Andrew M. Abrahams

Equation of state in a strong magnetic field - Finite temperature and gradient corrections

The equation of state for condensed matter in a strong magnetic field is constructed. The regime for which statistical models and spherical Wigner-Seitz lattice cells are valid approximations is treated. The equation of state for a free nonrelativistic homogeneous electron gas in a uniform magnetic field is examined as a function of temperature, after which this treatment is refined by incorporating Coulomb interactions in a magnetic Thomas-Fermi model which allows for finite temperature. Gradient corrections to the zero-temperature equation of state are then evaluated by constructing a magnetic Thomas-Fermi-Dirac-Weizsaecker model, these corrections having a considerable effect on the zero-pressure density for matter in strong magnetic fields. Finally, the hydrostatic equilibrium equation for the surface structure of a neutron star is integrated using the presently computed equations of state. 52 refs.

Black-hole collisions from Brill-Lindquist initial data: Predictions of perturbation theory.

The Misner initial value solution for two momentarily stationary black holes has been the focus of much numerical study. We report here analytic results for an astrophysically similar initial solution, that of Brill and Lindquist (BL). Results are given from perturbation theory for initially close holes and are compared with available numerical results. A comparison is made of the radiation generated from the BL and the Misner initial values, and the physical meaning is discussed.

Molecules and chains in a strong magnetic field - Statistical treatment

A Thomas-Fermi-Dirac-Weizsaecker statistical model is developed and employed to investigate diatomic molecules and infinite molecular chains in strong magnetic fields. The standard magnetic Thomas-Fermi-Dirac kinetic, potential, and exchange energy functionals are supplemented by a gradient correction to the kinetic energy. The numerical method used for solving this system in two spatial dimensions is detailed. Numerical solutions for a wide range of magnetic strengths and elements are presented to demonstrate the robustness, as well as the limitations, of the statistical approach. These calculations qualitatively reproduce many of the results of detailed quantum mechanical treatments. For example, the fractional binding energy is greatest for low atomic numbers and for strong magnetic fields.

Calculation of gravitational waveforms from black hole collisions and disk collapse: Applying perturbation theory to numerical spacetimes.

Many simulations of gravitational collapse to black holes become inaccurate before the total emitted gravitational radiation can be determined. The main difficulty is that a significant component of the radiation is still in the near-zone, strong field region at the time the simulation breaks down. We show how to calculate the emitted waveform by matching the numerical simulation to a perturbation solution when the final state of the system approaches a Schwarzschild black hole. We apply the technique to two scenarios: the head-on collision of two black holes and the collapse of a disk to a black hole. This is the first reasonably accurate calculation of the radiation generated from colliding black holes that form from matter collapse.

Collisions of boosted black holes: Perturbation theory prediction of gravitational radiation.

We consider general relativistic Cauchy data representing two nonspinning, equal-mass black holes boosted toward each other. When the black holes are close enough to each other and their momentum is sufficiently high, an encompassing apparent horizon is present so the system can be viewed as a single, perturbed black hole. We employ gauge-invariant perturbation theory, and integrate the Zerilli equation to analyze these time-asymmetric data sets and compute gravitational waveforms and emitted energies. When coupled with a simple Newtonian analysis of the infall trajectory, we find striking agreement between the perturbation calculation of emitted energies and the results of fully general relativistic numerical simulations of time-symmetric initial data.

Equation of state in metals and cold stars: Evaluation of statistical models

The Thomas-Fermi-Dirac (TFD) statistical model for electronic structures if refined by Weizsacker gradient and correlation energy corrections. The resulting FTD-λWc model is then used to calculate the nonrelativistic, cold matter equation of state (EOS) in the Wigner-Seitz spherical cell approximation. The correction terms are important mainly at low densities near matter equilibrium (i.e., zero pressure). Inclusion of the gradient term removes many of the unphysical features of the TFD model. Results are summarized for several elements of astrophysical and theoretical importance. The mass-radius relation of a low-mass white dwarf model is calculated using the TDF-λWc EOS.

Critical phenomena and relativistic gravitational collapse

Recent calculations have shown the existence of critical phenomena in general relativity associated with the collapse of wavepackets of massless fields that are near, in parameter space, the onset of black hole formation (the critical point). Two physically distinct systems have been explored: collapse of spherically-symmetric massless scalar field and collapse of vacuum, axisymmetric gravitational waves. Nonlinear effects dominate near the critical point. Black-hole mass serves as an order parameter and has a power-law dependence on critical separation in the supercritical region of parameter space. Remarkably, the values of the critical exponent of the power law are nearly identical in the two systems. The nonlinearity induces the fields to oscillate. Each successive oscillation is an echo, obeying a spatial and temporal scaling relation.

Disk collapse in general relativity.

The radial collapse of a homogeneous disk of collisionless particles can be solved analytically in Newtonian gravitation. To solve the problem in general relativity, however, requires the full machinery of numerical relativity. The collapse of a disk is the simplest problem that exhibits the two most significant and challenging features of strong-field gravitation: black hole formation and gravitational wave generation. We carry out dynamical calculations of several different relativistic disk systems. We explore the growth of ring instabilities in equilibrium disks, and how they are suppressed by sufficient velocity dispersion. We calculate waveforms from oscillating disks, and from disks that undergo gravitational collapse to black holes. Studies of disk collapse to black holes should also be useful for developing new techniques for numerical relativity, such as apparent horizon boundary conditions for black hole spacetimes.