We report a second example of critical behavior in gravitational collapse. Collapse of axisymmetric gravitational wave packets is computed numerically for a one-parameter family of initial data, A black hole first appears along the sequence at a critical parameter value p*. As with spherical scalarfield collapse, a power law is found to relate black-hole mass (the order parameter) and critical separation: MsH oc [pp*~S. The critical exponent is P 0.37, remarkably close to that observed by Choptuik. Near-critical evolutions produce echoes from the strong-field region which appear to exhibit scaling.

In this paper we present results from a new two-dimensional numerical relativity code used to study the interaction of gravitational waves with a black hole. The initial data correspond to a single black hole superimposed with time-symmetric gravitational waves (Brill waves). A gaugeinvariant method is presented for extracting the gravitational waves from the numerically generated spacetime. We show that the interaction between the gravitational wave and the black hole excites the quasinormal modes of the black hole. An extensive comparison of these results is made to blackhole perturbation theory. For low-amplitude initial gravitational waves, we find excellent agreement between the theoretically predicted t = 2 and t = 4 wave forms and the wave forms generated by the code. Additionally, a code test is performed wherein the propagation of the wave on the black-hole background is compared to the evolution predicted by perturbation theory. PACS number(s): 04.30.+~,95.3O.S1,97.60.Lf I. I N T R O D U C T I O N 45 3544

The evolution of physical and gauge degrees of freedom in the Einstein and Yang-Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can always be written in fluxconservative first-order symmetric hyperbolic form. This dynamical form is ideal for global analysis, analytic approximation methods such as gaugeinvariant perturbation theory, and numerical solution.

Nonspherical perturbation theory has been necessary to understand the meaning of radiation in spacetimes generated through fully nonlinear numerical relativity. Recently, perturbation techniques have been found to be successful for the time evolution of initial data found by nonlinear methods. Anticipating that such an approach will prove useful in a variety of problems, we give here both the practical steps, and a discussion of the underlying theory, for taking numerically generated data on an initial hypersurface as initial value data and extracting data that can be considered to be nonspherical perturbations.associated with curvilinear coordinate systems, and the effects of outer boundary conditions which are approximate. [3] We suggest here that at least part of the cure for this problem may lie in the use of the theory and techniques of nonspherical perturbations of the Schwarzschild spacetime ("NPS"). By this we mean the techniques for treating spacetimes as deviations, first order in some smallness parameter, from the Schwarzschild spacetime. These techniques differ from "linearized theory" which treats perturbations of the spacetime from Minkowski spacetime and which cannot describe black holes. The basic ideas and methods were set down by many authors and lead to "wave equations" for the even parity [4] and odd parity [5] perturbations.NPS has been used to compute outgoing radiation waveforms from a wide variety of black hole processes, including the scattering of waves [6], particles falling into a hole [7], and stellar collapse to form a hole [8]. The general scheme of NPS also underlies the techniques for extraction of radiation from numerically evolved spacetimes [9]. NPS computations have recently been used in conjunction with fully numerical evolution, as a code test [10] and as a strong-field radiation extraction procedure [3].Here we are interested in another sort of application of NPS theory. To understand such applications we consider an example: Two very relativistic neutron stars falling into each other, coalescing and forming a horizon, as depicted in Fig. 1. The curve "hypersurface," in Fig. 1, indicates a spacelike "initial" surface. The spacetime can be divided into three regions by this initial surface and the horizon. The early evolution, in region I, below the initial hypersurface, is highly dynamical and nonspherical. Spherical perturbation theory is clearly inapplicable. Above the initial surface the spacetime remains highly nonspherical in region II inside the event horizon, but outside the event horizon, in region III, it may be justified to consider the spacetime to be a perturbation of a Schwarzschild spacetime. This is essentially guaranteed if the initial hypersurface is chosen late enough, in some sense, after the formation of the horizon. The evolution in region III, then, is determined by cauchy data on the initial hypersurface exterior to the horizon. It is important to note that this is made possible by the fact that the horizon is a causal boundary which shields the outer...