The wave equation on the Schwarzschild metric II: Local decay for the spin 2 Regge Wheeler equation

Odd-type spin 2 perturbations of Einstein's equation can be reduced to the scalar Regge-Wheeler equation. We show that the weighted norms of solutions are in L^2 of time and space. This result uses commutator methods and applies uniformly to all relevant spherical harmonics.Comment: AMS-LaTeX, 8 pages with 1 figure. There is an errata to this paper at gr-qc/060807

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space

We provide a uniform decay estimate of Morawetz type for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. This estimate is both uniform in space and time, so in particular it implies a uniform bound on the sup norm of solutions which can be given in terms of certain inverse powers of the radial and advanced/retarded time coordinate variables. As a model application, we show these estimates give a very simple proof small amplitude scattering for nonlinear scalar fields with higher than cubic interactions.Comment: 24 page

Stability and Instability of Extreme Reissner-Nordstr\"om Black Hole Spacetimes for Linear Scalar Perturbations I

We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon. The fundamental new aspect of this problem is the degeneracy of the redshift on the event horizon. Several new analytical features of degenerate horizons are also presented.Comment: 37 pages, 11 figures; published version of results contained in the first part of arXiv:1006.0283, various new results adde

Strichartz estimates on Schwarzschild black hole backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place

Theories of practice and public health:understanding (un)healthy practices

Psychological understandings and individualistic theories of human behaviour and behaviour change have dominated both academic research and interventions at the ‘coalface’ of public health. Meanwhile, efforts to understand persistent inequalities in health point to structural factors, but fail to show exactly how these translate into the daily lives (and hence health) of different sectors of the population. In this paper, we suggest that social theories of practice provide an alternative paradigm to both approaches, informing significantly new ways of conceptualising and responding to some of the most pressing contemporary challenges in public health. We introduce and discuss the relevance of such an approach with reference to tobacco smoking, focusing on the life course of smoking as a practice, rather than on the characteristics of individual smokers or on broad social determinants of health. This move forces us to consider the material and symbolic elements of which smoking is comprised, and to follow the ways in which these elements have changed over time. Some of these developments have to do with the relation between smoking and other practices such as drinking alcohol, relaxing and socialising. We suggest that intervening in the future of smoking depends, in part, on understanding the nature of these alliances, and how sets of practices co-evolve. We conclude by reflecting on the implications of taking social practices as the central focus of public health policy, commenting on the benefits of such a paradigmatic turn, and on the challenges that this presents for established methods, policies and programmes