Marginally Outer Trapped Surfaces Research Articles

Symmetry and instability of marginally outer trapped surfaces

We consider an initial data set having a continuous symmetry and a marginally outer trapped surface (MOTS) that is not preserved by this symmetry. We show that such a MOTS is unstable except in an exceptional case. In non-rotating cases we provide a Courant-type lower bound on the number of unstable eigenvalues. These results are then used to prove the instability of a large class of exotic MOTSs that were recently observed in the Schwarzschild spacetime. We also discuss the implications for the apparent horizon in data sets with translational symmetry.

New spinorial mass-quasilocal angular momentum inequality for initial data with marginally future trapped surface

We prove a new geometric inequality that relates the Arnowitt–Deser–Misner mass of initial data to a quasilocal angular momentum of a marginally outer trapped surface (MOTS) inner boundary. The inequality is expressed in terms of a 1-spinor, which satisfies an intrinsic first-order Dirac-type equation. Furthermore, we show that if the initial data is axisymmetric, then the divergence-free vector used to define the quasilocal angular momentum cannot be a Killing field of the generic boundary.

Exotic marginally outer trapped surfaces in rotating spacetimes of any dimension

The recently developed MOTSodesic method for locating marginally outer trapped surfaces (MOTSs) was effectively restricted to non-rotating spacetimes. In this paper we extend the method to include (multi-)axisymmetric time slices of (multi-)axisymmetric spacetimes of any dimension. We then apply this method to study MOTSs in the BTZ, Kerr and Myers–Perry black holes. While there are many similarities between the MOTSs observed in these spacetimes and those seen in Schwarzschild and Reissner-Nordström, details of the more complicated geometries also introduce some new, previously unseen, behaviours.

Horizon area bound and MOTS stability in locally rotationally symmetric solutions

In this paper, we study the stability of marginally outer trapped surfaces (MOTSs), foliating horizons of the form , embedded in locally rotationally symmetric class II perfect fluid spacetimes. An upper bound on the area of stable MOTS is obtained. It is shown that any stable MOTS of the types considered in these spacetimes must be strictly stably outermost, that is, there are no MOTS ‘outside’ of and homologous to . Aspects of the topology of the MOTS, as well as the case when an extension is made to imperfect fluids, are discussed. Some non-existence results are also obtained. Finally, the ‘growth’ of certain matter and curvature quantities on certain unstable MOTS are provided under specified conditions.

Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3‑dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth ‘spacetime harmonic functions’ in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.

Rigidity of free boundary MOTS

The aim of this work is to present an initial data version of Hawking’s theorem on the topology of black hole spacetimes in the context of manifolds with boundary. More precisely, we generalize the results of G.J. Galloway and R. Schoen (Galloway et al., 2006) and G.J. Galloway (Galloway, 2008; Galloway, 2018) by proving that a compact free boundary stable marginally outer trapped surface (MOTS) Σ in an initial data set with boundary satisfying natural dominant energy conditions (DEC) is of positive Yamabe type, i.e. Σ admits a metric of positive scalar curvature with minimal boundary, provided Σ is outermost. To do so, we prove that if Σ is a compact free boundary stable MOTS which does not admit a metric of positive scalar curvature with minimal boundary in an initial data set satisfying the interior and the boundary DEC, then an outer neighborhood of Σ can be foliated by free boundary MOTS Σt, assuming that Σ is weakly outermost. Moreover, each Σt has vanishing outward null second fundamental form, is Ricci flat with totally geodesic boundary, and the dominant energy conditions saturate on Σt.

Interior marginally outer trapped surfaces of spherically symmetric black holes

There are notable similarities between the marginally outer trapped surfaces (MOTSs) present in the interior of a binary black hole merger and those present in the interior of the Schwarzschild black hole. Here we study the existence and properties of MOTSs with self-intersections in the interior of more general static and spherically symmetric black holes and coordinate systems. Our analysis is carried out in a parametrized family of Painlev\'e-Gullstrand-like coordinates that we introduce. First, for the Schwarzschild spacetime, we study the existence of these surfaces for various slicings of the spacetime finding them to be generic within the family of coordinate systems we investigate. Then, we study how an inner horizon affects the existence and properties of these surfaces by exploring examples: the Reissner-Nordstr\"om black hole and the four-dimensional Gauss-Bonnet black hole. We find that an inner horizon results in a finite number of self-intersecting MOTSs, but their properties depend sensitively on the interior structure of the black hole. By analyzing the spectrum of the stability operator, we show that our results for two-horizon black holes provide exact-solution examples of recently observed properties of unstable MOTSs present in the interior of a binary black hole merger, such as the sequence of bifurcations/annihilations that lead to the disappearance of apparent horizons.

Mean Curvature Flow in Null Hypersurfaces and the Detection of MOTS

We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.

Ultimate fate of apparent horizons during a binary black hole merger. II. The vanishing of apparent horizons

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.

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 nonspinning black holes. We show that, following the appearance of the outer common horizon and subsequent interpenetration of the original horizons, they continue to exist for a finite period of time before they are individually annihilated by unstable marginally outer trapped surfaces (MOTSs). The inner common horizon vanishes in a similar, though independent, way. This completes the understanding of the analog of the event horizon's "pair of pants" diagram for the apparent horizon. Our result is facilitated by a new method for locating 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.

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 [Phys. Rev. D 104, 084084 (2021)] they are applied to black hole mergers.

Stable surfaces and free boundary marginally outer trapped surfaces

We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of minimal and CMC surfaces. We prove two versions of Christodoulou–Yau estimate for \({\mathbf {H}}\)-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of \({\mathbf {H}}\)-stable surfaces. Moreover, we give a definition of capillary stability for MOTS with boundary. This notion of stability leads to an area upper bound inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. These are straightforward generalizations of Chen–Fraser–Pang and Carlotto–Franz results for free boundary minimal surfaces, respectively.

Quasi-Round MOTSs and Stability of the Schwarzschild Null Penrose Inequality

In Roesch (Proof of a null penrose conjecture using a new quasi-local mass, arXiv:1609.02875 [gr-qc], 2016), the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this paper, for a class of strictly stable Weakly Isolated Horizons, we show the existence of a unique foliation by quasi-round MOTS. In the second part, we show that any subsequent space-time perturbation continues to admit a quasi-round MOTS. Finally, for perturbations of the quasi-round MOTS in Schwarzschild, we identify sufficient conditions on the asymptotics of any past-pointing null hypersurface that yields the Null Penrose Inequality.

New spinorial approach to mass inequalities for black holes in general relativity

A new spinorial strategy for the construction of geometric inequalities involving the Arnowitt-Deser-Misner mass of black hole systems in general relativity is presented. This approach is based on a second order elliptic equation (the approximate twistor equation) for a valence 1 Weyl spinor. This has the advantage over other spinorial approaches to the construction of geometric inequalities based on the Sen-Witten-Dirac equation that it allows us to specify boundary conditions for the two components of the spinor. This greater control on the boundary data has the potential of giving rise to new geometric inequalities involving the mass. In particular, it is shown that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) which depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data in an appropriate way, one obtains a new nontrivial bounds of the mass in terms of the inner expansion of the MOTS. The analysis makes use of a new formalism for the $1+1+2$ decomposition of spinorial equations.

Emergence of Apparent Horizon in Gravitational Collapse

We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smooth (except at that point) and spacelike. In the proof we employ a scale critical trapped surface formation criterion established by An and Luk and a new type of quasilinear elliptic equation is studied. The main conclusion of this paper proves a conjecture of Ashtekar on black hole thermodynamics. And the spacetimes constructed here could also be viewed as (non-spherically symmetric) generalizations of the well-known Vaidya spacetime.

Inside the final black hole: puncture and trapped surface dynamics

A popular approach in numerical simulations of black hole binaries is to model black holes as punctures in the fabric of spacetime. The location and the properties of the black hole punctures are tracked with apparent horizons, namely outermost marginally outer trapped surfaces (MOTSs). As the holes approach each other, a common apparent horizon suddenly appears, engulfing the two black holes and signaling the merger. The evolution of common apparent horizons and their connection with gravitational wave emission have been studied in detail with the framework of dynamical horizons. We present a study of the dynamics of the MOTSs and their punctures in the interior of the final black hole. The study focuses on head-on mergers for various initial separations and mass ratios. We find that MOTSs intersect for most of the parameter space. We show that for those situations in which they do not, it is because of the singularity avoidance property of the moving puncture gauge condition used in the study. Although we are unable to carry out evolutions that last long enough to show the ultimate fate of the punctures, our results suggest that MOTSs always intersect and that at late times their overlap is only partial. As a consequence, the punctures inside the MOTSs, although close enough to each other to act effectively as a single puncture, do not merge.

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.

Interior of a Binary Black Hole Merger.

We find strong numerical evidence for a new phenomenon in a binary black hole spacetime, namely, the merger of marginally outer trapped surfaces (MOTSs). By simulating the head-on collision of two nonspinning unequal mass black holes, we observe that the MOTS associated with the final black hole merges with the two initially disjoint surfaces corresponding to the two initial black holes. This yields a connected sequence of MOTSs interpolating between the initial and final state all the way through the nonlinear binary black hole merger process. In addition, we show the existence of a MOTS with self-intersections formed immediately after the merger. This scenario now allows us to track physical quantities (such as mass, angular momentum, higher multipoles, and fluxes) across the merger, which can be potentially compared with the gravitational wave signal in the wave zone, and with observations by gravitational wave detectors. This also suggests a possibility of proving the Penrose inequality mathematically for generic astrophysical binary back hole configurations.

On the classification of MOTS in the de Sitter space

In this paper, we deal with marginally outer trapped surfaces (MOTS) immersed in the de Sitter space \(\mathbb {S}_1^{n+2}\). In this setting we are able to obtain a Simons formula for the null second fundamental form and under some appropriate constraints on the MOTS, we apply a weak maximum principle in order to guarantee that it must be either a totally geodesic submanifold or isometric to an open piece of an isoparametric submanifold with two distinct principal curvatures one of which is simple. In this last case, supposing that the initial data where the MOTS lying is a totally umbilical spacelike hypersurface of \(\mathbb {S}_1^{n+2}\), we conclude that it must be either isometric to a circular cylinder, a hyperbolic cylinder or a Clifford torus.

Rigidity of marginally outer trapped (hyper)surfaces with negative 𝜎-constant

In this paper we generalize a result of Galloway and Mendes in two different situations: in the first case for marginally outer trapped surfaces (MOTSs) of genus greater than 1 1 and, in the second case, for MOTSs of high dimension with negative σ \sigma -constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension 2 2 ) or for the volume (in dimension ≥ 3 \ge 3 ). These results are extensions of theorems of Nunes and Moraru to general (non-time-symmetric) initial data sets.