Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes

A variety of gravitational dynamics problems in asymptotically anti-de Sitter (AdS) spacetime are amenable to efficient numerical solution using a common approach involving a null slicing of spacetime based on infalling geodesics, convenient exploitation of the residual diffeomorphism freedom, and use of spectral methods for discretizing and solving the resulting differential equations. Relevant issues and choices leading to this approach are discussed in detail. Three examples, motivated by applications to non-equilibrium dynamics in strongly coupled gauge theories, are discussed as instructive test cases. These are gravitational descriptions of homogeneous isotropization, collisions of planar shocks, and turbulent fluid flows in two spatial dimensions.Comment: 70 pages, 19 figures; v4: fixed minus sign typo in last term of eqn. (3.47

Bosonization and Fermion Liquids in Dimensions Greater Than One

(Revised, with postscript figures appended, corrections and added comments.) We develop and describe new approaches to the problem of interacting Fermions in spatial dimensions greater than one. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting Fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing Fermion liquids. The complicated functional renormalization group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the Bosonization of higher dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase space and small-angle scattering .... (7 figures, appended as a postscript file at the end of the TeX file).Comment: 48 text pages, plain TeX, BUP-JBM-

Coupling of Linearized Gravity to Nonrelativistic Test Particles: Dynamics in the General Laboratory Frame

The coupling of gravity to matter is explored in the linearized gravity limit. The usual derivation of gravity-matter couplings within the quantum-field-theoretic framework is reviewed. A number of inconsistencies between this derivation of the couplings, and the known results of tidal effects on test particles according to classical general relativity are pointed out. As a step towards resolving these inconsistencies, a General Laboratory Frame fixed on the worldline of an observer is constructed. In this frame, the dynamics of nonrelativistic test particles in the linearized gravity limit is studied, and their Hamiltonian dynamics is derived. It is shown that for stationary metrics this Hamiltonian reduces to the usual Hamiltonian for nonrelativistic particles undergoing geodesic motion. For nonstationary metrics with long-wavelength gravitational waves (GWs) present, it reduces to the Hamiltonian for a nonrelativistic particle undergoing geodesic \textit{deviation} motion. Arbitrary-wavelength GWs couple to the test particle through a vector-potential-like field $N_a$, the net result of the tidal forces that the GW induces in the system, namely, a local velocity field on the system induced by tidal effects as seen by an observer in the general laboratory frame. Effective electric and magnetic fields, which are related to the electric and magnetic parts of the Weyl tensor, are constructed from $N_a$ that obey equations of the same form as Maxwell's equations . A gedankin gravitational Aharonov-Bohm-type experiment using $N_a$ to measure the interference of quantum test particles is presented.Comment: 38 pages, 7 figures, written in ReVTeX. To appear in Physical Review D. Galley proofs corrections adde

Discrete spherical means of directional derivatives and Veronese maps

We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using the Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation

The Standard Cosmological Model and CMB Anisotropies

This is a course on cosmic microwave background (CMB) anisotropies in the standard cosmological model, designed for beginning graduate students and advanced undergraduates. ``Standard cosmological model'' in this context means a Universe dominated by some form of cold dark matter (CDM) with adiabatic perturbations generated at some initial epoch, e.g., Inflation, and left to evolve under gravity alone (which distinguishes it from defect models). The course is primarily theoretical and concerned with the physics of CMB anisotropies in this context and their relation to structure formation. Brief presentations of the uniform Big Bang model and of the observed large--scale structure of the Universe are given. The bulk of the course then focuses on the evolution of small perturbations to the uniform model and on the generation of temperature anisotropies in the CMB. The theoretical development is performed in the (pseudo--)Newtonian gauge because it aids intuitive understanding by providing a quick reference to classical (Newtonian) concepts. The fundamental goal of the course is not to arrive at a highly exact nor exhaustive calculation of the anisotropies, but rather to a good understanding of the basic physics that goes into such calculations.Comment: Course given at the International School of Space Science: 3K Cosmology, held in L'Aquila, Italy, September 1998. 44 pages with 4 figure

The extensional viscosity of a dilute suspension of spherical particles at intermediate microscale Reynolds numbers

The extensional viscosity of a dilute suspension of spherical particles (rigid spheres, viscous drops or gas bubbles) is computed for the case when the Reynolds number of the microscale disturbance motion R is not restricted to be small, as in the classical analysis of Einstein and Taylor. However, the present theory is restricted to steady axisymmetric pure straining flow (uniaxial extension). The rate of energy dissipation is expressed using the Bobyleff-Forsythe formula and then conditionally convergent integrals are removed explicitly. The problem is thereby reduced to a determination of the flow around a particle, subject to pure straining at infinity, followed (for rigid particles) by an evaluation of the volume integral of the vorticity squared. In the case of fluid particles, further integrals over the volume and surface of the particle are required. In the present paper, results are obtained numerically for 1 [less-than-or-eq, slant] R [less-than-or-eq, slant] 1000 for a rigid sphere, for a drop whose viscosity is equal to the viscosity of the ambient fluid, and for an inviscid drop (gas bubble). For the last case, limiting results are also obtained for R [rightward arrow] [infinity] using Levich's approach. All of these results show a strain-thickening behaviour which increases with the viscosity of the particle. The possibility of experimental verification of the results, which is complicated by the inapplicability of the approximation of material frame-indifference in this case, is discussed