Formation Of Trapped Surfaces Research Articles

The fate of a quantum-corrected collapsing star in General Relativity

We incorporate some corrections inspired by loop quantum gravity into the concept of gravitational collapse and propose a complete model of the dynamic process. The model carries the essence of a mass-independent upper bound on the curvature scalars, originally found as a crucial feature of black holes in loop quantum gravity. The quantum-inspired interior is immersed in a geometry filled with null radiation, and they are matched at a distinct boundary hypersurface. The ultimate fate of the process depends on the inhomogeneities of the metric tensor coefficients. We find a critical parameter λ embedded in the inhomogeneity of the conformal factor of the interior metric. Examples with λ< 0 enforce an eventual collapse to singularity, and λ> 0 cases produce a non-singular collapse resulting in a loop-quantum-corrected Schwarzschild geometry modulo a conformal factor. Interestingly, for λ< 0 as well, there exist situations where the quantum effects are able to cause a bounce but fall short of preventing the ultimate formation of a singularity. The trapped surface formation condition is studied for the λ<0 case to infer about the visibility of the final singularity. Interestingly, we find a possibility of the formation of three horizons during the course of the collapse. Eventually, all of them merge into a single horizon, which envelopes the final singularity. For the non-singular case, there is a possibility that the sphere can evolve into a wormhole throat whose radius is found to be inversely proportional to the critical parameter λ. Depending on the nature of evolution and the shell regions, the collapsing shells violate some standard energy conditions, which can be associated with quantum-inspired corrections.

A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

In this study we show that, from arbitrarily dispersed initial data, both the focusing of electromagnetic fields and the concentration of gravitational waves could lead to the formation of trapped surfaces. We establish a scale–critical semi–global existence result from past null infinity for the Einstein–Maxwell system by assigning the signature for decay rates to both geometric quantities and Maxwell components. This result generalizes an approach of the first author in [4] on studying the Einstein vacuum equations to the physical Einstein-Maxwell system by employing additional elliptic estimates and geometric renormalizations and it also extends a result of Yu [18] and serves as the first scale-critical result for Einstein's equations coupled with matter.

A proof of the instability of AdS for the Einstein-massless Vlasov system

In recent years, the conjecture on the instability of Anti-de Sitter spacetime, put forward by Dafermos–Holzegel (Dynamic instability of solitons in 4 + 1 dimesnional gravity with negative cosmological constant, 2006. https://www.dpmms.cam.ac.uk/~md384/ADSinstability.pdf) and Dafermos (The Black Hole Stability problem, Talk at the Newton Institute, Cambridge, 2006. http://www-old.newton.ac.uk/webseminars/pg+ws/2006/gmx/1010/dafermos/) in 2006, has attracted a substantial amount of numerical and heuristic studies. Following the pioneering work (Phys Rev Lett 107(3):031102, 2011) of Bizon–Rostworowski, research efforts have been mainly focused on the study of the spherically symmetric Einstein-scalar field system. The first rigorous proof of the instability of AdS in the simplest spherically symmetric setting, namely for the Einstein-null dust system, was obtained in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). In order to circumvent problems associated with the trivial break down of the Einstein-null dust system occuring at the center r=0, Moschidis (2017) studied the evolution of the system in the exterior of an inner mirror placed at r=r_{0}, r_{0}>0. However, in view of additional considerations on the nature of the instability, it was necessary for Moschidis (2017) to allow the mirror radius r_{0} to shrink to 0 with the size of the initial perturbation; well-posedness in the resulting complicated setup (involving low-regularity estimates of uniform modulus with respect to r_{0}) was obtained in Moschidis (The Einstein-null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability, 2017. arXiv:1704.08685). In this paper, we establish the instability of AdS for the Einstein-massless Vlasov system in spherical symmetry; this will be the first proof of the AdS instability conjecture for an Einstein-matter system which is well-posed for regular initial data in the standard sense, without the addition of an inner mirror. The necessary well-posedness results for this system are obtained in our companion paper (Moschidis in The characteristic initial-boundary value problem for the Einstein-massless Vlasov system in spherical symmetry, 2018. arXiv:1812.04274). Our proof utilises an instability mechanism based on beam interactions which is superficially similar to the one appearing in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). However, new difficulties associated with the Einstein-massless Vlasov system (such as the need for control on the paths of non-radial geodesics in a large curvature regime) will force us to develop a different strategy of proof involving a novel configuration of beam interactions. One of the main novelties of our construction is the introduction of a multi-scale hierarchy of domains in phase space, on which the initial support of the Vlasov field f is localised. The propagation of this hierarchical structure of the support of f along the evolution will be crucial both for controlling the geodesic flow under minimal regularity assumptions and for guaranteeing the existence of the solution until the time of trapped surface formation.

Short-pulse method and trapped surface formation

In this paper, we survey the short-pulse method invented by Christodoulou, with which he proved that focusing of gravitational waves would lead to trapped surface formation. We will also discuss some recent generalizations by the author and collaborators.

Singularity resolution in gravitational collapse

We investigate the unhindered gravitational collapse of a homogeneous scalar field with nonzero potential, a two-dimensional analog of the Mexican hat-shaped Higgs field potential. The collapsing scalar field is surrounded by an exterior retarded (outgoing) generalized Vaidya spacetime. We prove that the density dependence on the scale factor cannot be expressed as an algebraic function in such a scenario. For a certain transcendental expression of the density of such a field as a function of scale factor, we then show that the collapse evolves to a singularity at an infinite comoving time, which is equivalent to saying that the singularity is avoided altogether. An ultra high density region of the order of Planck length can, however, be reached in a finite comoving time. The absence of the formation of trapped surfaces makes this ultra high density region globally visible.

Trapped Surface Formation for Spherically Symmetric Einstein–Maxwell-Charged Scalar Field System with Double Null Foliation

In this paper, we generalize a method introduced by Christodoulou for studying the Einstein-scalar field to prove a trapped surface formation criterion for the Einstein–Maxwell-charged scalar field system under spherical symmetry. If we further require the initial charge to be sufficiently small, we obtain an almost-scale-critical result in the perturbative regime. In the “Appendix”, we also include a proof of Christodoulou’s result on trapped surface formation for the Einstein-scalar field using double null coordinates, as well as a strengthed scale-critical criterion in the case of Minkowskian incoming characteristic initial data.

A Scale-Critical Trapped Surface Formation Criterion: A New Proof Via Signature for Decay Rates

We provide a self-contained proof of a trapped surface formation theorem, which simplifies the previous results by Christodoulou and by An–Luk. Our argument is based on a systematic approach for the scale-critical estimates in An–Luk and it connects Christodoulou’s short-pulse method and Klainerman–Rodnianski’s signature counting argument to the peeling properties previously studied in the small-data regime such as Klainerman–Nicolo. In particular this allows us to avoid elliptic estimates and geometric renormalizations, and gives us new technical improvements and simplifications.

Does the Loop Quantum μo Scheme Permit Black Hole Formation?

We explore the way different loop quantization prescriptions affect the formation of trapped surfaces in the gravitational collapse of a homogeneous dust cloud, with particular emphasis on the so-called μo scheme in which loop quantum cosmology was initially formulated. Its undesirable features in cosmological models led to the so-called improved dynamics or the μ¯ scheme. While the jury is still out on the right scheme for black hole spacetimes, we show that as far as black hole formation is concerned, the μo scheme has another, so far unknown, serious problem. We found that in the μo scheme, no trapped surfaces would form for a nonsingular collapse of a homogeneous dust cloud in the marginally bound case unless the minimum nonzero area of the loops over which holonomies are computed or the Barbero–Immirzi parameter decreases almost four times from its standard value. It turns out that the trapped surfaces in the μo scheme for the marginally bound case are also forbidden for an arbitrary matter content as long as the collapsing interior is isometric to a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We found that in contrast to the situation in the μo scheme, black holes can form in the μ¯ scheme, as well as other lattice refinements with a mass gap determined by quantum geometry.

What is a reduced boundary in general relativity?

The concept of boundary plays an important role in several branches of general relativity, e.g. the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped surfaces. On the other hand, in a branch of mathematics known as geometric measure theory, the usefulness has been discovered long ago of yet another concept, i.e. the reduced boundary of a finite-perimeter set. This paper proposes therefore a definition of finite-perimeter sets and their reduced boundary in general relativity. Moreover, a basic integral formula of geometric measure theory is evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry for the first time in the literature. This research prepares the ground for a measure-theoretic approach to several concepts in gravitational physics, supplemented by geometric insight. Moreover, such an investigation suggests considering the possibility that the in–out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole.

Marginally trapped surfaces in spherical gravitational collapse

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 matter, we analyze, how the nature of the marginally trapped surfaces modify as we change the energy momentum tensor. These studies reveal that depending on the mass function and the mass profile, it is possible to envisage situations where dynamical horizons, timelike tubes or isolated horizons may arise.

The formation of trapped surfaces in the gravitational collapse of spherically symmetric scalar fields with a positive cosmological constant

Given spherically symmetric characteristic initial data for the Einstein-scalar field system with a positive cosmological constant, we provide a criterion, in terms of the dimensionless size and dimensionless renormalized mass content of an annular region of the data, for the formation of a future trapped surface. This corresponds to an extension of Christodoulou’s classical criterion by the inclusion of the cosmological term.

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.

A Construction of Collapsing Spacetimes in Vacuum

In this paper, we construct a class of collapsing spacetimes in vacuum without any symmetries. The spacetime contains a black hole region which is bounded from the past by the future event horizon. It possesses a Cauchy hypersurface with trivial topology which is located outside the black hole region. Based on existing techniques in the literature, the spacetime can in principle be constructed to be past geodesically complete and asymptotic to Minkowski space. The construction is based on a semi-global existence result of the vacuum Einstein equations built on a modified version of the a priori estimates that were originally established by Christodoulou in his work on the formation of trapped surface, and a gluing construction carried out inside the black hole. In particular, the full detail of the a priori estimates needed for the existence is provided, which can be regarded as a simplification of Christodoulou's original argument.

Higher-dimensional charged LTB collapse in f(R) gravity

This paper deals with the study of the gravitational collapse of a charged perfect fluid in n+2 dimensions in f(R) gravity. The Darmois matching criterion is used to formulate the junction conditions between the Lemaitre-Tolman-Bondi metric (interior) and Reissner-Nordstrom metric (exterior) in n+2 dimensions while Senovilla junction conditions for f(R) gravity are also considered. The field equations are solved for the constant curvature which implies that matter variables are constant quantities. The expression for the time difference between the formation of trapped surfaces and singularity is established which supports the cosmic censorship hypothesis. We conclude that the electromagnetic field affects this time difference by reducing the repulsive effect of f R0) term. It is also found that the collapse rate of the charged fluid is faster as compared to the perfect fluid.

The intersection of a hyperplane with a lightcone in the Minkowski spacetime

Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green's function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.

Radiative gravitational collapse to spherical, toroidal and higher genus black holes

We derive the matching conditions between FLRW and generalised Vaidya spacetimes with spherical, planar or hyperbolic symmetry, across timelike hypersurfaces. We then construct new models of gravitational collapse of FLRW spacetimes with a negative cosmological constant having electromagnetic radiation in the exterior. The final state of the collapse are asymptotically AdS black holes with spherical, toroidal or higher genus topologies. We analyse the collapse dynamics including trapped surface formation, for various examples.

Cylindrically symmetric inhomogeneous dust collapse with a zero expansion component

We investigate a class of cylindrically symmetric inhomogeneous Λ-dust spacetimes which have a regular axis and some zero expansion component. For , we obtain new exact solutions to the Einstein equations and show that they are unique, within that class. For , we recover the Senovilla–Vera metric and show that it can be locally matched to an Einstein–Rosen type of exterior. Finally, we explore some consequences of the matching, such as trapped surface formation and gravitational radiation in the exterior.

Brown-York mass and the hoop conjecture in nonspherical massive systems

We discuss the relation between the concentration of the Brown-York mass and the formation of trapped surfaces in non-spherical massive systems. In particular, we formulate and prove a precise version of the Thorne hoop conjecture in conformally flat three-geometries sliced by equipotential foliation leaves. An intriguing relationship between the total rest mass and the Brown-York mass is shown. This is a further investigation of the previous work on the Brown-York mass hoop conjecture in spherical symmetry.

Holographic thermalization in a quark confining background

We study holographic thermalization of a strongly coupled theory inspired by two colliding shock waves in a vacuum confining background. Holographic thermalization means a black hole formation, in fact a trapped surface formation. As a vacuum confining background we considered a well know bottom-up AdS/QCD model that provides the Cornell potential as well as reproduces QCD beta-function. We perturb vacuum background by colliding domain shock waves, that are assumed to be holographically dual to heavy ions collisions. Our main physical assumption is that we can make a restriction on the time of a trapped surface production that makes a natural limitation on the size of the domain where the trapped surface is produced. This limits the intermediate domain where the main part of the entropy is produced. In this domain one can use an intermediate vacuum background as an approximation to the full confining background. In this intermediate background a dependence of the produced entropy on colliding energy is very similar to the experimental dependence of particles multiplicities on colliding ions energy obtained from RHIC and LHC. This permits us to conclude that the entropy produced in collisions of domain shock waves during a short time models rather well the experimental data.

Einstein–Cartan gravitational collapse of a homogeneous Weyssenhoff fluid

We consider the gravitational collapse of a spherically symmetric homogeneous matter distribution consisting of a Weyssenhoff fluid in the presence of a negative cosmological constant. Our aim is to investigate the effects of torsion and spin averaged terms on the final outcome of the collapse. For a specific interior spacetime setup, namely the homogeneous and isotropic FLRW metric, we obtain two classes of solutions to the field equations where depending on the relation between spin source parameters, $(i)$ the collapse procedure culminates in a spacetime singularity or $(ii)$ it is replaced by a non-singular bounce. We show that, under certain conditions, for a specific subset of the former solutions, the formation of trapped surfaces is prevented and thus the resulted singularity could be naked. The curvature singularity that forms could be gravitationally strong in the sense of Tipler. Our numerical analysis for the latter solutions shows that the collapsing dynamical process experiences four phases, so that two of which occur at the pre-bounce and the other two at post-bounce regimes. We further observe that there can be found a minimum radius for the apparent horizon curve, such that the main outcome of which is that there exists an upper bound for the size of the collapsing body, below which no horizon forms throughout the whole scenario.