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

Electromagnetic finite elements based on a four-potential variational principle

Electromagnetic finite elements based on a variational principle that uses the electromagnetic four-potential as a primary variable are derived. This choice is used to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of electromagnetic/mechanical systems that involve superconductors. The main advantages of the four-potential as a basis for finite element formulation are that the number of degrees of freedom per node remains modest as the problem dimensionally increases, that jump discontinuities on interfaces are naturally accommodated, and that statics as well as dynamics may be treated without any a priori approximations. The new elements are tested on an axisymmetric problem under steady state forcing conditions. The results are in excellent agreement with analytical solutions

Nonlinear Acoustics in Underwater and Biomedical Applications: Array Performance Degradation and Time Reversal Invariance

This dissertation describes a model for acoustic propagation in inhomogeneous flu- ids, and explores the focusing by arrays onto targets under various conditions. The work explores the use of arrays, in particular the time reversal array, for underwater and biomedical applications. Aspects of propagation and phasing which can lead to reduced focusing effectiveness are described. An acoustic wave equation was derived for the propagation of finite-amplitude waves in lossy time-varying inhomogeneous fluid media. The equation was solved numerically in both Cartesian and cylindrical geometries using the finite-difference time-domain (FDTD) method. It was found that time reversal arrays are sensitive to several debilitating factors. Focusing ability was determined to be adequate in the presence of temporal jitter in the time reversed signal only up to about one-sixth of a period. Thermoviscous absorption also had a debilitating effect on focal pressure for both linear and nonlinear propagation. It was also found that nonlinearity leads to degradation of focal pressure through amplification of the received signal at the array, and enhanced absorption in the shocked waveforms. This dissertation also examined the heating effects of focused ultrasound in a tissue-like medium. The application considered is therapeutic heating for hyperther- mia. The acoustic model and a thermal model for tissue were coupled to solve for transient and steady temperature profiles in tissue-like media. The Pennes bioheat equation was solved using the FDTD method to calculate the temperature fields in tissue-like media from focused acoustic sources. It was found that the temperature-dependence of the medium's background prop- erties can play an important role in the temperature predictions. Finite-amplitude effects contributed excess heat when source conditions were provided for nonlinear ef- fects to manifest themselves. The effect of medium heterogeneity was also found to be important in redistributing the acoustic and temperature fields, creating regions with hotter and colder temperatures than the mean by local scattering and lensing action. These temperature excursions from the mean were found to increase monotonically with increasing contrast in the medium's properties.Office of Naval Research (Code 321-TS

Braneworld black holes and black strings

This thesis involves the study of strong and weak gravity phenomenology within the braneworld paradigm. We begin with a general overview of the hypothesised concept of extra spatial dimensions and explain why they are so interesting. Turning next to the topic of classical four-dimensional black holes, we discuss their formation via gravitational collapse and indicate some of the strong observational evidence of their existence. We then merge the two independent theories of extra dimensions and black holes together to form braneworld black holes. Focusing our attention on two distinct braneworld scenarios, we examine the effects produced from either strong or weak gravity. The prospect of obtaining experimental verification of the existence of additional spacelike dimensions in the upcoming ground-based accelerators, makes the theoretical research of braneworld gravity within this thesis even more enticing. We start with a non-perturbative approach to look for exact, spherically symmetric star or black hole solutions on a Randall-Sundrum brane from the perspective of the five-dimensional spacetime. By fixing the background, we explore the permissible braneworld trajectories within it that correspond to a braneworld observer, the solutions of the brane Tolmann-Oppenheimer-Volkoff equations. A variety of static gravitating matter sources on the brane are obtained in a range of different backgrounds. Our final aim is a consistent brane embedding in a Schwarzschild- Anti de Sitter spacetime as these solutions are potential candidates for brane stars or black holes. The weak and dominant energy conditions determine the physically sensible solutions which have the interpretation of braneworld stars. We then study time-dependent trajectories as a possible description of time-dependent braneworld black holes. This work is then generalised by relaxing the simplifying assumption of Z(_2)-symmetry, previously imposed around the brane. Non-Z(_2) symmetric spacetimes are applicable in processes which concern only one side of the brane, for example black hole recoil or the emission of Hawking radiation. We determine that a subset of the allowed brane trajectories in an asymmetric background are exactly the same as the Z(_2)-symmetric case. Next, we explore perturbative gravity in the Hofava-Witten model of heterotic M-theory. The study of scalar and gravitational fluctuations determines that the radion mode is coupled to the bulk scalar field, indicating only one single degree of freedom. Our analysis also determines the instability of a black string. We then compute the complete mass spectrum of the graviton mode. Using the five-dimensional gravitational physics, we determine what the gravitational interaction an observer on the braneworld would perceive. This analysis involves the computation of the Newtonian potential between two test masses on the visible brane, together with the four-dimensional tensor structure of the massless graviton propagator. Finally, as an application to the earlier work, we comment on work which is in progress: the study of possible brane black hole solutions in low energy heterotic M-theory

An evaluation of the graviton propagator in de sitter space

The graviton propagator in four-dimensional de Sitter space is found in closed form. The vacuum state is taken to be the de Sitter-invariant “euclidean” or “Gibbons-Hawking” vacuum. The gauge-fixing term used is the standard choice introduced by Christensen and Duff. The propagator is given explicity in this gauge, and is found to be finite for points that are not null-related. The method used is new: mode-sums on the four-sphere are expressed as maximally symmetric bitensors. The result is then given in a completely geometric and coördinate-free form. This same method can easily be used for maximally symmetric spaces of general dimension, including anti-de Sitter space with supersymmetric boundary conditions, and for different choices of gauge

Topics in the Dynamics or General Relativity

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Introductory Lectures on Group Representations

Part I. Takeshi Hirai, Atomosphere in the theory of group representations [1]Part II. Nobuhiko Tatsuuma, General theory of unitary representation of locally compact groups [37

Absolute Value Boundedness, Operator Decomposition, and Stochastic Media and Equations

The research accomplished during this period is reported. Published abstracts and technical reports are listed. Articles presented include: boundedness of absolute values of generalized Fourier coefficients, propagation in stochastic media, and stationary conditions for stochastic differential equations