Burkhard Kleihaus, Jutta Kunz, Eugen Radu
We construct balanced black ring solutions in $d\geq 6$ spacetime dimensions, by solving the Einstein field equations numerically with suitable boundary conditions. The black ring solutions have a regular event horizon with $S^1\times S^{d-3}$ topology, and they approach the Minkowski background asymptotically. We analyze their global and horizon properties. The Smarr formula is well satisfied.
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We study axisymmetric self-gravitating non-linear $σ$-model in five-dimensional spacetime admitting three commutating Killing vector fields. We show that the only asymptotically flat black ring solution with a regular event horizon is the black ring characterized by mass and two angular momenta with constant mapping.
Using a solution generating technique, we derive a new exact solution describing a charged static black ring on the Taub-bolt gravitational instanton in five dimensions. Unlike the black ring constructed on the self-dual Taub-NUT instanton, it turns out that it is possible to find values of the parameters for which the static black ring is in equilibrium and the conical singularities disappear. We compute its conserved charges and discuss some of its thermodynamic properties.
Using the solitonic solution-generating technique we rederived the one-rotational five-dimensional black ring solution found by Emparan and Reall. The seed solution is not the Minkowski metric, which is the seed of $S^2$-rotating black ring. The obtained solution has more parameters than the Emparan and Reall's $S^1$-rotating black ring. We found the conditions of parameters to reduce the solution to the $S^1$-rotating black ring. In addition we examined the relation between the expressions of the metric in the prolate-spheroidal coordinates and in the canonical coordinates.
We present balanced black ring solutions of pure Einstein-Maxwell theory in five dimensions. The solutions are asymptotically flat, and their tension and gravitational self-attraction are balanced by the repulsion due to rotation and electrical charge. Hence the solutions are free of conical singularities and possess a regular horizon which exhibits the topology S1 x S2 of a torus. We discuss the global charges and the horizon properties of the solutions and show that they satisfy a Smarr relation. We construct these black rings numerically, restricting to the case of black rings with a rotati…
Previously the five dimensional $S^1$-rotating black rings have been superposed in a concentric way by some solitonic methods, and regular systems of two $S^1$-rotating black rings were constructed by the authors and then Evslin and Krishnan (we called these solutions "black di-rings"). In this place we show some characteristics of the solutions of five dimensional black di-rings, especially in thermodynamic equilibrium. After the summary of the di-ring expressions and their physical quantities, first we comment on the equivalence of the two different solution sets of the black di-rings. Then …
A black ring is a five-dimensional black hole with an event horizon of topology S1 x S2. We provide an introduction to the description of black rings in general relativity and string theory. Novel aspects of the presentation include a new approach to constructing black ring coordinates and a critical review of black ring microscopics.
Using a solution generating technique based on the symmetries of the dimensionally reduced Lagrangian we derive two exact solutions of the Einstein-Maxwell-Dilaton field equations in five dimensions. More specifically, these solutions describe general systems of non-extremally charged static asymptotically flat black bi/di-ring. We compute their conserved charges and investigate some of their properties. As expected, we find that in general there are conical singularities in between the black rings and for the static solutions considered here they cannot be completely eliminated.
We study non-axisymmetric linearised gravitational perturbations of the Emparan-Reall black ring using numerical methods. We find an unstable mode whose onset lies within the "fat" branch of the black ring and continues into the "thin" branch. Together with previous results using Penrose inequalities that fat black rings are unstable, this provides numerical evidence that the entire black ring family is unstable.
Singly-spinning Myers-Perry black holes in d>5 spacetime dimensions are unstable for sufficiently large angular momentum. We numerically construct (in d=6 and d=7) two new stationary branches of lumpy (rippled) black hole solutions which bifurcate from the onset of this ultraspinning instability. We give evidence that one of these branches connects through a topology-changing merger to black ring solutions which we also construct numerically. The other branch approaches a solution with large curvature invariants. We are also able to compare the d=7 ring solutions with results from finite-size …
We examine the dynamics of neutral black rings, and identify and analyze a selection of possible instabilities. We find the dominating forces of very thin black rings to be a Newtonian competition between a string-like tension and a centrifugal force. We study in detail the radial balance of forces in black rings, and find evidence that all fat black rings are unstable to radial perturbations, while thin black rings are radially stable. Most thin black rings, if not all of them, also likely suffer from Gregory-Laflamme instabilities. We also study simple models for stability against emission/a…
We present the general black ring solution in $U(1)^{3}$ supergravity in 5 dimensions with three independent dipole and electric charges. This immediately gives the general black ring solution in the minimal 5D supergravity as well.
We construct dynamical black ring solutions in the five dimensional Einstein-Maxwell system with a positive cosmological constant and investigate the geometrical structure. The solutions describe the physical process such that a thin black ring at early time shrinks and changes into a single black hole as time increase. We also discuss the multi-black rings and the coalescence of them.
In this paper, we investigate a spinning black ring and a charged black ring in the context of gravity's rainbow. By incorporating rainbow functions proposed by Amelino-Camelia, et al. in [arXiv:hep-th/9605211, arXiv:0806.0339v2] in the metric of the black rings, a considerable modification happens to their thermodynamical properties. We calculate corrections to the temperature, entropy and heat capacity of the black rings. These calculations demonstrate that the behavior of Hawking radiation changes considerably near the Planck scale in gravity's rainbow, where it is shown that black rings do…
The vacuum Einstein equations in five dimensions are shown to admit a solution describing an asymptotically flat spacetime regular on and outside an event horizon of topology S^1 x S^2. It describes a rotating ``black ring''. This is the first example of an asymptotically flat vacuum solution with an event horizon of non-spherical topology. There is a range of values for the mass and angular momentum for which there exist two black ring solutions as well as a black hole solution. Therefore the uniqueness theorems valid in four dimensions do not have simple higher dimensional generalizations. I…
We study bound orbits of a free particle around a singly rotating black ring. We find there exists chaotic motion of a particle which is gravitationally bound to the black ring by using the Poincare map.
We solve the recently-proposed equations describing non-BPS extremal multi-center configurations, and construct explicit solutions describing non-supersymmetric extremal black rings in Taub-NUT, as well as the seed solution for the most general extremal non-BPS under-rotating black hole in four dimensions. We also find solutions that contain both a black hole and a black ring, which descend to four-dimensional extremal non-BPS two-center black holes with generic charges.
The geodesic equation in the five-dimensional singly rotating black ring is non-integrable unlike the case of the Myers-Perry black hole. In the Newtonian limit of the black ring, its geodesic equation agrees with the equation of motion of a particle in the Newtonian potential due to a homogeneous ring gravitational source. In this paper, we show that the Newtonian equation of motion allows the separation of variables in the spheroidal coordinates, providing an non-trivial constant of motion quadratic in momenta. This shows that the Newtonian limit of a black ring recovers the symmetry of its …
We construct a five dimensional exact solution of the orthogonal black di-ring which has two black rings whose $S^1$-rotating planes are orthogonal. This solution has four free parameters which represent radii of and speeds of $S^1$-rotation of the black rings. We use the inverse scattering method. This method needs the seed metric. We also present a systematic method how to construct a seed metric. Using this method, we can probably construct other solutions having many black rings on the two orthogonal planes with or without a black hole at the center.
We propose a general framework for the study of asymptotically flat black objects with $k+1$ equal magnitude angular momenta in $d\geq 5$ spacetime dimensions (with $0\leq k\leq \big[\frac{d-5}{2} \big]$). In this approach, the dependence on all angular coordinates but one is factorized, which leads to a codimension-two problem. This framework can describe black holes with spherical horizon topology, the simplest solutions corresponding to a class of Myers-Perry black holes. A different set of solutions describes balanced black objects with $S^{n+1} \times S^{2k+1}$ horizon topology. The simpl…
We analyze the energy extraction by the Penrose process in higher dimensions. Our result shows the efficiency of the process from higher dimensional black holes and black rings can be rather high compared with than that in four dimensional Kerr black hole. In particular, if one rotation parameter vanishes, the maximum efficiency becomes infinitely large because the angular momentum is not bounded from above. We also apply a catastrophe theory to analyze the stability of black rings. It indicates a branch of black rings with higher rotational energy is unstable, which should be a different type…
Black rings are the black objects found in $D$ spacetime dimensional gravity when $D \geq 5$. These have event horizon topology $S^{D-3}\times S^1$. In this work the solutions of the large $D$ membrane paradigm dual to stationary black rings in Einstein-Maxwell theory with or without cosmological constant are studied. It is shown that the first order membrane equations can only admit static asymptotically flat black rings, and the equilibrium angular velocity for the asymptotically AdS black rings at large $D$ was obtained. The thermodynamic and dynamic stability of the asymptotically flat bla…
Configurations of multiple concentric black rings play an important role in determining the pattern of branchings, connections and mergers between different phases of higher-dimensional black holes. We examine them using both approximate and (in five dimensions) exact methods. By identifying the role of the different scales in the system, we argue that it is possible to have multiple black ring configurations in which all the rings have equal temperature and angular velocity. This allows us to correct and improve in a simple, natural manner, an earlier proposal for the phase diagram of singly-…
Black objects lose their mass and angular momenta through evaporation by Hawking radiation, and the investigation of their time evolution has a long history. In this paper, we study this problem for a five-dimensional doubly spinning black ring. The black ring is assumed to emit only massless scalar particles. We consider a thin black ring with a small thickness parameter, $λ\ll 1$, which can be approximated by a boosted Kerr string locally. We show that a thin black ring evaporates with fixing its thickness parameter $λ$. Further, in the case of an Emparan-Reall black ring, we derive analytic…
We initiate the study of dynamical instabilities of higher-dimensional black holes using the blackfold approach, focusing on asymptotically flat boosted black strings and singly-spinning black rings in $D\ge5$. We derive novel analytic expressions for the growth rate of the Gregory-Laflamme instability for boosted black strings and its onset for arbitrary boost parameter. In the case of black rings, we study their stability properties in the region of parameter space that has so far remained inaccessible to numerical approaches. In particular, we show that very thin (ultraspinning) black rings…
The main focus of this session was the presentation of new higher-dimensional black hole solutions, including black rings, black strings, and multi black holes, and the study of their properties. Besides new asymptotically flat and locally asymptotically flat black objects also new black holes with anti-de Sitter asymptotics were reported. The studies of their properties included the investigation of their stability, their thermodynamics, their analyticity and their existence. Furthermore, the geodesics in such higher-dimensional space-times were investigated.
We show how a recently discovered black ring solution with a rotating 2-sphere can be turned into two new solutions of Einstein-Maxwell-dilaton theory. The first is a four-dimensional solution describing a pair of oppositely charged, extremal black holes--known as a black dihole--undergoing uniform acceleration. The second is a five-dimensional solution describing a pair of concentric, static extremal black rings carrying opposite dipole charges--a so-called black di-ring. The properties of both solutions, which turn out to be formally very similar, are analyzed in detail. We also present, in …
In this paper, we investigate the holographic descriptions of two kinds of black rings, the neutral doubly rotating black ring and the dipole charged black ring. For generic nonextremal black rings, the information of holographic CFT duals, including the central charges and left- and right-moving temperatures, could be read from the thermodynamics at the outer and inner horizons, as suggested in arXiv:1206.2015. To confirm these pictures, we study the extreme black rings in the well-established formalism. We compute the central charges of dual CFTs by doing asymptotic symmetry group analysis i…
All known five dimensional, asymptotically flat, static black rings possess conical singularities. However, there is no fundamental obstruction forbidding the existence of balanced configurations, and we show that the Einstein--Klein-Gordon equations admit (numerical) solutions describing static asymptotically flat black rings, which are regular on and outside the event horizon. The scalar field is 'phantom', which creates the self-repulsion necessary to balance the black rings. Similar solutions are likely to exist in other spacetime dimensions, the basic properties of a line element describi…
We study the black ring solution in the Einstein-Gauss-Bonnet (EGB) theory at large $D$. By using the $1/D$ expansion in the near horizon region we derive the effective equations for the slowly rotating black holes in the EGB theory. The effective equations describe the non-linear dynamics of various stationary solutions, including the EGB black ring, the slowly rotating EGB black hole and the slowly boosted EGB black string. By different embeddings we construct these stationary solutions explicitly. By performing the perturbation analysis of the effective equations, we obtain the quasinormal …
We derive Hawking radiation of 5-dimensional black rings from gauge and gravitational anomalies using the method proposed by Robinson and Wilczek. We find as in the black hole case, the problem could reduce to a (1+1) dimensional field theory and the anomalies result in correct Hawking temperature for neutral,dipole and charged black rings.
We construct a supergravity solution describing a charged rotating black ring with S^2xS^1 horizon in a five dimensional asymptotically flat spacetime. In the neutral limit the solution is the rotating black ring recently found by Emparan and Reall. We determine the exact value of the lower bound on J^2/M^3, where J is the angular momentum and M the mass; the black ring saturating this bound has maximum entropy for the given mass. The charged black ring is characterized by mass M, angular momentum J, and electric charge Q, and it also carries local fundamental string charge. The electric charg…
We discuss geometrical properties of the horizon surface of five-dimensional rotating black holes and black rings. Geometrical invariants characterizing these 3D geometries are calculated. We obtain a global embedding of the 5D rotating black horizon surface into a flat space. We also describe the Kaluza-Klein reduction of the black ring solution (along the direction of its rotation) which relates this solution to the 4D metric of a static black hole distorted by the presence of external scalar (dilaton) and vector (`electromagnetic') field. The properties of the reduced black hole horizon and…
We study the effective theory of slowly rotating black holes at the infinite limit of the spacetime dimension D. This large D effective theory is obtained by integrating the Einstein equation with respect to the radial direction. The effective theory gives equations for non-linear dynamical deformations of the slowly rotating black hole as effective equations. The effective equations contain the slowly rotating Myers-Perry black hole, slowly boosted black string, non-uniform black string and black ring as stationary solutions. We obtain an analytic solution of the black ring by solving the eff…
The gravitational index of a small black ring
本文为五维弦理论中的小黑色环提出了一个引力路径积分鞍点解，该解在五维两个独立平面上旋转并具有有限面积事件视界，可写成三中心Bena-Warner解，用于计算超对称指标。
In this paper, the thermodynamic properties of a thin black ring in AdS space-time is explored when the size of the ring is comparable to quantum scales. The angular momentum to mass ratio of this system has an upper limit, which is the cosmological radius of the black ring. It is found that the small black ring will be thermodynamically stable due to the effects introduced by thermal fluctuations. However, we find that the black ring is less stable than thermal AdS. Thermodynamic analysis indicates that there is no critical point, but there is Hawking-Page transition to radiation, which is co…
We consider the problem of physical process version of the first law of black ring thermodynamics in n-dimensional Einstein gravity with additional (p+1)-form field strength and dilaton fields. The first order variations of mass, angular momentum and local charge for black ring was derived. By means of them we prove the physical process version of the first law of thermodynamics for stationary black rings.
The physical process version and the equilibrium state version of the first law of black ring thermodynamics in n-dimensional Einstein gravity with Chern-Simons term were derived. This theory constitutes the simplest generalization of the five-dimensional one admitting a stationary black ring solutions. The equilibrium state version of the first law of black ring mechanics was achieved by choosing any cross section of the event horizon to the future of the bifurcation surface.
We present the distorted five dimensional static black rings in Einstein-Maxwell-dilaton (EMd) gravity by using the sigma model structure of the dimensionally reduced field equations. We investigate how a static and neutral distribution of external matter distorts the black ring. In the case of a distorted five dimensional static black rings in EMd gravity the distortion parameters can be adjusted such that the solution is free of conical singularities in the region of the validity of the solution.
Despite all attempts, exact solutions for black rings in more than five dimensions remain elusive. In this paper we clarify some of the reasons for that, in particular we show that a peculiar symmetry of the five--dimensional black ring - separability of the base - cannot occur in dimensions higher than five. We also construct supersymmetric solutions that have symmetries of 5D supersymmetric black ring and show that they do not have regular horizons.
We study the geodesic motion of massless particles in singly rotating black ring spacetimes. We find stable stationary orbits of massless particles in toroidal spiral shape in the case that the thickness parameter of a black ring is less than a critical value. Furthermore, there exist nonstationary massless particles bounded in a finite region outside the horizon. This is the first example of stable bound orbits of massless particles around a black object.
We prove that five-dimensional extreme anti-de Sitter black ring solutions to the vacuum Einstein equations that admit biaxial symmetry do not exist. This is established by demonstrating the nonexistence of five-dimensional, biaxisymmetric, vacuum near-horizon geometries of extreme horizons with a nonzero cosmological constant and ring topology.
Necessary conditions for various algebraic types of the Weyl tensor are determined. These conditions are then used to find Weyl aligned null directions for the black ring solution. It is shown that the black ring solution is algebraically special, of type I_i, while locally on the horizon the type is II. One exceptional subclass - the Myers-Perry solution - is of type D.
It is known that the formation of apparent horizons with non-spherical topology is possible in higher-dimensional spacetimes. One of these is the black ring horizon with $S^1\times S^{D-3}$ topology where $D$ is the spacetime dimension number. In this paper, we investigate the black ring horizon formation in systems with $n$-particles. We analyze two kinds of system: the high-energy $n$-particle system and the momentarily-static $n$-black-hole initial data. In the high-energy particle system, we prove that the black ring horizon does not exist at the instant of collision for any $n$. But there…
We study the properties of the ergosurface of the Pomeransky-Senkov black rings, and show that it splits into an "inner"' and an "outer" region. As for the singular set, the topology of the "outer ergosurface" depends upon the value of parameters.
We examine bound orbits of particles around singly rotating black rings. We show that there exist stable bound orbits in toroidal spiral shape near the axis of the ring, and also exist stable circular orbits on the axis as special cases. The stable bound orbits can have arbitrary large size if the thickness of the ring is less than a critical value.
We show how any asymptotically flat supersymmetric solution of minimal d=5 supergravity with flat base space can be deformed into another supersymmetric asymptotically-Godel solution and apply this procedure to the recently found supersymmetric black-ring and black-string solutions.
We study Aichelburg-Sexl ultrarelativistic limits in higher dimensions. After reviewing the boost of D>=4 Reissner-Nordstrom black holes as a simple illustrative example, we consider the case of D=5 black rings, presenting new results for static charged rings.
We construct solutions for thin black rings in Anti-deSitter and deSitter spacetimes using approximate methods. Black rings in AdS exist with arbitrarily large radius and satisfy a bound |J| \leq LM, which they saturate as their radius becomes infinitely large. For angular momentum near the maximum, they have larger area than rotating AdS black holes. Thin black rings also exist in deSitter space, with rotation velocities varying between zero and a maximum, and with a radius that is always strictly below the Hubble radius. Our general analysis allows us to include black Saturns as well, which …
In this paper, we construct new solutions describing rotating black rings on Taub-NUT using the inverse-scattering method. These are five-dimensional vacuum space-times, generalising the Emparan-Reall and extremal Pomeransky-Sen'kov black rings to a Taub-NUT background space. When reduced to four dimensions in Kaluza-Klein theory, these solutions describe (possibly rotating) electrically charged black holes in superposition with a finitely separated magnetic monopole. Various properties of these solutions are studied, from both a five- and four-dimensional perspective.
Distorted black ring with rotation on the two-sphere
本文构建了仅在 $S^2$ 方向旋转且无锥形奇点的稳态黑洞环时空，并利用解生成技术添加外部引力场以消除渐近平坦黑洞环的锥形奇点。
We derive the first law of black rings thermodynamics in n-dimensional Einstein dilaton gravity with additional (p+1)-form field strength being the simplest generalization of five-dimensional theory containing a stationary black ring solution with dipole charge. It was done by means of choosing any cross section of the event horizon to the future of the bifurcation surface.
Using the inverse dimensional expansion method we study the elastic instability of black rings found recently in numerical analysis of fully nonlinear dynamical evolutions. In our analysis we should perform 1/D^1/2 expansions, not usual 1/D expansions, of the Einstein equations to capture this elastic instability of D dimensional black rings. By solving the Einstein equations at large D we obtain the effective equations for black rings, and the perturbation analysis of the large D effective equations with 1/D^1/2 expansions yields the formula for quasinormal mode frequencies. From this formula…
We argue that the Weyl coordinates and the rod-structure employed to construct static axisymmetric solutions in higher dimensional Einstein gravity can be generalized to the Einstein-Gauss-Bonnet theory. As a concrete application of the general formalism, we present numerical evidence for the existence of static black ring solutions in Einstein-Gauss-Bonnet theory in five spacetime dimensions. They approach asymptotically the Minkowski background and are supported against collapse by a conical singularity in the form of a disk. An interesting feature of these solutions is that the Gauss-Bonnet…
Light rings and shadows of static black holes in effective quantum gravity
该研究针对有效量子引力中两类保持广义协变性的静态黑洞，通过拓扑与反向光线追踪法分析其光环与阴影，发现量子参数对两类黑洞的光环、尺寸及透镜环的影响存在差异，可作为区分EQG与GR黑洞的依据。
We construct a supersymmetric black ring solution of SU(2) N=1, d=5 Super-Einstein-Yang-Mills (SEYM) theory by adding a distorted BPST instanton to an Abelian black ring solution of the same theory. The change cannot be observed from spatial infinity: neither the mass, nor the angular momenta or the values of the scalars at infinity differ from those of the Abelian ring. The entropy is, however, sensitive to the presence of the non-Abelian instanton, and it is smaller than that of the Abelian ring, in analogy to what happens in the supersymmetric coloured black holes recently constructed in th…
We calculate the quantum radiation from the five dimensional charged rotating black rings by demanding the radiation to eliminate the possible anomalies on the horizons. It is shown that the temperature, energy flux and angular-momentum flux exactly coincide with those of the Hawking radiation. The black rings considered in this paper contain the Myers-Perry black hole as a limit and the quantum radiation for this black hole, obtained in the literature, is recovered in the limit. The results support the picture that the Hawking radiation can be regarded as the anomaly eliminator on horizons an…
In order to address the issues raised by the recent discovery of non-uniqueness of black holes in five dimensions, we construct a solution of string theory at low energies describing a five-dimensional spinning black ring with three charges that can be interpreted as D1-brane, D5-brane, and momentum charges. The solution possesses closed timelike curves (CTCs) and other pathologies, whose origin we clarify. These pathologies can be avoided by setting any one of the charges, e.g. the momentum, to zero. We argue that the D1-D5-charged black ring, lifted to six dimensions, describes the thermal e…
We point out that 5D large radius doubly spinning black rings with rotation along S^1 and S^2 are afflicted by a robust instability. It is triggered by superradiant bound state modes. The Kaluza-Klein momentum of the mode along the ring is responsible for the bound state. This kind of instability in black strings and branes was first suggested by Marolf and Palmer and studied in detail by Cardoso, Lemos and Yoshida. We find the frequency spectrum and timescale of this instability in the black ring background, and show that it is active for large radius rings with large rotation along S^2. We i…
Greybody Factors Imprinted on Black Hole Ringdowns. II. Merging Binary Black Holes
本研究用含后牛顿修正的残余黑洞灰体因子建模可比质量比双黑洞并合的铃振引力波谱，基于SXS数据完成质量-自旋推断，验证了光环早于并合阶段形成的猜想，为研究双黑洞间距对光环形成的影响提供初步基础。
It has been shown recently that there is a large class of supersymmetric solutions of five-dimensional supergravity which generalize the supersymmetric black ring solution of Elvang et al. This class involves arbitrary functions. We show that most of these solutions do not have smooth event horizons, so they do not provide examples of black objects with infinite amounts of "hair".
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