Self-intersecting marginally outer trapped surfaces

We have shown previously that a merger of marginally outer trapped surfaces (MOTSs) occurs in a binary black hole merger and that there is a continuous sequence of MOTSs which connects the initial two black holes to the final one. In this paper, we confirm this scenario numerically and we detail further improvements in the numerical methods for locating MOTSs. With these improvements, we confirm the merger scenario and demonstrate the existence of self-intersecting MOTSs formed in the immediate aftermath of the merger. These results will allow us to track physical quantities across the non-linear merger process and to potentially infer properties of the merger from gravitational wave observations

Ultimate fate of apparent horizons during a binary black hole merger I: Locating and understanding axisymmetric marginally outer trapped surfaces

In classical numerical relativity, marginally outer trapped surfaces (MOTSs) are the main tool to locate and characterize black holes. For five decades it has been known that during a binary merger, a new outer horizon forms around the initial apparent horizons of the individual holes once they are sufficiently close together. However the ultimate fate of those initial horizons has remained a subject of speculation. Recent axisymmetric studies have shed new light on this process and this pair of papers essentially completes that line of research: we resolve the key features of the post-swallowing axisymmetric evolution of the initial horizons. This first paper introduces a new shooting-method for finding axisymmetric MOTSs along with a reinterpretation of the stability operator as the analogue of the Jacobi equation for families of MOTSs. Here, these tools are used to study exact solutions and initial data. In the sequel paper they are applied to black hole mergers

Ultimate fate of apparent horizons during a binary black hole merger II: Horizons weaving back and forth in time

In this second part of a two-part paper, we discuss numerical simulations of a head-on merger of two non-spinning black holes. We resolve the fate of the original two apparent horizons by showing that after intersecting, their world tubes "turn around" and continue backwards in time. Using the method presented in the first paper to locate these surfaces, we resolve several such world tubes evolving and connecting through various bifurcations and annihilations. This also draws a consistent picture of the full merger in terms of apparent horizons, or more generally, marginally outer trapped surfaces (MOTSs). The MOTS stability operator provides a natural mechanism to identify MOTSs which should be thought of as black hole boundaries. These are the two initial ones and the final remnant. All other MOTSs lie in the interior and are neither stable nor inner trapped

A Pair of Pants for the Apparent Horizon

We resolve the fate of the two original apparent horizons during the head-on merger of two non-spinning black holes, showing that these horizons exist for a finite amount of time before they individually "turn around" and move backward in time. This completes the understanding of the "pair of pants" diagram for the apparent horizon. Our result is facilitated by a new method for locating marginally outer trapped surfaces (MOTSs) based on a generalized shooting method. We also discuss the role played by the MOTS stability operator in discerning which among a multitude of MOTSs should be considered as black hole boundaries

Numerical Approach for Corvino-Type Gluing of Brill-Lindquist Initial Data

Building on the work of Giulini and Holzegel (2005), a new numerical approach is developed for computing Cauchy data for Einstein's equations by gluing a Schwarzschild end to a Brill-Lindquist metric via a Corvino-type construction. In contrast to, and in extension of, the numerical strategy of Doulis and Rinne (2016), the overdetermined Poisson problem resulting from the Brill wave ansatz is decomposed to obtain two uniquely solvable problems. A pseudospectral method and Newton-Krylov root finder are utilized to perform the gluing. The convergence analysis strongly indicates that the numerical strategy developed here is able to produce highly accurate results. It is observed that Schwarzschild ends of various ADM masses can be glued to the same interior configuration using the same gluing radius

What Happens to Apparent Horizons in a Binary Black Hole Merger?

We resolve the fate of the two original apparent horizons during the head-on merger of two non-spinning black holes. We show that following the appearance of the outer common horizon and subsequent inter-penetration of the original horizons, they continue to exist for a finite period of time before they are individually annihilated by unstable MOTSs. The inner common horizon vanishes in a similar, though independent, way. This completes the understanding of the analogue of the event horizon's "pair of pants" diagram for the apparent horizon. Our result is facilitated by a new method for locating marginally outer trapped surfaces (MOTSs) based on a generalized shooting method. We also discuss the role played by the MOTS stability operator in discerning which among a multitude of MOTSs should be considered as black hole boundaries.Comment: 6 pages, 5 figures, V2: final version with several changes (including changed title

Horizons in a binary black hole merger I: Geometry and area increase

Recent advances in numerical relativity have revealed how marginally trapped surfaces behave when black holes merge. It is now known that interesting topological features emerge during the merger, and marginally trapped surfaces can have self-intersections. This paper presents the most detailed study yet of the physical and geometric aspects of this scenario. For the case of a head-on collision of non-spinning black holes, we study in detail the world tube formed by the evolution of marginally trapped surfaces. In the first of this two-part study, we focus on geometrical properties of the dynamical horizons, i.e. the world tube traced out by the time evolution of marginally outer trapped surfaces. We show that even the simple case of a head-on collision of non-spinning black holes contains a rich variety of geometric and topological properties and is generally more complex than considered previously in the literature. The dynamical horizons are shown to have mixed signature and are not future marginally trapped everywhere. We analyze the area increase of the marginal surfaces along a sequence which connects the two initially disjoint horizons with the final common horizon. While the area does increase overall along this sequence, it is not monotonic. We find short durations of anomalous area change which, given the connection of area with entropy, might have interesting physical consequences. We investigate the possible reasons for this effect and show that it is consistent with existing proofs of the area increase law

Horizons in a binary black hole merger II: Fluxes, multipole moments and stability

We study in detail the dynamics and stability of marginally trapped surfaces during a binary black hole merger. This is the second in a two-part study. The first part studied the basic geometric aspects of the world tubes traced out by the marginal surfaces and the status of the area increase law. Here we continue and study the dynamics of the horizons during the merger, again for the head-on collision of two non-spinning black holes. In particular we follow the spectrum of the stability operator during the course of the merger for all the horizons present in the problem and implement systematic spectrum statistics for its analysis. We also study more physical aspects of the merger, namely the fluxes of energy which cross the horizon and cause the area to change. We construct a natural coordinate system on the horizon and decompose the various fields appearing in the flux, primarily the shear of the outgoing null normal, in spin weighted spherical harmonics. For each of the modes we extract the decay rates as the final black hole approaches equilibrium. The late part of the decay is consistent with the expected quasi-normal mode frequencies, while the early part displays a much steeper fall-off. Similarly, we calculate the decay of the horizon multipole moments, again finding two different regimes. Finally, seeking an explanation for this behavior, motivated by the membrane paradigm interpretation, we attempt to identify the different dynamical timescales of the area increase. This leads to the definition of a ``slowness parameter'' for predicting the onset of transition from a faster to a slower decay