Automatic adaptive grid refinement for the Euler equations

A method of adaptive grid refinement for the solution of the steady Euler equations for transonic flow is presented. Algorithm automatically decides where the coarse grid accuracy is insufficient, and creates locally uniform refined grids in these regions. This typically occurs at the leading and trailing edges. The solution is then integrated to steady state using the same integrator (FLO52) in the interior of each grid. The boundary conditions needed on the fine grids are examined and the importance of treating the fine/coarse grid inerface conservatively is discussed. Numerical results are presented

The Singularity in Generic Gravitational Collapse Is Spacelike, Local, and Oscillatory

A longstanding conjecture by Belinskii, Khalatnikov, and Lifshitz that the singularity in generic gravitational collapse is spacelike, local, and oscillatory is explored analytically and numerically in spatially inhomogeneous cosmological spacetimes. With a convenient choice of variables, it can be seen analytically how nonlinear terms in Einstein's equations control the approach to the singularity and cause oscillatory behavior. The analytic picture requires the drastic assumption that each spatial point evolves toward the singularity as an independent spatially homogeneous universe. In every case, detailed numerical simulations of the full Einstein evolution equations support this assumption.Comment: 7 pages includes 4 figures. Uses Revtex and psfig. Received "honorable mention" in 1998 Gravity Research Foundation essay contest. Submitted to Mod. Phys. Lett.

Hunting Local Mixmaster Dynamics in Spatially Inhomogeneous Cosmologies

Heuristic arguments and numerical simulations support the Belinskii et al (BKL) claim that the approach to the singularity in generic gravitational collapse is characterized by local Mixmaster dynamics (LMD). Here, one way to identify LMD in collapsing spatially inhomogeneous cosmologies is explored. By writing the metric of one spacetime in the standard variables of another, signatures for LMD may be found. Such signatures for the dynamics of spatially homogeneous Mixmaster models in the variables of U(1)-symmetric cosmologies are reviewed. Similar constructions for U(1)-symmetric spacetimes in terms of the dynamics of generic $T^2$-symmetric spacetime are presented.Comment: 17 pages, 5 figures. Contribution to CQG Special Issue "A Spacetime Safari: Essays in Honour of Vincent Moncrief

The Mellin Transform Technique for the Extraction of the Gluon Density

A new method is presented to determine the gluon density in the proton from jet production in deeply inelastic scattering. By using the technique of Mellin transforms not only for the solution of the scale evolution equation of the parton densities but also for the evaluation of scattering cross sections, the gluon density can be extracted in next-to-leading order QCD. The method described in this paper is, however, more general, and can be used in situations where a repeated fast numerical evaluation of scattering cross sections for varying parton distribution functions is required.Comment: 13 pages (LaTeX); 2 figures are included via epsfig; the corresponding postscript files are uuencode

Spatio-Temporal Scaling of Solar Surface Flows

The Sun provides an excellent natural laboratory for nonlinear phenomena. We use motions of magnetic bright points on the solar surface, at the smallest scales yet observed, to study the small scale dynamics of the photospheric plasma. The paths of the bright points are analyzed within a continuous time random walk framework. Their spatial and temporal scaling suggest that the observed motions are the walks of imperfectly correlated tracers on a turbulent fluid flow in the lanes between granular convection cells.Comment: Now Accepted by Physical Review Letter

A Radio Flare from GRB 020405: Evidence for a Uniform Medium Around a Massive Stellar Progenitor

We present radio observations of GRB 020405 starting 1.2 days after the burst, which reveal a rapidly-fading ``radio flare''. Based on its temporal and spectral properties, we interpret the radio flare as emission from the reverse shock. This scenario rules out a circumburst medium with a radial density profile \rho ~ r^{-2} expected around a mass-losing massive star, since in that case the reverse shock emission decays on the timescale of the burst duration t~100 s. Using published optical and X-ray data, along with the radio data presented here, we further show that a self-consistent model requires collimated ejecta with an opening angle of 6 degrees (t_j~0.95 days). As a consequence of the early jet break, the late-time (t>10 days) emission measured with the Hubble Space Telescope significantly deviates from an extrapolation of the early, ground-based data. This, along with an unusually red spectrum, F_\nu \~ \nu^{-3.9}, strengthens the case for a supernova that exploded at about the same time as GRB 020405, thus pointing to a massive stellar progenitor for this burst. This is the first clear association of a massive progenitor with a uniform medium, indicating that a \rho ~ r^{-2} profile is not a required signature, and in fact may not be present on the lengthscales probed by the afterglow in the majority of bursts.Comment: Submitted to ApJ; 14 pages, 2 tables, 3 figure

Evidence for an oscillatory singularity in generic U(1) symmetric cosmologies on T3×RT^3 \times R

A longstanding conjecture by Belinskii, Lifshitz, and Khalatnikov that the singularity in generic gravitational collapse is locally oscillatory is tested numerically in vacuum, U(1) symmetric cosmological spacetimes on $T^3 \times R$. If the velocity term dominated (VTD) solution to Einstein's equations is substituted into the Hamiltonian for the full Einstein evolution equations, one term is found to grow exponentially. This generates a prediction that oscillatory behavior involving this term and another (which the VTD solution causes to decay exponentially) should be observed in the approach to the singularity. Numerical simulations strongly support this prediction.Comment: 15 pages, Revtex, includes 12 figures, psfig. High resolution versions of figures 7, 8, 9, and 11 may be obtained from anonymous ftp to ftp://vela.acs.oakland.edu/pub/berger/u1genfig

Harmonic coordinate method for simulating generic singularities

This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.Comment: 5 pages, 4 figures, Revtex, discussion expanded, references adde

On Singularity Resolution in Quantum Gravity

We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schr\"odinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute "singularity resolution" in quantum gravity.Comment: 12 page

Optical properties of periodic systems within the current-current response framework: pitfalls and remedies

We compare the optical absorption of extended systems using the density-density and current-current linear response functions calculated within many-body perturbation theory. The two approaches are formally equivalent for a finite momentum $\mathbf{q}$ of the external perturbation. At $\mathbf{q}=\mathbf{0}$, however, the equivalence is maintained only if a small $q$ expansion of the density-density response function is used. Moreover, in practical calculations this equivalence can be lost if one naively extends the strategies usually employed in the density-based approach to the current-based approach. Specifically we discuss the use of a smearing parameter or of the quasiparticle lifetimes to describe the finite width of the spectral peaks and the inclusion of electron-hole interaction. In those instances we show that the incorrect definition of the velocity operator and the violation of the conductivity sum rule introduce unphysical features in the optical absorption spectra of three paradigmatic systems: silicon (semiconductor), copper (metal) and lithium fluoride (insulator). We then demonstrate how to correctly introduce lifetime effects and electron-hole interactions within the current-based approach.Comment: 17 pages, 6 figure