José Luis Jaramillo

New theoretical approaches to black holes

Abstract Quite recently, some new mathematical approaches to black holes have appeared in the literature. They do not rely on the classical concept of event horizon—which is very global, but on the local concept of hypersurfaces foliated by trapped surfaces. After a brief introduction to these new horizons, we focus on a viscous fluid analogy that can be developed to describe their dynamics, in a fashion similar to the membrane paradigm introduced for event horizons in the seventies, but with a significant change of sign of the bulk viscosity.

Mass and Angular Momentum in General Relativity

We present an introduction to mass and angular momentum in General Relativity. After briefly reviewing energy–momentum for matter fields, first in the flat Minkowski case (Special Relativity) and then in curved spacetimes with or without symmetries, we focus on the discussion of energy–momentum for the gravitational field. We illustrate the difficulties rooted in the Equivalence Principle for defining a local energy–momentum density for the gravitational field. This leads to the understanding of gravitational energy–momentum and angular momentum as nonlocal observables that make sense, at best, for extended domains of spacetime. After introducing Komar quantities associated with spacetime symmetries, it is shown how total energy–momentum can be unambiguously defined for isolated systems, providing fundamental tests for the internal consistency of General Relativity as well as setting the conceptual basis for the understanding of energy loss by gravitational radiation. Finally, several attempts to formulate quasi-local notions of mass and angular momentum associated with extended but finite spacetime domains are presented, together with some illustrations of the relations between total and quasi-local quantities in the particular context of black hole spacetimes. This article is not intended to be a rigorous and exhaustive review of the subject, but rather an invitation to the topic for nonexperts.

From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity

This paper reviews some aspects in the current relationship between mathematical and numerical general relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial-value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed in light of the interaction between numerical and mathematical relativists.

Improved constrained scheme for the Einstein equations: an approach to the uniqueness issue

The fully constrained formulation (FCF) proposed by Bonazzola, Gourgoulhon, Grand‐clement, and Novak is one of the constrained formulations of Einstein’s equations. It contains as an approximation the conformal flatness condition (CFC). The elliptic part of the FCF basically shares the same differential operators as the elliptic equations in the CFC scheme. We present here a reformulation of the elliptic sector of CFC that has the fundamental property of overcoming local uniqueness problems, and an extension of these ideas to FCF.

Spanish Relativity Meeting (ERE 2010): Gravity as a Crossroad in Physics

The 2010 edition of the Spanish Relativity Meeting (ERE2010) took place in Granada from 6–10 September 2010, and was hosted by the Instituto de Astrofisica de Andalucia (IAA - CSIC). This event represented the 34th edition of Encuentros Relativistas Espanoles (ERE), an international conference devoted to relativity and gravitation and organized every year by one of the Spanish groups working in this area. The particular scientific flavour of the 2010 edition was captured by the subtitle of the conference, Gravity as a Crossroad in Physics. Our underlying rationale was to present gravitational physics as a scientific locus for the interaction between (separate) communities in physics. It is a remarkable property of gravity that its specific problems provide a framework that calls for the interchange of ideas, concepts and methodologies from very different communities. In this edition we aimed to reflect this interdisciplinary perspective in the scientific programme. Each day during the week was devoted to a particular dialogue between two communities who share some of their ultimate goals, but differ in their conceptual background, methodology or technical approach. These dialogues were envisaged as opportunities to compare alternative viewpoints, maintaining a focus on their complementary nature. This led to the organization of the week as follows: Day 1: Fundamental vs Effective Approaches in Theoretical GravityThis day compared approaches to gravity that differ conceptually in their understanding of the nature of the basic physical degrees of freedom of the theory, namely confronting viewpoints supporting the fundamental status of such degrees of freedom with other research programs favouring some emergence mechanism. Gravitational analogues were also discussed on this day. Day 2: Geometric vs Quantum Field/String Theory Approaches to Quantum GravityThis day was focused on quantum gravity. A particular emphasis was placed on the comparison between geometric approaches to the quantization of general relativity (e.g. loop quantum gravity in the context of the canonical program) and approaches leaning on or evolving from a (quantum) field theory treatment of gravity (e.g. string/M-theory). Day 3: Theoretical Cosmology vs Physical CosmologyThis day addressed the current challenges in cosmology from a double perspective. On the one hand, offering an analysis of the large scale picture of the universe emerging from the accumulated body of observational data and, on the other hand, assessing the theoretical attempts to explain such a picture putting a special emphasis on the role of gravity. Day 4: Relativity vs AstrophysicsThis day was focused on astrophysical problems where general relativity plays a fundamental role. Challenges and difficulties encountered by relativists modelling specific astrophysical scenarios were disucssed as well as the problems found by astrophysicists needing general relativity as a key conceptual ingredient. Particular emphasis was placed on gravitational waves and compact objects. Day 5: Mathematical Relativity vs Numerical RelativityThis day discussed fundamental problems in general relativity, and more generally in gravity physics, where a close collaboration between relativists in the geometry/analysis community on the one hand, and relativists in the numerical community on the other hand, can prove to be particularly successful and insightful. The contributions in this volume have been organized in two blocks, corresponding to plenary and parallel sessions during the conference. In both cases we have kept the chronological order of the presented talks. The only exception to this rule is the parallel session dedicated to the memory of the late S Brian Edgar, labeled as IV.A during the conference, which we have placed immediately after the plenary session contributions. The result of the dialogue experience at the conference was extremely satisfactory and gratifying. Scientific sessions were thrilled by tantalizing and inspiring discussions, often continued in long walks around the Carmenes of the old city. In this spirit, we wish to thank all of the participants of the ERE meeting for their enthusiasm and especially the contributors to these proceedings for their synthesis effort. Granada, 25 July 2011 Victor Aldaya, Carlos Barcelo and Jose Luis Jaramillo Corrigendum added 2 June 2015: We acknowledge the funding provided by the MICINN (Ref: FIS2009-08009), Consejeria de Economia, Innovacion, Ciencia y Empleo, Junta de Andalucia (Ref: IAC10-I-7369) and CSIC for the organization of this meeting.

Excised black hole spacetimes: Quasilocal horizon formalism applied to the Kerr example

We present a numerical work aiming at the computation of excised initial data for black hole spacetimes in full general relativity, using the Dirac gauge in the context of a constrained formalism for the Einstein equations. Introducing the isolated horizon formalism for black hole excision, we especially solve the conformal metric part of the equations, and assess the boundary condition problem for it. In the stationary single black hole case, we present and justify a no-boundary treatment on the black hole horizon. We compare the data obtained with the well-known analytic Kerr solution in Kerr-Schild coordinates, and assess the widely used conformally flat approximation for simulating axisymmetric black hole spacetimes. Our method shows good concordance on physical and geometrical issues, with the particular application of the isolated horizon multipolar analysis to confirm that the solution obtained is indeed the Kerr spacetime. Finally, we discuss a previous suggestion in the literature for the boundary conditions for the conformal geometry on the horizon.

Trapping horizons as inner boundary conditions for black hole spacetimes

We present a set of inner boundary conditions for the numerical construction of dynamical black hole spacetimes, when employing a

Mathematical issues in a fully constrained formulation of the Einstein equations

3+1

Area Inequalities for Stable Marginally Trapped Surfaces

constrained evolution scheme and an excision technique. These inner boundary conditions are heuristically motivated by the dynamical trapping horizon framework and are enforced in an elliptic subsystem of the full Einstein equation. In the stationary limit they reduce to existing isolated horizon boundary conditions. A characteristic analysis completes the discussion of inner boundary conditions for the radiative modes.

Application of initial data sequences to the study of Black Hole dynamical trapping horizons

Bonazzola, Gourgoulhon, Grandclement, and Novak [Phys. Rev. D 70, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular, focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depends on the boundary conditions for the shift vector in the elliptic subsystem. This interplay between the hyperbolic and elliptic parts of the complete evolution system is used to assess the prescription of inner boundary conditions for the hyperbolic part when using an excision approach to black hole space-time evolutions.