5.2 The renormalisation group as 5 Aspects of current research5 Aspects of current research

5.1 Phase diagrams 

In analogy with critical phenomena in statistical mechanics, let us call a graph of the black hole threshold in the phase space of some self-gravitating system a phase diagram. The full phase space is infinite-dimensional, but one can plot a two-dimensional submanifold. In such a plot the black hole threshold is generically a line, analogous to the fluid/gas dividing line in the pressure/temperature plane.

Interesting phenomena can be expected in systems that admit more complicated phase diagrams. The massive complex scalar field for example, admits stable stars as well as black holes and flat space as possible end states. There are three phase boundaries, and these should intersect somewhere. A generic two-parameter family of initial data is expected to intersect each boundary in a line, and the three lines should meet at a triple point.

Similarly, many systems admit both type I and type II phase transitions, for example the massive real scalar field, and the SU (2) Yang-Mills field in spherical symmetry. In a two-dimensional family of initial data, these should again generically show up as lines, and generically these lines should intersect. Is the black hole mass at the intersection finite or zero? Is there a third line that begins where the type I and type II lines meet?

Choptuik, Hirschmann and Marsa [42] have investigated this for a specific two-parameter family of initial data for the spherical SU (2) Yang-Mills field, using a numerical evolution code that can follow the time evolutions for long after a black hole has formed. As known previously, the type I phase transition is mediated by the static Bartnik-McKinnon solution, which has one growing perturbation mode. The type II transition is mediated by a DSS solution with one growing mode. There is a third type of phase transition along a third line which meets the intersection of the type I and type II lines. On both sides of this ``type III'' phase transition the final state is a Schwarzschild black hole with zero Yang-Mills field strength, but the final state is distinguished by the value of the Yang-Mills gauge potential at infinity. (The system has two distinct vacuum states.) The critical solution is an unstable black hole with Yang-Mills hair, which collapses to a hairless Schwarzschild black hole with either vacuum state of the Yang-Mills field, depending on the sign of its one growing perturbation mode. The critical solution is not unique, but is a member of a 1-parameter family of hairy black holes parameterized by their mass. At the triple point the family ends in a zero mass black hole.

5.2 The renormalisation group as 5 Aspects of current research5 Aspects of current research

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de