Uniqueness of equilibrium on rings

We consider congestion games on networks with nonatomic users and user-specific costs. We are interested in the uniqueness property defined by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30 225-244] as the uniqueness of equilibrium flows for all assignments of strictly increasing cost functions. He settled the case with two-terminal networks. In the present work we characterize completely bidirectional rings for which the uniqueness property holds. The main result is that it holds precisely for nine networks and those obtained from them by elementary operations. For other bidirectional rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. Related results are also proven

Black Holes in Higher-Dimensional Gravity

These lectures review some of the recent progress in uncovering the phase structure of black hole solutions in higher-dimensional vacuum Einstein gravity. The two classes on which we focus are Kaluza-Klein black holes, i.e. static solutions with an event horizon in asymptotically flat spaces with compact directions, and stationary solutions with an event horizon in asymptotically flat space. Highlights include the recently constructed multi-black hole configurations on the cylinder and thin rotating black rings in dimensions higher than five. The phase diagram that is emerging for each of the two classes will be discussed, including an intriguing connection that relates the phase structure of Kaluza-Klein black holes with that of asymptotically flat rotating black holes.Comment: latex, 49 pages, 5 figures. Lectures to appear in the proceedings of the Fourth Aegean Summer School, Mytiline, Lesvos, Greece, September 17-22, 200

Black Rings, Supertubes, and a Stringy Resolution of Black Hole Non-Uniqueness

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 excitation of a supersymmetric D1-D5 supertube, which is in the same U-duality class as the D0-F1 supertube. We explain how the stringy microscopic description of the D1-D5 system distinguishes between a spherical black hole and a black ring with the same asymptotic charges, and therefore provides a (partial) resolution of the non-uniqueness of black holes in five dimensions.Comment: 33 pages, 1 figur

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.Comment: 39 pages, 17 figures; v2: conclusions concerning radial stability corrected + new appendix + refs added; v3: additional comments regarding stabilit

Rotating Circular Strings, and Infinite Non-Uniqueness of Black Rings

We present new self-gravitating solutions in five dimensions that describe circular strings, i.e., rings, electrically coupled to a two-form potential (as e.g., fundamental strings do), or to a dual magnetic one-form. The rings are prevented from collapsing by rotation, and they create a field analogous to a dipole, with no net charge measured at infinity. They can have a regular horizon, and we show that this implies the existence of an infinite number of black rings, labeled by a continuous parameter, with the same mass and angular momentum as neutral black rings and black holes. We also discuss the solution for a rotating loop of fundamental string. We show how more general rings arise from intersections of branes with a regular horizon (even at extremality), closely related to the configurations that yield the four-dimensional black hole with four charges. We reproduce the Bekenstein-Hawking entropy of a large extremal ring through a microscopic calculation. Finally, we discuss some qualitative ideas for a microscopic understanding of neutral and dipole black rings.Comment: 31 pages, 7 figures. v2: minor changes, added reference. v3: erroneous values of T_{ww} (eq.(3.39)) and n_p (eq.(5.20)) corrected, and accompanying discussion amended. In the journal version these corrections appear as an appended erratum. No major changes involve

Black Strings, Black Rings and State-space Manifold

State-space geometry is considered, for diverse three and four parameter non-spherical horizon rotating black brane configurations, in string theory and $M$-theory. We have explicitly examined the case of unit Kaluza-Klein momentum $D_1D_5P$ black strings, circular strings, small black rings and black supertubes. An investigation of the state-space pair correlation functions shows that there exist two classes of brane statistical configurations, {\it viz.}, the first category divulges a degenerate intrinsic equilibrium basis, while the second yields a non-degenerate, curved, intrinsic Riemannian geometry. Specifically, the solutions with finitely many branes expose that the two charged rotating $D_1D_5$ black strings and three charged rotating small black rings consort real degenerate state-space manifolds. Interestingly, arbitrary valued $M_5$-dipole charged rotating circular strings and Maldacena Strominger Witten black rings exhibit non-degenerate, positively curved, comprehensively regular state-space configurations. Furthermore, the state-space geometry of single bubbled rings admits a well-defined, positive definite, everywhere regular and curved intrinsic Riemannian manifold; except for the two finite values of conserved electric charge. We also discuss the implication and potential significance of this work for the physics of black holes in string theory.Comment: 41 pages, Keywords: Rotating Black Branes; Microscopic Configurations; State-space Geometry, PACS numbers: 04.70.-s Physics of black holes; 04.70.Bw Classical black holes; 04.70.Dy Quantum aspects of black holes, evaporation, thermodynamic

Quasilocal formalism and thermodynamics of asymptotically flat black objects

We study the properties of 5-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 necessarily 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.Comment: v2 clarifications on the advantage of using quasilocal formalism for black rings added, CQG versio