Universal scaling and echoing in the gravitational collapse of a complex scalar field

This paper studies the gravitational collapse of a complex scalar field at the threshold for black hole formation, assuming that the collapse is spherically symmetric and continuously self-similar. A new solution of the coupled Einstein-scalar field equations is derived, after a small amount of numerical work with ordinary differential equations. The universal scaling and echoing behavior discovered by Choptuik in spherically symmetrical gravitational collapse appears in a somewhat different form. The properties of the end state of the collapse are derived: The collapse leaves behind an irregular outgoing pulse of scalar radiation, with an approximately flat spacetime within it.