Unified approach to the entropy of an extremal rotating BTZ black hole: Thin shells and horizon limits

Using a thin shell, the first law of thermodynamics, and a unified approach, we study the thermodymanics and find the entropy of a (2+1)-dimensional extremal rotating Ba\~{n}ados-Teitelbom-Zanelli (BTZ) black hole. The shell in (2+1) dimensions, i.e., a ring, is taken to be circularly symmetric and rotating, with the inner region being a ground state of the anti-de Sitter (AdS) spacetime and the outer region being the rotating BTZ spacetime. The extremal BTZ rotating black hole can be obtained in three different ways depending on the way the shell approaches its own gravitational or horizon radius. These ways are explicitly worked out. The resulting three cases give that the BTZ black hole entropy is either the Bekenstein-Hawking entropy, $S=\frac{A_+}{4G}$, or it is an arbitrary function of $A_+$, $S=S(A_+)$, where $A_+=2\pi r_+$ is the area, i.e., the perimeter, of the event horizon in (2+1) dimensions. We speculate that the entropy of an extremal black hole should obey $0\leq S(A_+)\leq\frac{A_+}{4G}$. We also show that the contributions from the various thermodynamic quantities, namely, the mass, the circular velocity, and the temperature, for the entropy in all three cases are distinct. This study complements the previous studies in thin shell thermodynamics and entropy for BTZ black holes. It also corroborates the results found for a (3+1)-dimensional extremal electrically charged Reissner-Nordstr\"om black hole.Comment: 8 pages, 1 table, no figur

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

Quasiblack holes with pressure: General exact results

A quasiblack hole is an object in which its boundary is situated at a surface called the quasihorizon, defined by its own gravitational radius. We elucidate under which conditions a quasiblack hole can form under the presence of matter with nonzero pressure. It is supposed that in the outer region an extremal quasihorizon forms, whereas inside, the quasihorizon can be either nonextremal or extremal. It is shown that in both cases, nonextremal or extremal inside, a well-defined quasiblack hole always admits a continuous pressure at its own quasihorizon. Both the nonextremal and extremal cases inside can be divided into two situations, one in which there is no electromagnetic field, and the other in which there is an electromagnetic field. The situation with no electromagnetic field requires a negative matter pressure (tension) on the boundary. On the other hand, the situation with an electromagnetic field demands zero matter pressure on the boundary. So in this situation an electrified quasiblack hole can be obtained by the gradual compactification of a relativistic star with the usual zero pressure boundary condition. For the nonextremal case inside the density necessarily acquires a jump on the boundary, a fact with no harmful consequences whatsoever, whereas for the extremal case the density is continuous at the boundary. For the extremal case inside we also state and prove the proposition that such a quasiblack hole cannot be made from phantom matter at the quasihorizon. The regularity condition for the extremal case, but not for the nonextremal one, can be obtained from the known regularity condition for usual black holes.Comment: 18 pages, no figures; improved introduction, added references, calculations better explaine

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..