A DYNAMICAL SYSTEM APPROACH TO CRITICAL GRAVITATIONAL COLLAPSE

Abstract

The general dynamics of the spherically symmetric gravitational collapse of a massless scalar field is examined. We apply the Galerkin projection method to transform the partial differential equations ruling the dynamics of the collapse into a finite set of ordinary differential equations for the modal coefficients, after a convenient truncation procedure. We thus generate a dynamical system that not only reproduces the essential features of the dynamics of the gravitational collapse, even for a low order of truncation, but also discloses the structure of the dynamics. Each initial condition in modal space corresponds to a well defined spatial distribution of the scalar field. Numerical experiments with the system show that two main asymptotic configurations are possible: either the formation of a black hole or the dispersion of the scalar field leaving behind flat spacetime. Between the two asymptotic states there is a critical solution represented by a periodic orbit in the modal space with period Δ u ≈ 3.55. This periodic orbit acts as an intermediate attractor for the dynamics, being a limiting configuration for orbits generated from initial conditions in a critical surface of codimension one in the N-dimensional modal space. The compactness and fractality of this surface are briefly discussed.