An Analytic Model with Critical Behavior in Black Hole Formation

A simple analytic model is presented which exhibits a critical behavior in black hole formation, namely, collapse of a thin shell coupled with outgoing null fluid. It is seen that the critical behavior is caused by the gravitational nonlinearity near the event horizon. We calculate the value of the critical exponent analytically and find that it is very dependent on the coupling constants of the system.Comment: 21pp., ReVTeX, 7 figures (postscript, compressed and uuencoded), TIT/HEP-266/COSMO-4

Surface gravity in dynamical spherically symmetric spacetimes

A definition of surface gravity at the apparent horizon of dynamical spherically symmetric spacetimes is proposed. It is based on a unique foliation by ingoing null hypersurfaces. The function parametrizing the hypersurfaces can be interpreted as the phase of a light wave uniformly emitted by some far-away static observer. The definition gives back the accepted value of surface gravity in the static case by virtue of its nonlocal character. Although the definition is motivated by the behavior of outgoing null rays, it turns out that there is a simple connection between the generalized surface gravity, the acceleration of any radially moving observer, and the observed frequency change of the infalling light signal. In particular, this gives a practical and simple method of how any geodesic observer can determine surface gravity by measuring only the redshift of the infalling light wave. The surface gravity can be expressed as an integral of matter field quantities along an ingoing null line, which shows that it is a continuous function along the apparent horizon. A formula for the area change of the apparent horizon is presented, and the possibility of thermodynamical interpretation is discussed. Finally, concrete expressions of surface gravity are given for a number of four-dimensional and two-dimensional dynamical black hole solutions.Comment: 35 pages, revtex, 3 figures included using eps

Perturbations and Critical Behavior in the Self-Similar Gravitational Collapse of a Massless Scalar Field

This paper studies the perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (Roberts solution). The perturbation equations are derived and solved exactly. The perturbation spectrum is found to be not discrete, but occupying continuous region of the complex plane. The renormalization group calculation gives the value of the mass-scaling exponent equal to 1.Comment: 12 pages, RevTeX 3.1, 1 figur

Spherical Self-Similar Solutions in Einstein-Multi-Scalar Gravity

We consider a general non-linear sigma model coupled to Einstein gravity and show that in spherical symmetry and for a simple realization of self-similarity, the spacetime can be completely determined. We also examine some more specific matter models and discuss their relation to critical collapse.Comment: 11 pages, 1 figur

On critical behaviour in gravitational collapse

We give an approach to studying the critical behaviour that has been observed in numerical studies of gravitational collapse. These studies suggest, among other things, that black holes initially form with infinitesimal mass. We show generally how a black hole mass formula can be extracted from a transcendental equation. Using our approach, we give an explicit one parameter set of metrics that are asymptotically flat and describe the collapse of apriori unspecified but physical matter fields. The black hole mass formula obtained from this metric exhibits a mass gap - that is, at the onset of black hole formation, the mass is finite and non-zero.Comment: 11 pages, RevTex, 2 figures (available from VH

Self-Similar Collapse of Scalar Field in Higher Dimensions

This paper constructs continuously self-similar solution of a spherically symmetric gravitational collapse of a scalar field in n dimensions. The qualitative behavior of these solutions is explained, and closed-form answers are provided where possible. Equivalence of scalar field couplings is used to show a way to generalize minimally coupled scalar field solutions to the model with general coupling.Comment: RevTex 3.1, 15 pages, 3 figures; references adde

An extreme critical space-time: echoing and black-hole perturbations

A homothetic, static, spherically symmetric solution to the massless Einstein- Klein-Gordon equations is described. There is a curvature singularity which is central, null, bifurcate and marginally trapped. The space-time is therefore extreme in the sense of lying at the threshold between black holes and naked singularities, just avoiding both. A linear perturbation analysis reveals two types of dominant mode. One breaks the continuous self-similarity by periodic terms reminiscent of discrete self-similarity, with echoing period within a few percent of the value observed numerically in near-critical gravitational collapse. The other dominant mode explicitly produces a black hole, white hole, eternally naked singularity or regular dispersal, the latter indicating that the background is critical. The black hole is not static but has constant area, the corresponding mass being linear in the perturbation amplitudes, explicitly determining a unit critical exponent. It is argued that a central null singularity may be a feature of critical gravitational collapse.Comment: 6 revtex pages, 6 eps figure

Global tropical forest cover change assessment with medium spatial stellite imagery using a systematic sample grid - data, methods and first results.

At the Joint Research Centre (JRC) of the European Commission, a methodology has been developed to monitor the pan-tropical forest cover with remote sensing data for the years 1990-2000-2005 in Latin America, Southeast Asia and Africa on the basis of over 4000 sample units sample units with a dimension of 20 km by 20 km located at every full latitude and longitude degree confluence. From the Landsat Thematic Mapper (TM) and Enhanced Thematic Mapper (ETM) instruments, images with low cloud impact from the epochs around the years 1990, 2000 and 2005 were selected and subsets covering the sample units were cut-out, pre-processed, segmented and classified in five different land cover classes in order to build global and regional statistics on tropical forest cover change. The data was validated in three steps, internal correction of wrongly classified objects, external (national or regional) expert validation and internal harmonization of the data. In this paper, the data collection and the workflow of the forest cover change assessment for the epochs 1990 and 2000 is presented. Parts of the results for the Brazilian Amazon have been validated by comparing with interpretations of corresponding samples carried out by the Instituto Nacional de Pesquisas Espaciais (INPE), showing a very high correlation. Further, the figure produced by INPE through the PRODES program on gross deforestation for the years 1990-2000 was compared to the figure calculated on basis of the JRC results for the respective area, where the JRC estimate that was ca. 10% higher than the INPE estimate

Critical phenomena of collapsing massless scalar wave packets

An analytical model that represents the collapse of a massless scalar wave packet with continuous self-similarity is constructed, and critical phenomena are found. In the supercritical case, the mass of black holes is finite and has the form $M \propto (p - p^{*})^{\gamma}$, with $\gamma = 1/2$.Comment: Latex file, including 2 figures, avalaible upon reques

Continuous Self-Similarity Breaking in Critical Collapse

This paper studies near-critical evolution of the spherically symmetric scalar field configurations close to the continuously self-similar solution. Using analytic perturbative methods, it is shown that a generic growing perturbation departs from the critical Roberts solution in a universal way. We argue that in the course of its evolution, initial continuous self-similarity of the background is broken into discrete self-similarity with echoing period $\Delta = \sqrt{2}\pi = 4.44$, reproducing the symmetries of the critical Choptuik solution.Comment: RevTeX 3.1, 28 pages, 5 figures; discussion rewritten to clarify several issue