surface embedding plus $\Theta$ residual, area $A$, irreducible mass $M_{\rm irr}=\sqrt{A/(16\pi G^2)}$, quasi-local angular momentum with its axial-vector prescription, intrinsic scalar curvature and multipoles, and ingoing expansion. Record the principal stability eigenvalue when branch changes are important. Coordinate radius is useful for visualization but not as the main result.

For an event horizon, publish the generator mesh, seed time and seed family, null-condition residual, caustic/crossover handling, area history, and comparison with the marginal surfaces. Generator density is not a physical area density unless corrected for the evolving surface measure.

Benchmarks and convergence

A benchmark ladder begins with round Schwarzschild data, for which the standard slice has a marginal sphere of areal radius $2GM$. Next use a known Kerr slice, testing area and spin as well as the distorted coordinate shape. An independent reference implementation or an alternative algorithm should recover the same surface invariants from shared data. A dynamical spherical spacetime such as Vaidya tests time tracking and the signature of the marginal tube.

Run at least three spacetime resolutions and, independently, three surface angular resolutions. The horizon residual should decrease at the surface solver’s expected order until interpolation error dominates. Area, spin, multipoles, and formation time must converge under both refinements. Tightening the nonlinear tolerance cannot repair inaccurate interpolated geometry; demonstrate a plateau where physical outputs are insensitive to a smaller tolerance.

For event horizons, vary the late seed time and seed shape, generator resolution, temporal sampling, and interpolation order. Convergence should occur away from caustics; near generator entry, report localized errors rather than hiding them in a global area norm. Compare two independent late seeds to show backward attraction.

Failure modes and reproducibility boundary

Common MOTS failure modes include a bad center for a star-shaped ansatz, loss of angular resolution on a pinched surface, interpolation noise in second derivatives, a Jacobian becoming ill-conditioned near a stability eigenvalue crossing, convergence to the wrong branch, and no convergence despite a surface existing outside the search class. A coarse time cadence can turn smooth pair creation into an apparent jump. Gauge motion can move coordinate shapes without changing intrinsic geometry.

Event-horizon failure modes include insufficiently late seeding, missing generators at a merger crease set, temporal interpolation that violates nullness, and a simulation ending before the exterior settles. A retrospective surface found in truncated data is conditional on the assumed future completion.

We did not rerun the cited simulations or horizon-finder tests for this page. Reproduction requires the spacetime data or a documented generator, code version, parameter files, interpolation and solver tolerances, initial guesses or seed surfaces, and machine-readable embeddings and residuals at every resolution.

Check your understanding

Doubling angular resolution reduces the expansion residual by a factor of sixteen but changes the horizon area negligibly; doubling the spacetime resolution changes the area by $2\%$. Which error currently controls the area?

Solution

The spacetime discretization and interpolation error controls the area. The surface solve is already much more accurate than its input geometry. Further tightening the horizon residual would not address the observed $2\%$ change; the spacetime resolution study must be continued and its error propagated to the horizon observables.

References

• Gundlach, Carsten. “Pseudo-Spectral Apparent Horizon Finders: An Efficient New Algorithm.” Physical Review D 57 (1998): 863–875. doi:10.1103/PhysRevD.57.863.
• Thornburg, Jonathan. “Event and Apparent Horizon Finders for 3+1 Numerical Relativity.” Living Reviews in Relativity 10 (2007): 3. doi:10.12942/lrr-2007-3.