Carmen Rebelo

Thermodynamical description of stationary, asymptotically flat solutions with conical singularities

We examine the thermodynamical properties of a number of asymptotically flat, stationary (but not static) solutions having conical singularities, with both connected and non-connected event horizons, using the thermodynamical description recently proposed in [1]. The examples considered are the double-Kerr solution, the black ring rotating in either S 2 or S 1 and the black Saturn, where the balance condition is not imposed for the latter two solutions. We show that not only the BekensteinHawking area law is recovered from the thermodynamical description but also the thermodynamical angular momentum is the ADM angular momentum. We also analyse the thermodynamical stability and show that, for all these solutions, either the isothermal moment of inertia or the specific heat at constant angular momentum is negative, at any point in parameter space. Therefore, all these solutions are thermodynamically unstable in the grand canonical ensemble.

Radiation from a D-dimensional collision of shock waves: First order perturbation theory

We study the spacetime obtained by superimposing two equal Aichelburg-Sexl shock waves in D dimensions traveling, head-on, in opposite directions. Considering the collision in a boosted frame, one shock becomes stronger than the other, and a perturbative framework to compute the metric in the future of the collision is setup. The geometry is given, in first order perturbation theory, as an integral solution, in terms of initial data on the null surface where the strong shock has support. We then extract the radiation emitted in the collision by using a D-dimensional generalisation of the Landau-Lifschitz pseudo-tensor and compute the percentage of the initial centre of mass energy ϵ emitted as gravitational waves. In D = 4 we find ϵ = 25.0%, in agreement with the result of D’Eath and Payne [12]. As D increases, this percentage increases monotonically, reaching 40.0% in D = 10. Our result is always within the bound obtained from apparent horizons by Penrose, in D = 4, yielding 29.3%, and Eardley and Giddings [16], in D > 4, which also increases monotonically with dimension, reaching 41.2% in D = 10. We also present the wave forms and provide a physical interpretation for the observed peaks, in terms of the null generators of the shocks.

Dynamical and Thermodynamical Aspects of Interacting Kerr Black Holes

We consider the asymptotically flat double-Kerr solution for two equal mass black holes with either the same or opposite angular momentum and with a massless strut between them. For fixed angular momentum and mass, the angular velocity of two corotating Kerr black holes decreases as they approach one another, from the Kerr value at infinite separation to the value of a single Kerr black hole with twice the mass and the angular momentum at the horizons merging limit. We show that the ratio J/M 2 for extremal corotating Kerr black holes varies from unity at infinite separation to two at the merging limit. These results are interpreted in terms of rotational dragging and compared with the case of counterrotating Kerr black holes. We then analyze the merging of ergoregions. In the corotating case the merger point occurs at an angle of π/2, in agreement with recent general arguments. In the counterrotating case the ergoregions never merge. We study the horizon geometry for both cases as a function of the distance and provide embedding diagrams. Finally, we study the thermodynamical evolution of the corotating double-Kerr system, showing that, in the canonical ensemble, it is thermodynamically stable for fast spinning black holes. As for single Kerr black holes the stable and unstable phases are separated by a second order phase transition. We show that for large fixed angular momentum two Kerr black holes reach a minimum distance, before horizon merging has occurred, where the thermodynamical approximation breaks down. We also consider the microcanonical ensemble to study the maximal energy that can be extracted from the double-Kerr system as a function of the separation between the black holes.

A double Myers-Perry black hole in five dimensions

Using the inverse scattering method we construct a six-parameter family of exact, stationary, asymptotically flat solutions of the 4+1 dimensional vacuum Einstein equations, with U(1)2 rotational symmetry. It describes the superposition of two Myers-Perry black holes, each with a single angular momentum parameter, both in the same plane. The black holes live in a background geometry which is the Euclidean C-metric with an extra flat time direction. This background possesses conical singularities in two adjacent compact regions, each corresponding to a set of fixed points of one of the U(1) actions in the Cartan sub-algebra of SO(4). We discuss several aspects of the black holes geometry, including the conical singularities arising from force imbalance, and the torsion singularity arising from torque imbalance. The double Myers-Perry solution presented herein is considerably simpler than the four dimensional double Kerr solution and might be of interest in studying spin-spin interactions in five dimensional general relativity.

On the interaction between two Kerr black holes

The double-Kerr solution is generated using both a Backlund transformation and the Belinskii-Zakharov inverse-scattering technique. We build a dictionary between the parametrisations naturally obtained in the two methods and show their equivalence. We then focus on the asymptotically flat double-Kerr system obeying the axis condition which is 2 invariant; for this system there is an exact formula for the force between the two black holes, in terms of their physical quantities and the coordinate distance. We then show that the angular velocity of the two black holes decreases from the usual Kerr value at infinite distance to zero in the touching limit; the extremal limit of the two black holes is given by |J| = cM2, where c depends on the distance and varies from one to infinity as the distance decreases; for sufficiently large angular momentum the temperature of the black holes attains a maximum at a certain finite coordinate distance. All of these results are interpreted in terms of the dragging effects of the system.

Radiation from aD-dimensional collision of shock waves: Higher-order setup and perturbation theory validity

The collision of two D-dimensional, ultra-relativistic particles, described in General Relativity as Aichelberg-Sexl shock waves, is inelastic. In first order perturbation theory, the fraction of the initial centre of mass energy radiated away was recently shown to be 1/2 - 1/D. Here, we extend the formalism to higher orders in perturbation theory, and derive a general expression to extract the inelasticity, valid non-perturbatively, based on the Bondi mass loss formula. Then, to clarify why perturbation theory captures relevant physics of a strong field process in this problem, we provide one variation of the problem where the perturbative framework breaks down: the collision of ultra-relativistic charged particles. The addition of charge, and the associated repulsive nature of the source, originates an extra radiation burst, which we argue to be an artifact of the perturbative framework, veiling the relevant physics.

Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

An explicit global and unique isometric embedding into hyperbolic 3-space, H{sup 3}, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H{sup 3} of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the U(1) isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into H{sup 3}, for arbitrary values of the angular momentum. For this example, considering a quotient of H{sup 3} by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding that cannot be made global.

Factor Analysis to Support the Visualization and Interpretation of Clusters of Portal Users

Clusterings based on many variables are difficult to visualize and interpret. We present a methodology based on factor analysis (FA) which can be used for that purpose. FA generates a small set of variables which encode most of the information in the original variables. We apply the methodology to segment the users of a Web portal, using access log data. It not only makes it simpler to visualize and understand the clusters which are obtained on the original variables but it also helps the analyst in selecting some of the original variables for further analysis of those clusters

Radiation from a D-Dimensional Collision of Shock Waves: A Summary of the First Order Results

We describe how to set up a perturbative framework to compute the metric in the future of a D-dimensional collision of two high speed black holes, by superimposing two equal Aichelburg–Sexl shock waves traveling, head-on, in opposite directions. We then estimate the radiation emitted in the collision using a D-dimensional generalisation of the Landau–Lifschitz pseudo-tensor—workable in a first order approach—and compute the percentage of the initial centre of mass energy emitted as gravitational waves. We shall see that our first order results are always within the bound obtained from apparent horizons computations.

Radiation from a D-Dimensional Collision of Shock Waves: Numerics and a Charged Case

We describe the generalisation to higher orders, of a perturbative framework to find the metric after the collision of two Aichelburg–Sexl gravitational shock waves in D-dimensions. A central challenge is to estimate the amount of gravitational radiation emitted in the collision, at higher orders. We present an adaptation of the Bondi mass loss formula in D-dimensions which is valid non-perturbatively, for axially symmetric asymptotically flat space-times. This is shown to coincide with the Landau–Lifshitz pseudo tensor result at first order in perturbation theory. We also discuss the validity of the method with a collision of charged shocks.