Existence and stability of marginally trapped surfaces in black-hole spacetimes

Abstract

Marginally outer trapped surfaces (MOTSs, or marginal surfaces in short) are routinely used in numerical simulations of black hole spacetimes. They are an invaluable tool for locating and characterizing black holes quasi-locally in real time while the simulation is ongoing. It is often believed that a MOTS can behave unpredictably under time evolution; an existing MOTS can disappear, and a new one can appear without any apparent reason. In this paper we show that in fact the behavior of a MOTS is perfectly predictable and its behavior is dictated by a single real parameter, the stability parameter, which can be monitored during the course of a numerical simulation. We demonstrate the utility of the stability parameter to fully understand the variety of marginal surfaces that can be present in binary black hole initial data. We also develop a new horizon finder capable of locating very highly distorted marginal surfaces and we show that even in these cases, the stability parameter perfectly predicts the existence and stability of marginal surfaces.

Self-intersecting marginally outer trapped surfaces

“…via a pseudospectral method, which is what we chose. For our present application, we have implemented two additional features compared to what was used in [33]. These features are meant to deal with two additional complications that we must necessarily deal with: i) surfaces which have a very narrow "neck" (almost like a figure-eight), and in some instances have features like cusps and self intersections.…”

“…For this purpose, motivated by the methods used in [49], we employ bi-spherical coordinates [50]. ii) Unlike in [33] where the MOTS finder was applied to analytical initial data, we now have to deal with numerically generated data on a finite mesh. This requires the use of interpolation schemes some of which were already used in [34].…”

“…The reason for this is that, until recently, it was not known how marginal surfaces behave across the merger; near the merger the marginal surfaces are extremely distorted and previous numerical methods were not successful in tracking such highly distorted surfaces. Using improved numerical methods [33], we have recently shown the first evidence for the existence of a continuous sequence of marginal surfaces which interpolates between the two disjoint initial black holes and the single final remnant black hole [34]. This is the analog of the "pair of pants" picture for event horizons.…”

“…However, for the nonlinear search for a MOTS S, the expansion Θ ( ) and its derivatives have to be computed on a set of points x n ∈ R 2 along trial surfaces S i , c.f. [33], Section III.B. This requires evaluating the components of the metric h ij , its first and second spatial derivatives, the extrinsic curvature K ij and its first spatial derivatives at the points x n which generally do not coincide with any of the grid points of the simulation.…”

“…With the addition of numerical simulations, the task for our MOTS finder has become more general compared to the purely time-symmetric cases considered in [33]. Therefore, and in light of the surprising result of a self-intersecting MOTS, it is important to validate the method and test it for correctness in an analytic case where the result is known.…”