Abstract:
As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term 'critical phenomena'. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. This review gives an introduction to the phenomena, tries to summarize the essential features of what is happening, and then presents extensions and applications of this basic scenario. Critical phenomena are of interest particularly for creating surprising structure from simple equations, and for the light they throw on cosmic censorship and the generic dynamics of general relativity.

Abstract:
Lecture given at the workshop "Mathematical aspects of theories of gravitation", Stefan Banach International Mathematical Centre, 7 March 1996. A mini-introduction to critical phenomena in gravitational collapse is combined with a more detailed discussion of the regularity of the ``critical spacetimes'' dominating these phenomena.

Abstract:
We confirm recent numerical results of echoing and mass scaling in the gravitational collapse of a spherical Yang-Mills field by constructing the critical solution and its perturbations as an eigenvalue problem. Because the field equations are not scale-invariant, the Yang-Mills critical solution is asymptotically, rather than exactly, self-similar, but the methods for dealing with discrete self-similarity developed for the real scalar field can be generalized. We find an echoing period Delta = 0.73784 +/- 0.00002 and critical exponent for the black hole mass gamma = 0.1964 +/- 0.0007.

Abstract:
It is pointed out that the entropy of a membrane which is quantized perturbatively around a background position of fixed radius in a black hole spacetime is equal to the Bekenstein-Hawking entropy, if 1) the membrane surface is the horizon surface plus one Planck unit, and 2) its temperature is the Hawking temperature. (This is a comment on gr-qc 9411037.)

Abstract:
I construct a spherically symmetric solution for a massless real scalar field minimally coupled to general relativity which is discretely self-similar (DSS) and regular. This solution coincides with the intermediate attractor found by Choptuik in critical gravitational collapse. The echoing period is Delta = 3.4453 +/- 0.0005. The solution is continued to the future self-similarity horizon, which is also the future light cone of a naked singularity. The scalar field and metric are C1 but not C2 at this Cauchy horizon. The curvature is finite nevertheless, and the horizon carries regular null data. These are very nearly flat. The solution has exactly one growing perturbation mode, thus confirming the standard explanation for universality. The growth of this mode corresponds to a critical exponent of gamma = 0.374 +/- 0.001, in agreement with the best experimental value. I predict that in critical collapse dominated by a DSS critical solution, the scaling of the black hole mass shows a periodic wiggle, which like gamma is universal. My results carry over to the free complex scalar field. Connections with previous investigations of self-similar scalar field solutions are discussed, as well as an interpretation of Delta and gamma as anomalous dimensions.

Abstract:
This paper is an application of the ideas of the Born-Oppenheimer (or slow/fast) approximation in molecular physics and of the Isaacson (or short-wave) approximation in classical gravity to the canonical quantization of a perturbed minisuperspace model of the kind examined by Halliwell and Hawking. Its aim is the clarification of the role of the semiclassical approximation and the backreaction in such a model. Approximate solutions of the quantum model are constructed which are not semiclassical, and semiclassical solutions in which the quantum perturbations are highly excited.

Abstract:
The numerical relativity session at GR18 was dominated by physics results on binary black hole mergers. Several groups can now simulate these from a time when the post-Newtonian equations of motion are still applicable, through several orbits and the merger to the ringdown phase, obtaining plausible gravitational waves at infinity, and showing some evidence of convergence with resolution. The results of different groups roughly agree. This new-won confidence has been used by these groups to begin mapping out the (finite-dimensional) initial data space of the problem, with a particular focus on the effect of black hole spins, and the acceleration by gravitational wave recoil to hundreds of km/s of the final merged black hole. Other work was presented on a variety of topics, such as evolutions with matter, extreme mass ratio inspirals, and technical issues such as gauge choices.

Abstract:
By fine-tuning generic Cauchy data, critical phenomena have recently been discovered in the black hole/no black hole "phase transition" of various gravitating systems. For the spherisymmetric real scalar field system, we find the "critical" spacetime separating the two phases by demanding discrete scale-invariance, analyticity, and an additional reflection-type symmetry. The resulting nonlinear hyperbolic boundary value problem, with the rescaling factor Delta as the eigenvalue, is solved numerically by relaxation. We find Delta = 3.4439 +/- 0.0004.

Abstract:
Choptuik has demonstrated that naked singularities can arise in gravitational collapse from smooth, asymptotically flat initial data, and that such data have codimension one in spherical symmetry. Here we show, for perfect fluid matter with equation of state $p=\rho/3$, by perturbing around spherical symmetry, that such data have in fact codimension one in the full phase space, at least in a neighborhood of spherically symmetric data.

Abstract:
We review the problem of finding an apparent horizon in Cauchy data (Sigma, g_ab, K_ab) in three space dimensions without symmetries. We describe a family of algorithms which includes the pseudo-spectral apparent horizon finder of Nakamura et al. and the curvature flow method proposed by Tod as special cases. We suggest that other algorithms in the family may combine the speed of the former with the robustness of the latter. A numerical implementation for Cauchy data given on a grid in Cartesian coordinates is described, and tested on Brill-Lindquist and Kerr initial data. The new algorithm appears faster and more robust than previous ones.