Instabilities Of Geometrodynamic Evolution In The Hamiltonian Formulation Of General Relativity

Abstract

Difficulties with instability of numerical geometrodynamic evolution associate with the modeling of gravity waves generated by colliding black holes. This modeling is an integral part of the gravity wave detection effort (ground-based detectors LIGO and VIRGO and future space-based detector LISA). However, all algorithms have proven largely unsuccessful as they amplify (exponentially) nonphysical constraint-violating modes such that the numerical codes crash almost immediately after the modes appear. Equivalently, these models violate initial constraints (essentially enforcing conservation of energy and momentum) because of numerical errors and, subsequently, either imperfect numerical techniques or inadequate formulation of Hamilton's evolution equations cause rapid drift of the solutions off the constraint shell.

Another possibility seriously considered in these models concerns the inadequacy of Hamilton's equations off shell. This has resulted in hyperbolic reformulations of Einstein's equations that exclude acausal modes of the solutions (modes propagating between the normal and the null of a nullcone or out of the null cone), leading to more stable solutions. The development has resulted in marginally better algorithms. However, difficulties remain (this does not yield a sufficiently stable code). Improved understanding of the nature of instabilities is needed.

This research investigates instabilities caused by violation of the initial-value constraints. It considers the observation that the drift off shell of a source-free gravitational field equates with a transition from a source-free field to a field with a source. The simplest model permits, as a source, a scalar field that is not necessarily subject to standard energy conditions. Such a field resembles the one that is used by cosmologists for the description of inflation and acceleration phenomena or, historically preceding it, the C-field introduced originally by Hoyle and Narlikar. Unlike the approach taken in cosmology, this work considers associated exponential instability in the Hamiltonian formulation (ADM and its hyperbolic modifications). Particularly, this work looks at a field theory with a Lagrangian for a pre-existing source (or sourceless, though this does not change the formulation) and considers how introducing the C-field modifies the evolution of the system.

Study of the mechanism of instability might lead to better numerical evolution schemes, but this development is beyond the scope of the suggested work.