We study the power-law tails in the evolution of massless fields around a fixed background geometry corresponding to a black hole. We give analytical arguments for their existence at scri+, at the future horizon, and at future timelike infinity. We confirm their existence with numerical integrations of the curved spacetime wave equation on the background of a Schwarzschild and a Reissner-Nordstrom black hole. These results are relevant to studies of mass inflation and the instability of Cauchy horizons. The analytic arguments also suggest the behavior of the full nonlinear dynamics, which we study numerically in a companion paper. PACS number(s): 04.30.Db, 04.25.Dm, 04.40.Dg

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.

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We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a λ-system as suggested by Brodbeck et al. We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.

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. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.

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.Comment: RevTex, 26 galley or 53 preprint pages, 3 EPS figures, 2 table

We compare the predictions of linearized theory for the radiation produced in the collapse of a spherically symmetric scalar field with a full numerical integration of the Einstein equations. We find power-law tails and quasinormal ringing remarkably similar to predictions of linearized theory even in cases where nonlinearities are crucial. We also show that power-law tails develop even when the collapsing scalar field fails to produce a black hole.

Small non-spherical perturbations of a spherically symmetric but time-dependent background spacetime can be used to model situations of astrophysical interest, for example the production of gravitational waves in a supernova explosion. We allow for perfect fluid matter with an arbitrary equation of state p = p(ρ, s), coupled to general relativity. Applying a general framework proposed by Gerlach and Sengupta, we obtain covariant field equations, in a 2+2 reduction of the spacetime, for the background and a complete set of gauge-invariant perturbations, and then scalarize them using the natural frame provided by the fluid. Building on previous work by Seidel, we identify a set of true perturbation degrees of freedom admitting free initial data for the axial and for the l ≥ 2 polar perturbations. The true degrees of freedom are evolved among themselves by a set of coupled wave and transport equations, while the remaining degrees of freedom can be obtained by quadratures. The polar l = 0, 1 perturbations are discussed in the same framework. They require gauge fixing and do not admit an unconstrained evolution scheme.