Session E10: Focus Session: Numerical Relativity

Solvig Einstein's Equations Using Spectral Methods

Spectral methods provide a promising approach for numerical solutions of the vacuum Einstein equations for spacetimes containing black holes. Besides offering exponential convergence for smooth solutions, these methods greatly simplify the treatment of boundaries, especially boundaries of excised regions inside black holes. A spectral code for solving multidimensional hyperbolic and elliptic equations is described, and preliminary simulations of binary black hole spacetimes using this code are presented.   

Constructing Helically Symmetric Spacetimes

A number of people have recently begun work toward the construction of helically symmetric spacetimes that model binary systems of compact objects. Imposing helical symmetry will allow us to numerically solve the full set of Einstein equations, and the resulting spacetimes could yield accurate initial data for inspiral simulations. One of us (Uryu) has completed a code for constructing binary neutron stars, however convergence has not yet been attained. Starting from a 3D linear scalar field, we have explored the convergence properties of a series of increasingly complex toy problems.   

Dynamical evolution of quasi-circular binary black hole data

We present fully nonlinear dynamical evolutions of binary black hole data, whose orbital parameters are specified via the effective potential method for determining quasi-circular orbits. The cases studied range from the Cook-Baumgarte innermost stable circular orbit (ISCO) to significantly beyond that separation. In all cases we find the black holes to coalesce (as determined by the appearance of a common apparent horizon) in less than half an orbital period, indicating that the holes are not in quasi-circular orbits but are in fact nearly plunging together. We have studied the dynamics of the final black hole and determined its physical parameters, such as spin, mass and oscillation frequency.   

A New Code for Relativistic Magnetohydrodynamics in Dynamical Spacetimes

Several problems at the forefront of theoretical astrophysics require treatment of magnetic phenomena in full general relativity (GR). Such problems include the origin of gamma-ray bursts, magnetic braking of differential rotation in nascent neutron stars arising from stellar core collapse or binary merger, the formation of disks around newly-formed black holes, etc. To tackle these interesting, unsolved problems involving both GR and magnetohydrodynamics (MHD), we have developed a GR-MHD code capable of evolving magnetized fluids in dynamical spacetimes. We evolve the metric by integrating the BSSN equations, and use a high resolution shock capturing scheme to handle the MHD. We will describe the formulation of the coupled Einstein-Maxwell-MHD evolution equations and the algorithms used to integrate them. We will also present some code tests, giving particular attention to tests involving MHD waves induced by a gravitational wave.   

Why Enforce the Hamiltonian Constraint in Numerical Relativity?

The indefinite sign of the Hamiltonian constraint means that solutions to Einstein's equations must achieve a delicate balance --- often among almost canceling large terms. If numerical errors cause violation of the Hamiltonian constraint, the failure of the delicate balance can lead to qualitatively wrong behavior rather than just a lesser accuracy. This issue is different from instabilities caused by constraint-violating modes. Examples of numerical simulations of collapsing cosmological spacetimes exhibiting local mixmaster dynamics with and without Hamiltonian constraint enforcement will be presented. Criteria to relate a measure of constraint violation to the time scale for the development of wrong behavior will be discussed.   

Towards Accurate Modeling of Binary Black Holes

Massive black holes (MBHs) that are believed to reside at the centers of all galaxies with bulges will form a binary and coalesce into each other following a galactic merger. The final stage of MBH binary evolution is a strong source of low frequency gravitational waves for the joint NASA/ESA LISA mission. The merging phase of the coalescence will provide an excellent opportunity to investigate Einstein's theory of general relativity in a strong nonlinear regime. Encouraging progress has been made recently in numerical relativity simulations of orbiting/merging black holes; however a long term and accurate evolution of a realistic situation has yet to be demonstrated. Using mesh refinement techniques, we focus on accurate simulations of the source (merging binary black holes) and accurate extraction of the generated gravitational waves. We study a head-on collision as the first model problem for a thorough validation of our approach. I will discuss some of the issues involved and present recent results.   

A global binary black hole metric via asymptotic matching of post-Newtonian and black hole perturbation expansions

We present an astrophysically realistic approximate global metric for a binary black hole spacetime that could be used to construct initial data for numerical relativity. This metric is obtained by asymptotically matching a post-Newtonian metric for a binary system to a perturbed Schwarzschild metric for each hole. In the {\it internal zone} near each black hole, the metric is given by the Schwarzschild solution plus a quadrupolar perturbation corresponding to an external gravitational field. In the {\it near zone}, well outside each black hole but less than a reduced wavelength from the center of mass of the binary, the metric is given by a post-Newtonian expansion including the lowest-order deviations from flat spacetime. When the near zone overlaps each internal zone in a {\it buffer zone}, the post-Newtonian and perturbed Schwarzschild metrics can be asymptotically matched to each other. By demanding matching (over the 3-volume of the buffer zone) rather than patching (choosing a particular 2-surface in the buffer zone), we guarantee that the metric and all of its derivatives are asymptotic to each other in the buffer zone.