Thin Black Ring Research Articles

The stable trapping phenomenon for black strings and black rings and its obstructions on the decay of linear waves

The geometry of solutions to the higher dimensional Einstein vacuum equations presents aspects that are absent in four dimensions, one of the most remarkable being the existence of stably trapped null geodesics in the exterior of asymptotically flat black holes. This paper investigates the stable trapping phenomenon for two families of higher dimensional black holes, namely black strings and black rings, and how this trapping structure is responsible for the slow decay of linear waves on their exterior. More precisely, we study decay properties for the energy of solutions to the scalar, linear wave equation $\Box_{g_{\textup{ring}}} \Psi=0$, where $g_{\textup{ring}}$ is the metric of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class $\mathfrak{g}$ of black ring metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black ring exterior $(\mathcal{D},g_{\textup{ring}})$, with $g_{\textup{ring}}\in\mathfrak{g}$. The proof generalizes the perturbation argument and quasimode construction of Holzegel--Smulevici \cite{SharpLogHolz} to the case of a non-separable wave equation and crucially relies on the presence of stably trapped null geodesics on $\mathcal{D}$. As a by-product, the same logarithmic lower bound can be established for any five-dimensional black string. Our result is the first mathematically rigorous statement supporting the expectation that black rings are dynamically unstable to generic perturbations. In particular, we conjecture a new \textit{nonlinear} instability for five-dimensional black strings and thin black rings which is already present at the level of scalar perturbations and clearly differs from the mechanism driven by the well-known Gregory--Laflamme instability.

Note on the thermodynamic stability of a black ring at quantum scales

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 confirmed by the Gibbs free energy analysis.

Instabilities of thin black rings: closing the gap

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 ≥ 5. 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 exhibit a Gregory-Laflamme instability, giving strong evidence that black rings are unstable in the entire range of parameter space. For very thin rings, we show that the growth rate of the instability increases with increasing non-axisymmetric mode m while for thicker rings, there is competition between the different modes. However, up to second order in the blackfold approximation, we do not observe an elastic instability, in particular for large modes m ≫ 1, where this approximation has higher accuracy. This suggests that the Gregory-Laflamme instability is the dominant instability for very thin black rings. Additionally, we find a long-lived mode that describes a wiggly time-dependent deformation of a black ring. We comment on disagreements between our results and corresponding ones obtained from a large D analysis of black ring instabilities.

Einstein-Gauss-Bonnet black rings at large D

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 modes of the EGB black ring. We find that thin EGB black ring becomes unstable against non-axisymmetric perturbation. Furthermore, we numerically evolve the effective equations in a particular case to study the final state of the instability, and find that the thin black ring becomes the stable non-uniform black ring at late time, which gives a relative strong evidence to support the conjecture given in [25].

Black ring entropy from the Weyl tensor

A black ring is an asymptotically flat vacuum solution of the Einstein equations with an event horizon of topology $S^1\times S^2$. A connection between the black ring entropy and its Weyl tensor $C_{\mu\nu\lambda\rho}$ is explored by interpreting the Weyl scalar invariant $C_{\mu\nu\lambda\rho} C^{\mu\nu\lambda\rho}$ as the entropy density in 5-dimensional space-time. We calculate the proper volume integral of $C_{\mu\nu\lambda\rho} C^{\mu\nu\lambda\rho}$ for a neutral black ring and prove that it is proportional to the entropy of a thin black ring. Similar calculations are extended to more general cases: the black string, the black ring with two angular momenta, and the black ring with a cosmological constant. The proportionality still maintains or is valid at least at the leading order.

Charged black rings at large D

We study the charged slowly rotating black holes in the Einstein-Maxwell theory in the large dimensions (D). By using the 1/D expansion in the near regions of the black holes we obtain the effective equations for the charged slowly rotating black holes. The effective equations capture the dynamics of various stationary solutions, including the charged black ring, the charged slowly rotating Myers-Perry black hole and the charged slowly boosted black string. Via different embeddings we construct these stationary solutions explicitly. For the charged black ring at large D, we find that the charge lowers the angular momentum due to the regularity condition on the solution. By performing the perturbation analysis of the effective equations, we obtain the quasinormal modes of the charge perturbation and the gravitational perturbation analytically. Like the neutral case the charged thin black ring suffers from the Gregory-Laflamme-like instability under the non-axisymmetric perturbations, but the charge weakens the instability. Besides, we find that the large D analysis always respects the cosmic censorship.

Black rings at large D

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 effective equations. Furthermore, by perturbation analysis of the effective equations, we find a quasinormal mode condition of the black ring in analytic way. As a result we confirm that thin black ring is unstable against non-axisymmetric perturbations. We also include 1/D corrections to the effective equations and discuss the effects by 1/D corrections.

Black rings in global anti-de Sitter space

We construct five dimensional black rings in global anti-de Sitter space using numerical methods. These rings satisfy the BPS bound |J| < Mℓ, but the angular velocity always violates the Hawking-Reall bound |Ω H ℓ| ≤ 1, indicating that they should be unstable under superradiance. At high temperatures, the limit |Ω H ℓ| ↘ 1 is attained by thin rings with an arbitrarily large radius. However, at sufficiently low temperatures, this limit is saturated by a new kind of rings, whose outer circle can still be arbitrarily long while the hole in the middle does not grow proportionally. This gives rise to a membrane-like horizon geometry, which does not have an asymptotically flat counterpart. We find no evidence for thin AdS black rings whose transverse S 2 is much larger than the radius of AdS, ℓ, and thus these solutions never fall into the hydrodynamic regime of the dual CFT. Thermodynamically, we find that AdS black rings never dominate the grand canonical ensemble. The behaviour of our solutions in the microcanonical ensemble approaches known perturbative results in the thin-ring limit.

Time evolution of a thin black ring via Hawking radiation

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, $\ensuremath{\lambda}\ensuremath{\ll}1$, which can be approximated by a boosted Kerr string locally. We show that a thin black ring evaporates with fixing its thickness parameter $\ensuremath{\lambda}$. Further, in the case of an Emparan-Reall black ring, we derive analytic formulas for the time evolution, which has one parameter to be evaluated numerically. We find that the lifetime of a thin black ring is shorter by a factor of $O({\ensuremath{\lambda}}^{2})$ compared to a five-dimensional Schwarzschild black hole with the same initial mass. We also study detailed properties of the Hawking radiation from the thin black ring, including the energy and angular spectra of emitted particles.

How fluids bend: the elastic expansion for higher-dimensional black holes

Hydrodynamics can be consistently formulated on surfaces of arbitrary co-dimension in a background space-time, providing the effective theory describing long-wavelength perturbations of black branes. When the co-dimension is non-zero, the system acquires fluid-elastic properties and constitutes what is called a fluid brane. Applying an effective action approach, the most general form of the free energy quadratic in the extrinsic curvature and extrinsic twist potential of stationary fluid brane configurations is constructed to second order in a derivative expansion. This construction generalizes the Helfrich-Canham bending energy for fluid membranes studied in theoretical biology to the case in which the fluid is rotating. It is found that stationary fluid brane configurations are characterized by a set of 3 elastic response coefficients, 3 hydrodynamic response coefficients and 1 spin response coefficient for co-dimension greater than one. Moreover, the elastic degrees of freedom present in the system are coupled to the hydrodynamic degrees of freedom. For co-dimension-1 surfaces we find a 8 independent parameter family of stationary fluid branes. It is further shown that elastic and spin corrections to (non)-extremal brane effective actions can be accounted for by a multipole expansion of the stress-energy tensor, therefore establishing a relation between the different formalisms of Carter, Capovilla-Guven and Vasilic-Vojinovic and between gravity and the effective description of stationary fluid branes. Finally, it is shown that the Young modulus found in the literature for black branes falls into the class predicted by this approach - a relation which is then used to make a proposal for the second order effective action of stationary blackfolds and to find the corrected horizon angular velocity of thin black rings.

Quasilocal formalism and thermodynamics of asymptotically flat black objects

We study the properties of five-dimensional black objects by using the renormalized boundary stress tensor for locally asymptotically flat spacetimes. This provides a more refined form of the quasilocal formalism, which is useful for a holographic interpretation of asymptotically flat gravity. We apply this technique to examine the thermodynamic properties of black holes, black rings and black strings. The advantage of using this method is that we can go beyond the ‘thin ring’ approximation and compute the boundary stress tensor for any general (thin or fat) black ring solution. We argue that the boundary stress tensor encodes the necessary information to distinguish between black objects with different horizon topologies in the bulk. We also study in detail the susy black ring and clarify the relation between the asymptotic charges and the charges defined at the horizon. Furthermore, we obtain the balance condition for ‘thin’ dipole black rings.

Quasilocal equilibrium condition for black ring

We use the conservation of the renormalized boundary stress-energy tensor to obtain the equilibrium condition for a general (thin or fat) black ring solution. We also investigate the role of the spatial stress in the thermodynamics of deformation within the quasilocal formalism of Brown and York and discuss the relation with other methods. In particular, we discuss the quantum statistical relation for the unbalanced black ring solution.

Dynamical black rings with a positive cosmological constant

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 increases. We also discuss the multiblack rings and the coalescence of them.

Black rings in (Anti)-de Sitter space

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| ≤ 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 we discuss briefly. We present a simple physical argument why supersymmetric AdS black rings must not be expected: they do not possess the necessary pressure to balance the AdS potential. We discuss the possible existence or absence of `large AdS black rings' and their implications for a dual hydrodynamic description. An analysis of the physical properties of rotating AdS black holes is also included.

Phases of five-dimensional black holes

We argue that the configurations that approach maximal entropy in five-dimensional asymptotically flat vacuum gravity, for fixed mass and angular momentum, are `black Saturns' with a central, close to static, black hole and a very thin black ring around it. For any value of the angular momentum, the upper bound on the entropy is equal to the entropy of a static black hole of the same total mass. For fixed mass, spin and area there are families of multi-ring solutions with an arbitrarily large number of continuous parameters, so the total phase space is infinite-dimensional. Somewhat surprisingly, the phases of highest entropy are not in thermal equilibrium. Imposing thermodynamical equilibrium drastically reduces the phase space to a finite, small number of different phases.

Dynamics and stability of black rings

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/absorption of massless particles. Our results point to the conclusion that most neutral black rings suffer from classical dynamical instabilities, but there may still exist a small range of parameters where thin black rings are stable. We also discuss the absence of regular real Euclidean sections of black rings, and thermodynamics in the grand-canonical ensemble.

Anomalous induction of brightness and surface qualities: a new illusion due to radial lines and chromatic rings.

When a chromatic (eg light-blue) annulus surrounds the central gap of an Ehrenstein figure so as to connect the inner ends of the radial lines, a striking new lightness effect emerges: the central white disk has both a self-luminous quality (brighter than in the regular Ehrenstein figure) and a surface quality (dense, paste-like). Self-luminous and surface qualities do not ordinarily appear co-extensively: hence, the brightness induction is called anomalous. In experiment 1, subjects separately scaled self-luminous and surface properties, and in experiment 2, brightness was nulled by physically darkening the central gap. Experiments 3 and 4 were designed to evaluate the importance of chromatic versus achromatic properties of the annulus; other aspects of the annulus (width or the inclusion of a thin black ring inside or outside the chromatic annulus) were tested in experiments 5-7. In experiments 8-12, subjects rated the brightness of modified Ehrenstein figures varying the radial lines (number, length, width, contrast, arrangement). Variation of these parameters generally affected brightness enhancement in the Ehrenstein figure and anomalous brightness induction in a similar manner, but was stronger for the latter effect. On the basis of these results, anomalous brightness induction is attributed to a surface induction process triggered by an interaction between illusory brightness enhancement (due to the radial lines) and border ownership (due to the blue annulus).