Race, Gender, Sexuality, Ability, Identity and Cycling, Blog 5

Student blog posts from the Great VCU Bike Race Book

First Law of Black Hole Mechanics as a Condition for Stationarity

In earlier work [arXiv:1302.1237], we provided a Hilbert manifold structure for the phase space for the Einstein-Yang-Mills equations, and used this to prove a condition for initial data to be stationary. Here we use the same phase space to consider the evolution of initial data exterior to some closed 2-surface boundary, and establish a condition for stationarity in this case. It is shown that the differential relationship given in the first law of black hole mechanics is exactly the condition required for the initial data to be stationary; this was first argued non-rigorously by Sudarsky and Wald in 1992. Furthermore, we give evidence to suggest that if this differential relationship holds then the boundary surface is the bifurcation surface of a bifurcate Killing horizon.Comment: 20 page

A note on mass-minimising extensions

A conjecture related to the Bartnik quasilocal mass, is that the infimum of the ADM energy, over an appropriate space of extensions to a compact 3-manifold with boundary, is realised by a static metric. It was shown by Corvino [Comm. Math. Phys. 214(1), (2000)] that if the infimum is indeed achieved, then it is achieved by a static metric; however, the more difficult question of whether or not the infimum is achieved, is still an open problem. Bartnik [Comm. Anal. Geom. 13(5), (2005)] then proved that critical points of the ADM mass, over the space of solutions to the Einstein constraints on an asymptotically flat manifold without boundary, correspond to stationary solutions. In that article, he stated that it should be possible to use a similar construction to provide a more natural proof of Corvino's result. In the first part of this note, we discuss the required modifications to Bartnik's argument to adapt it to include a boundary. Assuming that certain results concerning a Hilbert manifold structure for the space of solutions carry over to the case considered here, we then demonstrate how Bartnik's proof can be modified to consider the simpler case of scalar-flat extensions and obtain Corvino's result. In the second part of this note, we consider a space of extensions in a fixed conformal class. Sufficient conditions are given to ensure that the infimum is realised within this class.Comment: 17 pages. Substantial changes to Section 3. Updated to agree with published versio

Sustainable Professional Practice

The project aims were to examine the ways in which the two distance learning social work programmes (The Open University (OU)and Charles Sturt University(CSU)) operate - looking at pedagogies and in particular how learning and teaching works in the practicum. Exchange visits were organised with Associate Professor Bowles spending 10 days at The Open University in June 2009, and Mick McCormick pending time at CSU in August/September 2009. Our aim was to investigate the different and similar ways in which we approached the teaching of practitioners in social work. We gained many benefits from our contacts and had opportunity to meet and work with social work academics and input to teaching and learning within respective academic institutions. Many of our findings are reflected in recently published work, or work in publication. Findings also have an ongoing impact on the production, delivery and review of our respective practice learning social work programmes

On a Minkowski-like inequality for asymptotically flat static manifolds

The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R^n; for example, such an inequality holds for surfaces in spatial Schwarzschild and AdS-Schwarzschild manifolds. In this note, we adapt a recent analysis of Y. Wei to prove a Minkowski-like inequality for general static asymptotically flat manifolds.Comment: 10 pages. Proc. Amer. Math. Soc. V4: Fixed typo in eq (1.1

Propeller dynamic and aeroelastic effects

Various aspects of propeller blade dynamics are considered including those factors which are exciting the blades and the dynamic response of the blades to the excitations. Methods for treating this dynamic system are described and problems are discussed which may arise with advanced turboprop designs employing thin, swept blades