Galerkin method in the gravitational collapse: A dynamical system approach

We study the general dynamics of the spherically symmetric gravitational collapse of a massless scalar field. We apply the Galerkin projection method to transform a system of partial differential equations into a set of ordinary differential equations for modal coefficients, after a convenient truncation procedure, largely applied to problems of turbulence. In the present case, we have generated a finite dynamical system that reproduces the essential features of the dynamics of the gravitational collapse, even for a lower order of truncation. Each initial condition in the space of modal coefficients corresponds to a well defined spatial distribution of the scalar field. Numerical experiments with the dynamical system show that, depending on the strength of the scalar field packet, the formation of black holes or the dispersion of the scalar field leaving behind flat spacetime are the two main outcomes. We also find numerical evidence that between both asymptotic states there is a critical solution represented by a limit cycle in the modal space with a period Δu~3.55.