Session D10: Techniques of Numerical Relativity

Generalized Harmonic Gauge Drivers

Recent work on the development of gauge drivers for the generalized harmonic Einstein system will be presented. These new gauge drivers allow a large class of gauge (i.e. coordinate) conditions to be imposed while keeping the generalized harmonic representation of the Einstein system hyperbolic. This class of gauge conditions includes as special cases many of the standard conditions used in numerical relativity: e.g. Bona-Masso slicing, Gamma-drivers, etc. New gauge-controlling boundary conditions will be discussed, along with numerical results that illustrate the effectiveness of this new gauge driver system.   

Evaluating event horizon finding techniques

Event horizons are the defining physical features of black hole spacetimes, and are of considerable interest in studying black hole dynamics. Because of their global nature, event horizons can only be determined after the end of a numerical simulation. Methods of finding event horizons in numerically-generated spacetimes are based on the fact that outgoing null geodesics near an event horizon converge exponentially to the horizon when followed backwards in time. Two existing methods in the literature are discussed: an individual-geodesic method and a level-set method. A third and intermediate method, a surface-flow method, is presented here and implemented numerically alongside the geodesic method. Both single black holes and head-on black hole mergers are explored as test cases. The most robust method for black hole mergers is found to be the individual-geodesic method. The presented techniques are remarkably accurate and allow to track the event-horizon through several quasi-normal oscillations during ringdown of the binary merger.   

Gauge conditions for numerical relativity

The standard 1+log slicing and Gamma driver shift conditions, coupled with the BSSN evolution equations, have been remarkably successful for the binary black hole problem. I will discuss some features of these gauge conditions, why they work and how they might be modified for use with other formulations of the evolution equations.   

Probing the Binary Black Hole Merger Regime with Scalar Perturbations

We present results obtained by scattering a scalar field off the curved background of a coalescing binary black hole system. A massless scalar field is evolved on a fixed background provided by hypersurfaces generated from a binary black hole inspiral. We show that the scalar field scattered from the merger region exhibits quasinormal ringing once a common apparent horizon surrounds the two black holes. This occurs earlier than the onset of the perturbative regime as measured by the start of the quasinormal ringing in the gravitational waveforms. We also use the scalar quasinormal frequency to associate a mass and a spin with each hypersurface, and observe the compatibility of this constraint with the horizon mass and spin computed from the dynamical horizon framework.   

Rapidly spinning binary black hole initial data

Numerical simulations of binary-black-hole spacetimes must begin with initial data that both satisfy the constraint equations of general relativity and lead to evolutions with the desired physical properties. We use the extended conformal thin sandwich equations to construct constraint-satisfying binary-black-hole initial data with i) nearly-maximal spins aligned with the orbital angular momentum and ii) low orbital eccentricities. Specifically, we construct conformally-flat data with dimensionless spins larger than 0.97 and conformally-curved data with spins larger than 0.995.   

Stability of Iterative Algorithms for Rotating Neutron Stars

Similar methods have been used to construct models of rapidly rotating stars, in Newtonian and relativistic contexts. The choice of method has been based on numerical experiments, which indicate that particular methods converge quickly to a solution, while others diverge. The theory underlying these differences, however, has not been understood. In an attempt to provide a better theoretical understanding, we analytically examine the behavior of different iterative schemes near an exact solution. We find the spectrum of the linearized iteration operator and show for self-consistent field methods that iterative instability corresponds to unstable modes of this operator.   

Numerical Relativity from a Gauge Theory Perspective

We present some recent results in a program to discretize the first-order Hilbert-Palatini action for gravity on a simplicial complex as a first step towards computing numerical relativity simulations in a fully gauge-covariant manner. The tetrad and spin connection, the dynamical variables of this theory, discretize naturally on the struts of the complex, and the resulting action is both locally Lorentz and diffeomorphically invariant. Because constraints are associated with these symmetry transformations, the evolutions which result from the Euler-Lagrange procedure are exactly constraint-preserving. This discretization procedure introduces extra degrees of freedom, in much the same way as lattice quantum gauge theory simulations, but we expect theoretically that these will be irrelevant at physical scales in our simulations. We are presently attempting to verify this computationally.