We study properties of stable, strictly stable and locally outermost marginally outer trapped surfaces in spacelike hypersurfaces of spacetimes possessing certain symmetries such as isometries, homotheties and conformal Killings. We first obtain results for general diffeomorphisms in terms of the so-called metric deformation tensor and then particularize to different types of symmetries. In particular, we find restrictions at the surfaces on the vector field generating the symmetry. Some consequences are discussed. As an application we present a result on non-existence of stable marginally outer trapped surfaces in slices of FLRW.
We investigate toroidal Marginally Outer Trapped Surfaces (MOTS) and Marginally Outer Trapped Tubes (MOTT) in closed Friedmann-Lemaitre-Robertson-Walker (FLRW) geometries. They are constructed by embedding Constant Mean Curvature (CMC) Clifford tori in a FLRW spacetime. This construction is used to assess the quality of certain isoperimetric inequalities, recently proved in axial symmetry. Similarly to spherically symmetric MOTS existing in FLRW spacetimes, the toroidal ones are also unstable.
In this paper, we explore the formation of the marginally trapped and marginally anti-trapped surfaces that arise from the evolution of homogeneous dust in D-dimensions with and without the cosmological constant, this is seen through the analytical expressions for such surfaces. We obtain closed form expressions for the Norm of the normal to the Horizon that decides their causal nature and also look at several interesting features of these surface evolution that are significantly different from the four dimensional counterpart. We obtain the expressions for the Ashtekar-Badrikrihnan's Area-bal…
Let $M$ be a locally rotationally symmetric spacetime with at least one of the rotation or spatial twist being non-zero. It is proved that $M$ cannot admit a non-minimal marginally trapped tube of the form $χ=X(t)$.
We review the basic setup of Kaluza-Klein theory, namely a 5-dimensional vacuum with a cyclic isometry, which corresponds to Einstein-Maxwell-dilaton theory in 4-dimensional spacetime. We first recall the behaviour of Killing horizons and its generators under bundle lift and projection. We then show that the property of compact surfaces of being (stably) marginally trapped is preserved under lift and projection provided the appropriate ("Pauli-") conformal scaling is used for the spacetime metric. We also discuss and compare recently proven area inequalities for stable axially symmetric 2-dime…
To better understand the allowed range of black hole geometries, we study Weyl-distorted Schwarzschild solutions. They always contain trapped surfaces, a singularity and an isolated horizon and so should be understood to be (geometric) black holes. However we show that for large distortions the isolated horizon is neither a future outer trapping horizon (FOTH) nor even a marginally trapped surface: slices of the horizon cannot be infinitesimally deformed into (outer) trapped surfaces. We consider the implications of this result for popular quasilocal definitions of black holes.
We discuss a family of inequalities involving the area, angular momentum and charges of stably outermost marginally trapped surfaces in generic non-vacuum dynamical spacetimes, with non-negative cosmological constant and matter sources satisfying the dominant energy condition. These inequalities provide lower bounds for the area of spatial sections of dynamical trapping horizons, namely hypersurfaces offering quasi-local models of black hole horizons. In particular, these inequalities represent particular examples of the extension to a Lorentzian setting of tools employed in the discussion of …
我们调查的一般行为的边缘被困管（大致随时间演变的表观视野）使用简单的，球对称的例子，尘埃和标量场崩溃/吸积到预先存在的黑洞。我们发现，在适当的物理条件下，边缘捕获管的演化可能是空的，类时的，或类空的，进一步的边缘捕获的两个球体的横截面可能会扩大或收缩的面积。当落入黑洞的物质满足$ρ- P \leq 1/A$时，就会发生类空膨胀，其中$A$是视界的面积，而$ρ$和$P$分别是物质的密度和压力。当$(ρ- P)$大于这个截止值时，就会发生类时演化，因此对于大型黑洞来说，这将是更常见的。物理上，它们对应于视界“跳跃”，因为极端条件迫使新视界在旧视界之外形成。
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-lin…
In the present paper we provide new examples of marginally trapped surfaces and tubes in FLRW spacetimes by using a basic relation between these objects and CMC surfaces in 3-manifolds. We also provide a new method to construct marginally trapped surfaces in closed FLRW spacetimes, which is based on the classical Hopf map. The utility of this method is illustrated by providing marginally trapped surfaces crossing expanding and collapsing regions of a closed FLRW spacetime. The approach introduced in this paper is also extended to twisted spaces.
A marginally outer trapped hypersurface is a generalization of minimal hypersurfaces originated from general relativity. We show a curvature estimate for stable marginally outer trapped hypersurfaces up to the free boundary satisfying a uniform area bound. Our proof is based on an iteration argument. The curvature estimate was previously known via a blowup argument for stable minimal hypersurfaces.
We accurately compute the scalar 2-curvature, the Weyl scalars, associated quasi-local spin, mass and higher multipole moments on marginally trapped surfaces in numerical 3+1 simulations. To determine the quasi-local quantities we introduce a new method which requires a set of invariant surface integrals, allowing for surface grids of a few hundred points only. The new technique circumvents solving the Killing equation and is also an alternative to approximate Killing vector fields. We apply the method to a perturbed non-axisymmetric black hole ringing down to Kerr and compare the quasi-local …
We examine potential deformations of inner black hole and cosmological horizons in Reissner-Nordström de-Sitter spacetimes. While the rigidity of the outer black hole horizon is guaranteed by theorem, that theorem applies to neither the inner black hole nor past cosmological horizon. Further for pure deSitter spacetime it is clear that the cosmological horizon can be deformed (by translation). For specific parameter choices, it is shown that both inner black hole and cosmological horizons can be infinitesimally deformed. However these do not extend to finite deformations. The corresponding res…
We sharpen the known inequalities $A Λ\le 4π(1-g)$ and $A\ge 4πQ^2$ between the area $A$ and the electric charge $Q$ of a stable marginally outer trapped surface (MOTS) of genus g in the presence of a cosmological constant $Λ$. In particular, instead of requiring stability we include the principal eigenvalue $λ$ of the stability operator. For $Λ^{*} = Λ+ λ> 0$ we obtain a lower and an upper bound for $ Λ^{*} A$ in terms of $ Λ^{*} Q^2$ as well as the upper bound $ Q \le 1/(2\sqrt{Λ^{*}})$ for the charge, which reduces to $ Q \le 1/(2\sqrtΛ)$ in the stable case $λ\ge 0$. For $Λ^{*} < 0$ there r…
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-swallow…
In this paper we survey some recent advances in the analysis of marginally outer trapped surfaces (MOTS). We begin with a systematic review of results by Schoen and Yau on Jang's equation and its relationship with MOTS. We then explain recent work on the existence, regularity, and properties of MOTS and discuss the consequences for the trapped region. We include an outlook with some directions for future research.
We study the location of marginally trapped surfaces in space-times resulting from an axial deformation of static isotropic systems, and show that the Misner-Sharp mass evaluated on the corresponding undeformed spherically symmetric space provides the correct gravitational radius to locate the spheroidal horizon.
We construct an example of a spherically symmetric black hole interior in which there is NO (spherically symmetric) marginally trapped tube asymptotic to the event horizon. The construction uses a self-gravitating massive scalar field matter model, and the key condition we impose is that the scalar field $ϕ$ be bounded below by a positive constant along the event horizon.
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 \emph{stability…
We derive inequalities between the area, the angular momentum and the charges for axisymmetric closed outermost stably marginally outer trapped surfaces, embedded in dynamical and, in general, non-axisymmetric spacetimes satisfying the Einstein-Maxwell-dilaton-matter equations. In proving the inequalities we assume that the dilaton potential is nonnegative and that the matter energy-momentum tensor satisfies the dominant energy condition.
In this paper, we analyse the causal aspects of evolving marginally trapped surfaces in a D-dimensional spherically symmetric spacetime, sourced by perfect fluid with a cosmological constant. The norm of the normal to the marginally trapped tube is shown to be the product of lie derivatives of the expansion parameter of future outgoing null rays along the incoming and outgoing null directions. We obtain a closed form expression for this norm in terms of principal density, pressure, areal radius and cosmological constant. For the case of a homogeneous fluid distribution, we obtain a simple form…
This paper deals with a detail study of gravitational collapse of dust and viscous fluids under the assumptions of spherical symmetry. Our main goal is to closely analyze the horizons which arise during this gravitational phenomenon. To this end, we examine the formation and evolution of trapped surfaces in these spacetimes, with special attention to trapped regions and cylinders foliated by marginally trapped surfaces. The time evolution of trapped surfaces, collapsing shell as well as the event horizon are identified analytically as well as numerically. Using different density profiles of ma…
It is a standard fact that trapped or marginally trapped surfaces are not visible from conformal infinity, under the usual set of conditions on matter fields and the conformal completion, provided that the cosmological constant is non-positive. In this note we show that the situation is more delicate in the presence of a positive cosmological constant: we present examples of visible marginally trapped surfaces, and we provide a set of natural conditions which guarantee non-visibility.
We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initial data is sufficiently small and decays like O(r^{-1/2}), and the potential function V is nonnegative with bounded second derivative. This result can be loosely interpreted as a statement about the stability of `nice' asymptotic behavior of marginally trapped tubes under certain…
When a lump of matter falls into a black hole it is expected that a marginally trapped tube when hit moves outwards everywhere, even in regions not yet in causal contact with the infalling matter. But to describe this phenomenon analytically in 3+1 dimensions is difficult since gravitational radiation is emitted. By considering a particle falling into a toy model of a black hole in 2+1 dimensions an exact description of this non-local behaviour of a marginally trapped tube is found.
The behavior of quasi-local black hole horizons in a binary black hole merger is studied numerically. We compute the horizon multipole moments, fluxes and other quantities on black hole horizons throughout the merger. These lead to a better qualitative and quantitative understanding of the coalescence of two black holes; how the final black hole is formed, initially grows and then settles down to a Kerr black hole. We calculate the rate at which the final black hole approaches equilibrium in a fully non-perturbative situation and identify a time at which the linear ringdown phase begins. Final…
We provide a general formalism that allows to analyze the phenomenon of tunneling in arbitrary spacetimes. We show that a flux of particles produced by tunneling through general marginally trapped surfaces may be perceived by some privileged observers. We discuss how this particle perception can be related to Hawking/Unruh radiation in specific cases. Our approach naturally leads to an expression for the effective surface gravity of marginally trapped surfaces. The procedure is applicable to general astrophysical and cosmological dynamical situations. Some practical examples for known and new …
It has been proposed that the areas of marginally trapped or anti-trapped surfaces (also known as leaves of holographic screens) may encode some notion of entropy. To connect this to AdS/CFT, we study the case of marginally trapped surfaces anchored to an AdS boundary. We establish that such boundary-anchored leaves lie between the causal and extremal surfaces defined by the anchor and that they have area bounded below by that of the minimal extremal surface. This suggests that the area of any leaf represents a coarse-grained von Neumann entropy for the associated region of the dual CFT. We fu…
As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy momentum term. In this note we consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. The "canonical" dynamical horizon in Vaidya spacetime and certain spacelike hypersurfaces in Nariai spacetime provide illustrations of the…
In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. In particular, we prove a splitting result for 3-dimensional initial data sets analogous to a result of Bray, Brendle and Neves [10] concerning a…
Marginally outer trapped surfaces are widely considered as the best quasi-local replacements for event horizons of black holes in General Relativity. However, this equivalence is far from being proved, even in stationary and static situations. In this paper we study an important aspect of this equivalence, namely whether classic uniqueness theorems of static black holes can be extended to static spacetimes containing weakly outer trapped surfaces or not. Our main theorem states that, under reasonable hypotheses, a static spacetime satisfying the null energy condition and containing an asymptot…
The aim of this paper is to collect some facts about the blowup of Jang's equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponential convergence near a strictly stable MOTS.
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 dyna…
Seven different refinements of trapped surfaces are proposed, each intended as potential stability conditions. This article concerns spherical symmetry, but each condition can be generalized. Involute trapped spheres satisfy a similar condition to minimal trapped spheres, which are are strictly minimal with respect to the Kodama vector. There is also a weaker version of involute trapped. Outer trapped spheres have positive surface gravity. Increasingly (future, respectively past) trapped spheres generate spheres which are more trapped in a (future, respectively past) causal direction, with thr…
We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a "no KIDs" condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data containing marginally outer trapped surfaces lead to geodesically incomplete spacetimes.
We prove area inequalities for stable marginally outer trapped surfaces in Einstein-Maxwell-dilaton theory. Our inspiration comes on the one hand from a corresponding upper bound for the area in terms of the charges obtained recently by Dain, Jaramillo and Reiris [1] in the pure Einstein-Maxwell case without symmetries, and on the other hand from Yazadjiev's inequality [2] in the axially symmetric Einstein-Maxwell-dilaton case. The common issue in these proofs and in the present one is a functional ${\mathscr W}$ of the matter fields for which the stability condition readily yields an {\it upp…
We present numerical simulations of a Kerr black hole perturbed by a pulse of ingoing gravitational radiation. For strong perturbations we find up to five concentric marginally outer trapped surfaces. These trapped surfaces appear and disappear in pairs, so that the total number of such surfaces at any given time is odd. The world tubes traced out by the marginally outer trapped surfaces are found to be spacelike during the highly dynamical regime, approaching a null hypersurface at early and late times. We analyze the structure of these marginally trapped tubes in the context of the dynamical…
