The Constraints in Spherically Symmetric General Relativity I --- Optical Scalars, Foliations, Bounds on the Configuration Space Variables and the Positivity of the Quasi-Local Mass

We examine the constraints of spherically symmetric general relativity with one asymptotically flat region, exploiting both the traditional metric variables and variables constructed from the optical scalars. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints which are related by time reversal. We introduce a one-parameter family of linear extrinsic time foliations of spacetime. The values of the parameter yielding globally valid gauges correspond to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasi-local mass on spheres of fixed proper radius which we prove is positive when the constraints are satisfied. Underpinning the proof are various local bounds on the configuration variables. We prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we provide a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.Comment: 45 pages, Plain Tex, 1 figure available from the authors

Hamilton's equations for a fluid membrane

Consider a homogenous fluid membrane described by the Helfrich-Canham energy, quadratic in the mean curvature of the membrane surface. The shape equation that determines equilibrium configurations is fourth order in derivatives and cubic in the mean curvature. We introduce a Hamiltonian formulation of this equation which dismantles it into a set of coupled first order equations. This involves interpreting the Helfrich-Canham energy as an action; equilibrium surfaces are generated by the evolution of space curves. Two features complicate the implementation of a Hamiltonian framework: (i) The action involves second derivatives. This requires treating the velocity as a phase space variable and the introduction of its conjugate momentum. The canonical Hamiltonian is constructed on this phase space. (ii) The action possesses a local symmetry -- reparametrization invariance. The two labels we use to parametrize points on the surface are themselves physically irrelevant. This symmetry implies primary constraints, one for each label, that need to be implemented within the Hamiltonian. The two lagrange multipliers associated with these constraints are identified as the components of the acceleration tangential to the surface. The conservation of the primary constraints imply two secondary constraints, fixing the tangential components of the momentum conjugate to the position. Hamilton's equations are derived and the appropriate initial conditions on the phase space variables are identified. Finally, it is shown how the shape equation can be reconstructed from these equations.Comment: 24 page

Energy-Efficient Management of Data Center Resources for Cloud Computing: A Vision, Architectural Elements, and Open Challenges

Cloud computing is offering utility-oriented IT services to users worldwide. Based on a pay-as-you-go model, it enables hosting of pervasive applications from consumer, scientific, and business domains. However, data centers hosting Cloud applications consume huge amounts of energy, contributing to high operational costs and carbon footprints to the environment. Therefore, we need Green Cloud computing solutions that can not only save energy for the environment but also reduce operational costs. This paper presents vision, challenges, and architectural elements for energy-efficient management of Cloud computing environments. We focus on the development of dynamic resource provisioning and allocation algorithms that consider the synergy between various data center infrastructures (i.e., the hardware, power units, cooling and software), and holistically work to boost data center energy efficiency and performance. In particular, this paper proposes (a) architectural principles for energy-efficient management of Clouds; (b) energy-efficient resource allocation policies and scheduling algorithms considering quality-of-service expectations, and devices power usage characteristics; and (c) a novel software technology for energy-efficient management of Clouds. We have validated our approach by conducting a set of rigorous performance evaluation study using the CloudSim toolkit. The results demonstrate that Cloud computing model has immense potential as it offers significant performance gains as regards to response time and cost saving under dynamic workload scenarios.Comment: 12 pages, 5 figures,Proceedings of the 2010 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 2010), Las Vegas, USA, July 12-15, 201

Extended objects with edges

We examine, from a geometrical point of view, the dynamics of a relativistic extended object with loaded edges. In the case of a Dirac-Nambu-Goto [DNG] object with DNG edges, the worldsheet $m$ generated by the parent object is, as in the case without boundary, an extremal timelike surface in spacetime. Using simple variational arguments, we demonstrate that the worldsheet of each edge is a constant mean curvature embedded timelike hypersurface on $m$, which coincides with its boundary, $\partial m$. The constant is equal in magnitude to the ratio of the bulk to the edge tension. The edge, in turn, exerts a dynamical influence on the motion of the parent through the boundary conditions induced on $m$, specifically that the traces of the projections of the extrinsic curvatures of $m$ onto $\partial m$ vanish.Comment: 13 pages, latex, published in Phys. Rev. D55, 2388 (1997