Entropy of an extremal electrically charged thin shell and the extremal black hole

There is a debate as to what is the value of the the entropy $S$ of extremal black holes. There are approaches that yield zero entropy $S=0$, while there are others that yield the Bekenstein-Hawking entropy $S=A_+/4$, in Planck units. There are still other approaches that give that $S$ is proportional to $r_+$ or even that $S$ is a generic well-behaved function of $r_+$. Here $r_+$ is the black hole horizon radius and $A_+=4\pi r_+^2$ is its horizon area. Using a spherically symmetric thin matter shell with extremal electric charge, we find the entropy expression for the extremal thin shell spacetime. When the shell's radius approaches its own gravitational radius, and thus turns into an extremal black hole, we encounter that the entropy is $S=S(r_+)$, i.e., the entropy of an extremal black hole is a function of $r_+$ alone. We speculate that the range of values for an extremal black hole is $0\leq S(r_+) \leq A_+/4$.Comment: 11 pages, minor changes, added references, matches the published versio

An Object-Based Approach to Modelling and Analysis of Failure Properties

In protection systems, when traditional technology is replaced by software, the functionality and complexity of the system is likely to increase. The quantitative evidence normally provided for safety certification of traditional systems cannot be relied upon in software-based systems. Instead there is a need to provide qualitative evidence. As a basis for the required qualitative evidence, we propose an object-based approach that allows modelling of both the application and software domains. From the object class model of a system and a formal specification of the failure properties of its components, we generate a graph of failure propagation over object classes, which is then used to generate a graph in terms of object instances in order to conduct fault tree analysis. The model is validated by comparing the resulting minimal cut sets with those obtained from the fault tree analysis of the original system. The approach is illustrated on a case study based on a protection system from..

The Two-Dimensional Analogue of General Relativity

General Relativity in three or more dimensions can be obtained by taking the limit $\omega\rightarrow\infty$ in the Brans-Dicke theory. In two dimensions General Relativity is an unacceptable theory. We show that the two-dimensional closest analogue of General Relativity is a theory that also arises in the limit $\omega\rightarrow\infty$ of the two-dimensional Brans-Dicke theory.Comment: 8 pages, LaTeX, preprint DF/IST-17.9

Conformal entropy from horizon states: Solodukhin's method for spherical, toroidal, and hyperbolic black holes in D-dimensional anti-de Sitter spacetimes

A calculation of the entropy of static, electrically charged, black holes with spherical, toroidal, and hyperbolic compact and oriented horizons, in D spacetime dimensions, is performed. These black holes live in an anti-de Sitter spacetime, i.e., a spacetime with negative cosmological constant. To find the entropy, the approach developed by Solodukhin is followed. The method consists in a redefinition of the variables in the metric, by considering the radial coordinate as a scalar field. Then one performs a 2+(D-2) dimensional reduction, where the (D-2) dimensions are in the angular coordinates, obtaining a 2-dimensional effective scalar field theory. This theory is a conformal theory in an infinitesimally small vicinity of the horizon. The corresponding conformal symmetry will then have conserved charges, associated with its infinitesimal conformal generators, which will generate a classical Poisson algebra of the Virasoro type. Shifting the charges and replacing Poisson brackets by commutators, one recovers the usual form of the Virasoro algebra, obtaining thus the level zero conserved charge eigenvalue L_0, and a nonzero central charge c. The entropy is then obtained via the Cardy formula.Comment: 21 page

Two-dimensional gravitation and Sine-Gordon-Solitons

Some aspects of two-dimensional gravity coupled to matter fields, especially to the Sine-Gordon-model are examined. General properties and boundary conditions of possible soliton-solutions are considered. Analytic soliton-solutions are discovered and the structure of the induced space-time geometry is discussed. These solutions have interesting features and may serve as a starting point for further investigations.Comment: 23 pages, latex, references added, to appear in Phys.Rev.

The Three-Dimensional BTZ Black Hole as a Cylindrical System in Four-Dimensional General Relativity

It is shown how to transform the three dimensional BTZ black hole into a four dimensional cylindrical black hole (i.e., black string) in general relativity. This process is identical to the transformation of a point particle in three dimensions into a straight cosmic string in four dimensions.Comment: Latex, 9 page

Documented international enquiry on solid sedimentary fossil fuels; Coal: definitions, classifications, reserves-resources and energy potential

This paper deals with all solid sedimentary fossil fuels, i.e. coal, the main one for geological reserves and resources, peat, and oil shales. Definitions of coal ( < 50% ash) and coal seam (thickness and depth limits) are examined in view of an international agreement regarding new concepts for a common reserves and resources evaluation using the same nomenclature. The 50% ash limit, already adopted by UN-ECE for coal definition, allows the creation of a new category—the organic shales (50–75% ash)—comprising energetic materials still valuable for thermal use (coal shales) or to be retorted for oil production (oil shales). Geological relations between coals, oil shales, solid bitumen, liquid hydrocarbons, natural gas, and coalbed methane are also examined together with environmental problems. As a final synthesis of all topics, the paper discusses the problems related with a modern geological classification of all solid sedimentary fuels based on: various rank parameters (moisture content, calorific value, reflectance), maceral composition, and mineral matter content (and washability). Finally, it should be pointed out that the paper is presented as series of problems, some of them old ones, but never resolved until now. In order to facilitate the next generation of coal geologists to resolve these problems on the basis of international agreements, all sections begin with documented introductions for further questions opening an international enquiry. The authors hope that the answers will be abundant enough and pertinent to permit synthetic international solutions, valuable for the new millennium, with the help of interested consulted authorities, international pertinent organisations, and regional experts. D 2002 Elsevier Science B.V. All rights reserved