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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 definite spatial distribution of 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
found numerical evidence that between both asymptotic states, there is a
critical solution represented by a limit cycle in the modal space with period
.Comment: 9 pages, revtex4, 10 ps figures; Phys. Rev. D, in pres