The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.

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Stationary Black Holes: Uniqueness and Beyond

These notes represent approximately one semester’s worth of lectures on introductory general relativity for beginning graduate students in physics Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: gravitational radiation, black holes, and cosmology Individual chapters, and potentially updated versions, can be found at http://itpucsbedu/~carroll/notes/

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Lecture Notes on General Relativity

A classification scheme for boundary points of incomplete space-times is described. For all classes explicit examples are presented to illustrate the different behaviour of the geometry near those boundary points.

Singular space-times

Singularities in conformally flat spacetimes

We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space-time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space-time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc.

The interior of charged black holes and the problem of uniqueness in general relativity

Geometry and physics: an overview 1. The background manifold structure 2. Differentiation 3. The curvature 4. Space-time and tetrad formalism 5. Spinors and the classification of the Weyl tensor 6. Coupling between fields and geometry 7. Dynamics on curved manifolds 8. Geometry of congruences 9. Physical measurements in space-time 10. Spherically symmetric solutions 11. Axially symmetric solutions References Notation Index.

Relativity on Curved Manifolds

The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

The Analysis of Space-Time Singularities

The analysis of space-time singularities.

A set of conditions for the reasonableness of space-time is proposed and investigated. Using these, together with strong causality and an assumption of genericness, it is shown that future timelike or null geodesically incomplete space-times contain either curvature or intermediate singularities, or primordial singularities.

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Space-time singularities

The UK produces an estimated 2 Mt of waste electrical and electronic equipment (WEEE) annually and the management of this waste has become a foremost environmental issue in the UK. Whilst the collection, transportation and treatment of WEEE contributes to climate change due to its considerable energy and material requirements, the effective recovery and reuse or recycling of WEEE can contribute towards a net climate benefit. Here, we present a combined material flow analysis and carbon footprint approach (based on a bespoke calculator tool) for quantifying the flows of WEEE through a national waste management system and evaluating their potential climate impacts. We apply this approach to analyse the WEEE management system for the UK from 2010 to 2030 using prospective scenario analysis and assess the carbon footprint of their management pathways. Reuse was identified as the most favourable end-of-life management option in terms of potential climate impact, followed by recycling, with landfill identified as being the least favourable option. Overall, current end-of-life management practices for WEEE in the UK were found to result in a net positive (i.e. beneficial) climatic effect, although this saving was found to reduce when WEEE recycled as non-obligated WEEE was not included. Overall, we recommend that future national policies should focus on formalising indirect WEEE collection pathways to help increase overall collection rates and, thus, reuse and recycling activities.

C. J. S. Clarke

Papers

Relativity on Curved Manifolds

The Analysis of Space-Time Singularities

The analysis of space-time singularities.

Space-time singularities

Evaluating the carbon footprint of WEEE management in the UK