Self-similarity in Einstein-Maxwell-dilaton theories and critical collapse

We study continuously self-similar solutions of four-dimensional Einstein-Maxwell-dilaton theory, with an arbitrary dilaton coupling. Self-similarity is an emergent symmetry of gravitational collapse near the threshold of black hole formation. The resulting `critical collapse' picture has been intensively studied in the past for self-gravitating scalar fields or perfect fluids, but little is known concerning other systems. Here we assess the impact of gauge fields on critical collapse, in the context of low-energy string theories. Matter fields need not inherit the symmetries of a spacetime. We determine the homothetic conditions that scale-invariance of the metric imposes on the dilaton and electromagnetic fields, and we obtain their general solution. The inclusion of a potential for the dilaton is compatible with the homothetic conditions if and only if it is of the Liouville type. By imposing also spherical symmetry, a detailed analysis of critical collapse in these systems is possible by casting the field equations as an autonomous system. We find analytically that Choptuik's critical exponent depends on the dilaton coupling. Despite this and the presence of two novel fixed points, the electromagnetic field necessarily vanishes for the critical solution.