4.7 Black hole charge and angular momentum

Given the scaling power law for the black hole mass in critical collapse, one would like to know what happens if one takes a generic one-parameter family of initial data with both electric charge and angular momentum (for suitable matter), and fine-tunes the parameter p to the black hole threshold. Does the mass still show power-law scaling? What happens to the dimensionless ratios and Q / M, with L being the black hole angular momentum and Q its electric charge? Tentative answers to both questions have been given using perturbations around spherically symmetric uncharged collapse.

4.7.1 Charge

Gundlach and Martín-García [78] have studied scalar massless electrodynamics in spherical symmetry. Clearly, the real scalar field critical solution of Choptuik is a solution of this system too. Less obviously, it remains a critical solution within massless (and in fact, massive) scalar electrodynamics in the sense that it still has only one growing perturbation mode within the enlarged solution space. Some of its perturbations carry electric charge, but as they are all decaying, electric charge is a subdominant effect. The charge of the black hole in the critical limit is dominated by the most slowly decaying of the charged modes. From this analysis, a universal power-law scaling of the black hole charge

was predicted. The predicted value of the critical exponent (in scalar electrodynamics) was subsequently verified in collapse simulations by Hod and Piran [86]. (The mass scales with as for the uncharged scalar field.) General considerations using dimensional analysis led Gundlach and Martín-García to the general prediction that the two critical exponents are always related, for any matter model, by the inequality

This has not yet been verified in any other matter model.

4.7.2 Angular momentum

Gundlach's results on nonspherically symmetric perturbations around spherical critical collapse of a perfect fluid [77] allow for initial data, and therefore black holes, with infinitesimal angular momentum. All nonspherical perturbations decrease towards the singularity. The situation is therefore similar to scalar electrodynamics versus the real scalar field. The critical solution of the more special model (here, the strictly spherically symmetric fluid) is still a critical solution within the more general model (a slightly nonspherical and slowly rotating fluid). In particular, axial perturbations (also called odd-parity perturbations) with angular dependence l =1 (i.e.\ dipole) will determine the angular momentum of the black hole produced in slightly supercritical collapse. Using a perturbation analysis similar to that of Gundlach and Martín-García [78], Gundlach [75] (see correction in [70]) has derived the angular momentum scaling law

 

For the range 0.123< k < 0.446 of equations of state, the angular momentum exponent is related to the mass exponent by

 

In particular for k =1/3, one gets . An angular momentum exponent was derived for the massless scalar field in [63] using second-order perturbation theory. Both results have not yet been tested against numerical collapse simulations.

Critical Phenomena in Gravitational Collapse Carsten Gundlach http://www.livingreviews.org/lrr-1999-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de