Seeking for toroidal event horizons from initially stationary BH configurations

We construct and evolve non-rotating vacuum initial data with a ring singularity, based on a simple extension of the standard Brill-Lindquist multiple black-hole initial data, and search for event horizons with spatial slices that are toroidal when the ring radius is sufficiently large. While evolutions of the ring singularity are not numerically feasible for large radii, we find some evidence, based on configurations of multiple BHs arranged in a ring, that this configuration leads to singular limit where the horizon width has zero size, possibly indicating the presence of a naked singularity, when the radius of the ring is sufficiently large. This is in agreement with previous studies that have found that there is no apparent horizon surrounding the ring singularity when the ring's radius is larger than about twice its mass.Comment: 24 pages, 14 figure

From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article 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 on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit

Self-force: Computational Strategies

Building on substantial foundational progress in understanding the effect of a small body's self-field on its own motion, the past 15 years has seen the emergence of several strategies for explicitly computing self-field corrections to the equations of motion of a small, point-like charge. These approaches broadly fall into three categories: (i) mode-sum regularization, (ii) effective source approaches and (iii) worldline convolution methods. This paper reviews the various approaches and gives details of how each one is implemented in practice, highlighting some of the key features in each case.Comment: Synchronized with final published version. Review to appear in "Equations of Motion in Relativistic Gravity", published as part of the Springer "Fundamental Theories of Physics" series. D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, Springer, 201