Critical gravitational collapse with angular momentum

Critical collapse in general relativity refers to phenomena that occur at the threshold, in the space of initial data, between data that lead to black hole formation (collapse) and those that do not. Regular initial data can be classified as supercritical or subcritical according to whether or not they form a black hole. We refer to the boundary between supercritical and subcritical data as the black-hole threshold, or the critical surface. In typeII critical collapse, the black-hole mass formed by supercritical data becomes arbitrarily small as the threshold is approached, and scales as a universal power of distance from this threshold. The exponent in these power laws is referred to as the critical exponent. Critical collapse was first reported in the seminal work of Choptuik [1], who performed numerical time evolutions of a massless scalar field in spherical symmetry. Soon afterwards, similar results were reported for a radiation fluid, i.e. a perfect fluid with the ultra-relativistic equation of state P = ρ/3 [2] (where P is the pressure and ρ the total energy density), and for axisymmetric gravitational waves in vacuum [3]. The literature on numerous further numerical experiments as well as theoretical derivations of the scaling laws is reviewed in [4]. In [5], one of us (CG) showed that the spherically symmetric, continuously self-similar critical solution for perfect fluid collapse with the equation of state P = κρ has only a single (l = 0) unstable mode for the range 1/9 < κ . 0.49, which includes radiation fluids with κ = 1/3. Based on this, and the more general theory given in [6], CG predicted power-law scaling for the blackhole mass and angular momentum for initial data with small deviations from spherical symmetry, and computed numerical values for the critical exponents. In [7], the other one of us (TWB), together with Montero, carried out the first critical collapse simulations of a radiation fluid in the absence of spherical symmetry. More recently, we generalized these simulations to study critical collapse with angular momentum [8]. Specifically, we considered a two-parameter family of initial data describing rotating radiation fluids, with one parameter η controlling the strength of the initial data and a second parameter Ω their angular momentum. These simulations confirmed the critical exponents found in [6] and provided evidence for their universality. In Sec. II we provide a self-contained derivation of the scaling laws in rotating critical collapse, and in Sec. III we demonstrate agreement with the numerical results of [8] for radiation fluids with κ = 1/3. Sec. IV contains a summary and discussion of our results.