Browsing by Browse by FOR 2008 "010504 Mathematical Aspects of General Relativity"

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a three-dimensional CR structure, which we call FRT metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of three-dimensional CR structures in terms of the Ricci curvature of the FRT metrics in the spirit of the results by Lewandowski et al. (Class Quantum Gravity 7(11):L241–L246, 1990) and also Hill et al. (Indiana Univ Math J 57(7):3131–3176, 2008. https://doi.org/10.1512/iumj.2008.57.3473).

In earlier work, 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 [S. McCormick, Adv. Theor. Math. Phys. 18, 799 (2014)]. 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 nonrigorously by Sudarsky and Wald [Phys. Rev. D 46, 1453 (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.

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 (Commun Math Phys 214(1):137-189, 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 (Commun Anal Geom 13(5):845-885, 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.

We use the techniques of Bartnik [5] to show that the space of solutions to the Einstein-Yang-Mills constraint equations on an asymptotically flat manifold with one end and zero boundary components, has a Hilbert manifold structure; the Einstein-Maxwell system can be considered as a special case. This is equivalent to the property of linearisation stability, which was studied in depth throughout the 70s [1, 2, 9, 11, 13, 18, 19]. This framework allows us to prove a conjecture of Sudarsky and Wald [22], namely that the validity of the first law of black hole thermodynamics is a suitable condition for stationarity. Since we work with a single end and no boundary conditions, this is equivalent to critical points of the ADM mass subject to variations fixing the Yang Mills charge corresponding exactly to stationary solutions. The natural extension to this work is to prove the second conjecture from [22], which is the case where an interior boundary is present; this will be addressed in future work.

Objective: To investigate effects of using different methods to identify under-reporters on trends in dietary intake and physical activity level (PAL) in non-under-reporters. Design: Dietary intake and activity pattern were recorded by 887 female university students from 1988 to 2003. Energy expenditure (EE) and PAL were measured using a factorial method. Nine methods were used to identify under-reporters. Trends between 1988 and 2003 were determined from linear regression of median data for each year. Results: Prevalence of under-reporting varied with the method used, ranging from 22.2% to 66.7%. Trends that were consistent using all methods were a decrease in per cent of energy from fat and no change in alcohol intake or PAL (P < 0.05). Trends in energy and macronutrient intakes varied with the method used. Using the four methods that excluded the most under-reporters there were increases in energy and gram carbohydrate (CHO) intakes (P < 0.05), but no change in gram fat intake. Using the four methods that excluded the fewest under-reporters there was no change in energy or gram CHO intake, but a decrease in gram fat intake (P < 0.05). Conclusion: Trends in energy, fat and CHO intakes in non-under-reporters varied with the method used to identify under-reporters.