6 pages, no figures. New references included, typos corrected, explanatory comments added. Much shorter version, in order to match EPL length restrictions. To be published in EPL
A reformulation of the Hoop Conjecture based on the concept of trapped circle is presented. The problems of severe compactness in every spatial direction, and of how to superpose the hoops with the surface of the black hole, are resolved. A new conjecture concerning "peeling" properties of dynamical/trapping horizons is propounded. A novel geometric Hoop inequality is put forward. The possibility of carrying over the results to arbitrary dimension is discussed.
The hoop conjecture is well confirmed in momentarily static spaces, but it has not been investigated systematically for the system with relativistic motion. To confirm the hoop conjecture for non-time-symmetric initial data, we consider the initial data of two colliding black holes with momentum and search an apparent horizon that encloses two black holes. In testing the hoop conjecture, we use two definitions of gravitational mass : one is the ADM mass and the other is the quasi-local mass defined by Hawking. Although both definitions of gravitational mass give fairly consistent picture of th…
By analyzing the apparent horizon (AH) formation in the collision of two pp-waves with rectangular sources in four dimensions, we study to what extent the AH can be distorted without violating the energy conditions. It is shown that the highly distorted AH can form in this system although it cannot be arbitrarily long. The hoop conjecture is examined for the formation of such highly distorted AHs, and our result gives a strong support to the hoop conjecture. We also point out the possible relation between the AH topology theorem and the hoop conjecture.
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.
The famous hoop conjecture by Thorne has been claimed to be\ violated in curved spacetimes coupled to linear electrodynamics. Hod \cite{Hod:2018} has recently refuted this claim by clarifying the status and validity of the conjecture appropriately interpreting the gravitational mass parameter $M$. However, it turns out that partial violations of the conjecture might seemingly occur also in the well known regular curved spacetimes of gravity coupled to \textit{nonlinear electrodynamic}s. Using the interpretation of $M$ in a generic form accommodating nonlinear electrodynamic coupling, we illust…
Gravitational collapse of radiation in an anti-de Sitter background is studied. For the spherical case, the collapse proceeds in much the same way as in the Minkowski background, i.e., massless naked singularities may form for a highly inhomogeneous collapse, violating the cosmic censorship, but not the hoop conjecture. The toroidal, cylindrical and planar collapses can be treated together. In these cases no naked singularity ever forms, in accordance with the cosmic censorship. However, since the collapse proceeds to form toroidal, cylindrical or planar black holes, the hoop conjecture in an …
I propose a sharp form of Thorne's hoop conjecture which relates Birkhoff's invariant $β$ for an outermost apparent horizon to its $ADM$ mass, $ β\le 4 πM_{ADM}$. I prove the conjecture in the case of collapsing null shells and provide further evidence from exact rotating black hole solutions. Since $β$ is bounded below by the length $l$ of the shortest non-trivial geodesic lying in the apparent horizon, the conjecture implies $l \le 4 πM_{ADM}$. The Penrose conjecture, $\sqrt{πA} \le 4 πM_{ADM}$, and Pu's theorem imply this latter consequence for horizons admitting an antipodal isometry. Quit…
Recently, with numerical methods, Hod clarified the validity of Thorne hoop conjecture for spatially regular static charged fluid spheres, which were considered as counterexamples against the hoop conjecture. In this work, we provide an analytical proof on Thorne hoop conjecture in the spatially regular static charged fluid sphere spacetimes.
We numerically investigated the sequences of initial data of thin spindle and thin ring in five-dimensional space-time in the context of the cosmic censorship conjecture. We modeled the matter in non-rotating homogeneous spheroidal or toroidal configurations under the momentarily static assumption, solved the Hamiltonian constraint equation, and searched the apparent horizons. We discussed when $S^3$ (black hole) or $S^1 \times S^2$ (black ring) horizons ("black objects") are formed. By monitoring the location of the maximum Kretchmann invariant, an appearance of `naked singularity' or `naked …
We analyze momentarily static initial data sets of the gravitational field produced by two-point sources in five-dimensional Kaluza-Klein spacetimes. These initial data sets are characterized by the mass, the separation of sources and the size of a extra dimension. Using these initial data sets, we discuss the condition for black hole formation, and propose a new conjecture which is a hybrid of the four-dimensional hoop conjecture and the five-dimensional hyperhoop conjecture. By using the new conjecture, we estimate the cross section of black hole formation due to collisions of particles in K…
We investigate the validity of Thorne's hoop conjecture in non-axisymmetric spacetimes by examining the formation of apparent horizons numerically. If spaces have a discrete symmetry about one axis, we can specify the boundary conditions to determine an apparent horizon even in non-axisymmetric spaces. We implement, for the first time, the ``hoop finder'' in non-axisymmetric spaces with a discrete symmetry. We construct asymptotically flat vacuum solutions at a moment of time symmetry. Two cases are examined: black holes distributed on a ring, and black holes on a spherical surface. It turns o…
The Thorne hoop conjecture is an attempt to make precise the notion that gravitational collapse occurs if enough energy is compressed into a small enough volume, with the `size' being defined by the circumference. We can make a precise statement of this form, in spherical symmetry, using the Brown-York mass as our measure of the energy. Consider a spherical 2-surface in a spherically symmetric spacetime. If the Brown-York mass, $M_{BY}$, and the circumference, $C$, satisfy $C < 2πM_{BY}$, then the system must either have emerged from a white hole or will collapse into a black hole. We show tha…
For horizonless spherical stars with uniform charge density, the hoop conjecture was tested based on the interior solution. In this work, we are interested in more general horizonless spherical charged stars. We test hoop conjecture using the exterior solution since all types of interior solutions correspond to the same exterior Reissner-Nordsr$\ddot{o}$m solution. Our analysis shows that the hoop conjecture is violated for very compact stars if we express the conjecture with the total ADM mass. And the hoop conjecture holds if we express the conjecture using the mass in the sphere.
We show that four-dimensional Lorentzian metrics admitting a global spacelike Lie group of isometries, $G_{1}={\mathbb R}$, which obey the Einstein equations for vacuum and certain types of matter, cannot contain apparent horizons. The assumed global isometry allows for the dimensional reduction of the (3+1) system to a (2+1) picture, wherein the four-dimensional metric fields act formally as matter fields. A theorem by Ida allows one to check for the absence of apparent horizons in the dimensionally reduced spacetime, with the four-dimensional results following from the topological product na…
We study the condition of black hole formation in five-dimensional space-time. We analytically solve the constraint equations of five-dimensional Einstein equations for momentarily static and conformally flat initial data of a spheroidal mass. We numerically search for an apparent horizon in various initial hypersurfaces and find both necessary and sufficient conditions for the horizon formation in terms of inequalities relating a geometric quantity and a mass defined in an appropriate manner. In the case of infinitely thin spheroid, our results suggest a possibility of naked singularity forma…
We propose a unified version of hoop conjecture valid for various black holes and horizonless compact stars. This conjecture is expressed by the mass to circumference ratio $4πM_{in}/C\leqslant 1$, where C is the circumference of the smallest ring that can engulf the object in all azimuthal directions and $M_{in}$ is the mass within the engulfing sphere.
A precise formulation of the hoop conjecture for four-dimensional spacetimes proposes that the Birkhoff invariant βfor an apparent horizon in a spacetime with mass M should satisfy β\le 4πM. The invariant βis the least maximal length of any sweepout of the 2-sphere apparent horizon by circles. An analogous conjecture in five spacetime dimensions was recently formulated, asserting that the Birkhoff invariant βfor S^1\times S^1 sweepouts of the apparent horizon should satisfy β\le (16/3)πM. Although this hoop inequality was formulated for conventional five-dimensional black holes with 3-sphere h…
We investigate the apparent horizon formation for high-energy head-on collisions of particles in multi-dimensional spacetime. The apparent horizons formed before the instance of particle collision are obtained analytically. Using these solutions, we discuss the feature of the apparent horizon formation in the multi-dimensional spacetime from the viewpoint of the hoop conjecture.
According to a variant of the hoop conjecture, if we localize two particles within the Schwarzschild radius corresponding to their center of mass energy, then a black hole will form. Despite a large body of work on the formation of primordial black holes, so far this conjecture has not been generalized to expanding spacetimes. We derive a formula which gives the distance within which two particles must be localized to give a black hole, and which crucially depends on the expansion rate of the background space. In the limit of a very slow expansion, we recover the flat spacetime case. In the op…
It is conjectured that stationary black holes are characterized by the inverse hoop relation ${\cal A}\leq {\cal C}^2/π$, where ${\cal A}$ and ${\cal C}$ are respectively the black-hole surface area and the circumference length of the smallest ring that can engulf the black-hole horizon in every direction. We explicitly prove that generic Kerr-Newman-(anti)-de Sitter black holes conform to this conjectured area-circumference relation.
We propose here a quantum hoop conjecture which states: the de Broglie wavelength of a quantum system cannot be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the quantum system. Based on this conjecture, we find an upper bound for the wave number (or the momentum) of a particle, which offers a natural cutoff for the vacuum energy of a scalar field.
Testing generalized spacetimes for black holes using the Hod function representation of the hoop conjecture
Zel’dovich International Center for Astrophysics, M. Akmullah Bashkir State Pedagogical University（泽尔多维奇天体物理学国际中心，M.阿克穆拉巴什基尔国立师范大学）；Department of Physics & Astronomy, Bashkir State University（巴什基尔国立大学物理与天文系）；Salavat Industrial College（萨拉瓦特工业学院）
The initial data of gravity for a cylindrical matter distribution confined on the brane is studied in the framework of the single brane Randall-Sundrum scenario. In this scenario, 5-dimensional aspect of gravity appears in the short range gravitational interaction. We found that the sufficiently thin configuration of matter leads to the formation of the marginal surface even if the configuration is infinitely long. This means that the hoop conjecture proposed by Thorne does not hold in the Randall-Sundrum scenario; Even if a mass $M$ does not get compacted into a region whose circumference in …
We address the issue of (quantum) black hole formation by particle collision in quantum physics. We start by constructing the horizon wave-function for quantum mechanical states representing two highly boosted non-interacting particles that collide in flat one-dimensional space. From this wave-function, we then derive a probability that the system becomes a black hole as a function of the initial momenta and spatial separation between the particles. This probability allows us to extend the hoop conjecture to quantum mechanics and estimate corrections to its classical counterpart.
On the Kalb-Ramond modified Lorentz violating hairy black holes and Thorne's hoop conjecture
Zel’dovich International Center for Astrophysics, Bashkir State Pedagogical University（泽尔多维奇天体物理学国际中心，巴什基尔国立师范大学）；High Energy Cosmic Ray Research Center, University of North Bengal（北孟加拉大学高能宇宙射线研究中心）；Ufa University of Science & Technology, Sterlitamak Campus（乌法科学与技术大学，斯特利塔马克校区）
We consider a closed region $R$ of 3d quantum space modeled by $SU(2)$ spin-networks. Using the concentration of measure phenomenon we prove that, whenever the ratio between the boundary $\partial R$ and the bulk edges of the graph overcomes a finite threshold, the state of the boundary is always thermal, with an entropy proportional to its area. The emergence of a thermal state of the boundary can be traced back to a large amount of entanglement between boundary and bulk degrees of freedom. Using the dual geometric interpretation provided by loop quantum gravity, we interprete such phenomenon…
The hoop conjecture, introduced by Thorne almost five decades ago, asserts that black holes are characterized by the mass-to-circumference relation $4π{\cal M}/{\cal C}\geq1$, whereas horizonless compact objects are characterized by the opposite inequality $4π{\cal M}/{\cal C}<1$ (here ${\cal C}$ is the circumference of the smallest ring that can engulf the self-gravitating compact object in all azimuthal directions). It has recently been proved that a necessary condition for the validity of this conjecture in horizonless spacetimes of spatially regular charged compact objects is that the mass…
Assuming the hoop conjecture in classical general relativity and quantum mechanics, any observer who attempts to perform an experiment in an arbitrarily small region will be stymied by the formation of a black hole within the spatial domain of the experiment. This behavior is often invoked in arguments for a fundamental minimum length. Extending a proof of the hoop conjecture for spherical symmetry to include higher curvature terms we investigate this minimum length argument when the gravitational couplings run with energy in the manner predicted by asymptotically safe gravity. We show that ar…
A dynamically transversely trapping surface (DTTS) is a new concept of an extension of a photon sphere that appropriately represents a strong gravity region and has close analogy with a trapped surface. We study formation of a marginally DTTS in time-symmetric, conformally flat initial data with two black holes, with a spindle-shaped source, and with a ring-shaped source, and clarify that $\mathcal{C}\lesssim 6πGM$ describes the condition for the DTTS formation well, where $\mathcal{C}$ is the circumference and $M$ is the mass of the system. This indicates that an understanding analogous to th…
The initial data of gravity for a cylindrical matter distribution confined to a brane are studied in the framework of the single-brane Randall-Sundrum scenario. In this scenario, the 5-dimensional nature of gravity appears in the short-range gravitational interaction. We find that a sufficiently thin configuration of matter leads to the formation of a marginal surface, even if the configuration is infinitely long. This implies that the hoop conjecture proposed by Thorne does not hold on the brane: Even if a mass $M$ does not become compacted into a region whose circumference ${\cal C}$ in ever…
We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any auxiliary assumptions such as maximality, time-symmetry, or special extrinsic foliations, and most importantly they can easily be generalized to the nonspherical case once an existence theory for a modified version of the Jang equation is developed. Moreover, our methods also yield…
A recent precise formulation of the hoop conjecture in four spacetime dimensions is that the Birkhoff invariant $β$ (the least maximal length of any sweepout or foliation by circles) of an apparent horizon of energy $E$ and area $A$ should satisfy $β\le 4 πE$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies $\ell^2 \le πA$. We have tested these conjectures on the horizons of all four-charged rotating black hole solutions of ungauged supergravity theories and find that they alway…
Bekenstein's generalized second law (GSL) of thermodynamics asserts that the sum of black-hole entropy, $S_{\text{BH}}=Ac^3/4\hbar G$ (here $A$ is the black-hole surface area), and the ordinary entropy of matter and radiation fields in the black-hole exterior region never decreases. We here re-analyze an intriguing gedanken experiment which was designed by Bekenstein to challenge the GSL. In this historical gedanken experiment an entropy-bearing box is lowered into a charged Reissner-Nordström black hole. For the GSL to work, the resulting increase in the black-hole surface area (entropy) must…
In this letter we provide a reformulation of the strong cosmic censor conjecture taking into account recent results on Malament--Hogarth space-times. We claim that the strong version of the cosmic censor conjecture can be formulated by postulating that a physically relevant space-time is either globally hyperbolic or possesses the Malament--Hogarth property. But it is known that a Malament--Hogarth space-time in principle is capable for performing non-Turing computations such as checking consistency of ZFC set theory. In this way we get an intimate conjectured link between the cosmic censorshi…
A brief, and certainly not exhaustive, survey is provided of some recent results and conjectures in four and higher spacetime dimensions, such as the Hoop Conjecture, relating the geometry of event horizons to dynamical quantities such as the total energy or mass of the spacetime containng the black hole. As a measure of the size of a hoop one may take Birkhoff's invariant based on sweepouts by circles or higher dimensional analogues or one may take the length of the shortest non-trivial closed geodesic. Also discussed are whether one can hear the shape of a black hole and to what extent one m…
Using Thorne's hoop conjecture, it is argued that the energies of the Hawking quanta emitted from canonical Schwarzschild black holes are bounded from above by the simple quantum relation ${\cal E}<{\cal E}_{\text{max}}={\hbar^{2/3}}/M^{1/3}$. In particular, it is shown that, due to non-linear (self-gravity) effects of the tunneling quanta, higher energy field modes are re-absorbed (rather than escape to infinity) by the black hole.
