Mathematical Relativity in Lisbon

Alan Rendall (Albert-Einstein-Institut, Potsdam, Germany)
Dynamics of linearized cosmological perturbations

In cosmology an important role is played by homogeneous and isotropic solutions of the Einstein-Euler equations and linearized perturbations of these. In this talk I present results of Paul Allen and myself on the asymptotic behaviour of certain linearized solutions (so-called "scalar perturbations") both in the approach to the initial singularity and at late times. The main equation of interest is a linear hyperbolic equation whose coefficients depend only on time. Expansions for the solutions are obtained in both asymptotic regimes. In both cases it is shown how general solutions can be parametrized by certain functions which are coefficients in the asymptotic expansion. The central technique in the proofs is that of energy estimates. An interesting feature of the results is a bifurcation observed in the late-time asymptotics for an asymptotically polytropic equation of state, occurring for the polytropic index corresponding to a radiation fluid.

Carlos Herdeiro (Universidade do Porto, Portugal)
Physical properties of the double Kerr solution

I will describe how a solution of the four dimensional vacuum Einstein equations describing two Kerr black holes held in equilibrium by a strut, called the double Kerr solution, is generated. For this purpose I will review briefly Weyl solutions and the inverse scattering technique, which explores the integrability of the vacuum Einstein equations with 2 commuting Killing vector fields (in four dimensions). I will then describe some physical properties of two special cases of the solution, dubbed counter-rotating and co-rotating case, namely: (1) Force and 'torque' balance; (2) Mutual rotational dragging: angular velocities and extremality condition; (3) Ergoregions merging; (4) Horizon geometry. I will close with some remarks about equilibrium of vacuum black holes in higher dimensions.

Hans Ringström (KTH, Stockholm, Sweden)
Future asymptotics of Bianchi type VI₀ vacuum solutions

The talk, which is based on joint work with Mark Heinzle, concerns the future asymptotic behaviour of solutions of Bianchi type VI₀. However, the questions are motivated by a desire to understand the asymptotic behaviour in the direction towards the singularity of solutions of Bianchi type VIII. In particular, they are motivated by a desire to understand whether particle horizons form or not in this class of spacetimes. The talk will to a large extent consist of a discussion of the concept of a particle horizon, the importance of the formation of particle horizons and what is known in the spatially homogeneous vacuum case. After that, the results concerning Bianchi VI₀ will be presented.

José Sande Lemos (Instituto Superior Técnico, Lisbon, Portugal)
Mass formula and entropy of black holes and quasiblack holes

A quasiblack hole, either nonextremal or extremal, can be broadly defined as the limiting configuration of a body when its boundary approaches the body's own quasihorizon. We first consider the mass contributions and the mass formula for a quasiblack hole. The analysis involves careful scrutiny of the surface stresses when the limiting configuration is reached. It is then shown that there exists a strict correspondence between the mass formulas for quasiblack holes and pure black holes. This perfect parallelism exists in spite of the difference in derivation and meaning of the formulas in both cases. Furthermore, for extremal quasiblack holes the finite surface stresses give zero contribution to the total mass. This leads to a very special version of the Abraham-Lorentz electron in general relativity in which the total mass has pure electromagnetic origin in spite of the presence of bare stresses. Second, we trace the origin of the black hole entropy S by considering a quasiblack hole. We show that if the body is thermal with the temperature taking the Hawking value at the quasihorizon limit, it follows, in the non-extremal case, from the first law of thermodynamics, that the entropy approaches the Bekenstein-Hawking value S=A/4. The entropy comes from the quasihorizon surface. For extremal quasiblack holes the result S=0 is found. The key role is played by the surface stresses on the quasihorizon, in that, in the non-extremal case they diverge but give finite contribution to S, while they are finite but give zero contribution to the entropy in extremal case. Thus, any distribution of matter inside the surface leads to the same universal value for the entropy in the quasihorizon limit. This can be of some help in the understanding of black hole entropy.

José Senovilla (Universidad del País Vasco, Bilbao, Spain)
The location of trapped surfaces

The possible location of future-trapped closed surfaces in spherically symmetric spacetimes is considered. The question of whether or not future-trapped compact surfaces can penetrate into flat portions of a spacetime will be discussed. In asymptotically flat dynamical cases, future-trapped surfaces do extend beyond any dynamical/trapping horizon, however they do not reach as far as the event horizon.

Marc Mars (Universidad de Salamanca, Spain)
Marginally outer trapped surfaces: evolution and properties under spacetime symmetries

Marginally outer trapped surfaces (MOTS) are usually considered as suitable quasi-local replacements of black holes. For this to be confirmed, an adequate understanding of their evolution and their behaviour in equilibrium configurations is necessary. In this talk I will discuss several results obtained in the last few years about existence and evolution of marginally outer trapped surfaces in the 3+1 setting and present some restrictions on MOTS imposed by presence of symmetries in the spacetime.

Miguel Sanchez (Universidad de Granada, Spain)
Boundaries of spacetimes: causal and conformal approaches

Among the notions of "boundary of a spacetime" used in Mathematical Relativity, the conformal and causal ones become specially interesting. The former has been the most useful boundary from the practical viewpoint. Nevertheless, this boundary has limitations, because it can be defined only in a rather restrictive class of spacetimes, and further restrictions are necessary in order to ensure its uniqueness and reasonable behavior. The notion of causal boundary has been the most promising alternative from a fundamental viewpoint. The problem of finding a precise and consistent definition has puzzled the specialists in the last three decades, but recent progress has allowed to find a new definition of the causal boundary which is expected to be definitive. In this talk, the development and subsequent problems on boundaries of spacetimes will be reviewed. The new definition of the causal boundary is emphasized, and compared with the conformal one.

Piotr Chruściel (University of Oxford, UK)
On five dimensional black holes with three commuting Killing vectors

We will review what is, and what isn't, known about the global structure of five dimensional black rings, black saturns, and similar five dimensional black hole solutions.

Simone Calogero (Universidad de Granada, Spain)
Bianchi Cosmologies with Anisotropic Matter

In this talk I will present recent results with Mark Heinzle on the dynamics of spatially homogeneous cosmological models with anisotropic matter. The discussion will be restricted to locally rotationally symmetric Bianchi models of class A. The matter source is not given explicitely, but only through a set of mild and physically motivated assumptions that are satisfied by important examples of matter models (Vlasov matter, magnetic fields, elastic materials). We prove that there exist several interesting asymptotic behaviors of the solutions which are compatible with the energy conditions. In particular, we prove the existence of a generic class of Bianchi IX solutions that do not recollapse in the future, despite the fact that they satisfy the strong energy condition by a wide margin.