Loading...
Please wait, while we are loading the content...
Similar Documents
Entropy using Path Integrals for Quantum Black Hole Models
| Content Provider | Semantic Scholar |
|---|---|
| Author | Obregón, Octavio Sabido, Marta Tkach, Vladimir I. |
| Copyright Year | 2000 |
| Abstract | Canonical quantum gravity has been used in the search for eigenvalue equations that could describe black holes. In this paper we choose one of the simplest of these quantum equations to show how the usual Feynman’s path integral approach can be applied to get the corresponding statistical properties. We get a logarithmic correction to the Bekenstein-Hawking entropy as already obtained by other authors by other means. PACS numbers: 04.60.Ds,04.60.Kz, 04.70.-s, 04.70.Dy Typeset using REVTEX E-mail: octavio@ifug3.ugto.mx E-mail: msabido@ifug3.ugto.mx E-mail: vladimir@ifug1.ugto.mx. 1 In the early seventies, in his insightful work Bekenstein [1] proposed the quantization of a black hole. He suggested that its surface gravity is proportional to its temperature and that the area of its event horizon is proportional to its entropy. In his remarkable work he conjectured that the horizon area of non-extremal black holes plays the role of a classical adiabatic invariant. He concluded that the horizon area should have a discrete spectrum with uniformly spaced eigenvalues. Another result at that same time, was the discovery of the mechanical laws in the framework of general relativity, that govern non-extremal black holes by [2]. These laws have a striking analogy with those of thermodynamics. This similarity was not well understood until Hawking, in his seminal work [3] a year later, discovered that black holes evaporate, they radiate as black bodies with a temperature proportional to the surface gravity. One can then argue that the laws of black hole mechanics have a real thermodynamical meaning and that the entropy of a black hole is one fourth of its horizon area. These remarkable results of nearly three decades ago, point out to a deeper relation among classical gravitation, quantum mechanics and statistical properties. Hawking’s semiclassical calculations allowed to interpret the relation between classical black hole mechanics and its thermodynamics leading to the celebrated entropy area expression. During these years, different approaches have emerged to try to understand the interplay between the quantum and classical descriptions, and the statistical properties of black holes. We briefly mention the main quantum gravity non-perturbative formalisms. String theory, whose building blocks are essentially D-branes [4], loop quantum gravity [5], and canonical quantum gravity, having as elementary constituents the quantum excitations of the geometry itself . These proposals emerge essentially from different principles. String theory is sought to be fundamental, it provides the precise expression for the temperature and entropy. Loop and canonical gravity do not provide the corresponding proportionality constants unambiguously. These, however, deal directly with the curved black hole geometry. In string theory, one carries out, for example, the calculation of Hawking ́s radiation in flat space. Moreover, in string theory these calculations have been only possible for extremal and nearly extremal black holes [4]. In this paper, we are interested in canonical quantum gravity treatments where a Hamiltonian quantum theory of spherically symmetric vacuum spacetime can be defined [6] (for our purposes we could also consider a ball or shell of dust collapsing to a black hole [7]). On the other hand, supposing a uniformly spaced area spectrum [8], the Schwarzschild black hole has been treated as a microcanonical ensemble [9]. Also considering a mini-superspace approach and by means of statistical techniques previously used in hadron physics, the usual area of black hole thermodynamics is recovered [10]. Moreover, assuming that the area spectrum of the black hole is uniformly spaced, a grand canonical ensemble has been considered with the ADM mass (the Hamiltonian) and the horizon area as separately observables [11]. It is argued that in this way the partition function is not divergent. However, the result, a logarithmic correction to the Bekenstein-Hawking entropy, is the same already obtained by Kastrup [12], by means of an analytic continuation approach and later by Mäkela and Repo [13] for the emitted radiation. These authors considered solely as physical observable the energy levels. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/gr-qc/0003023v1.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/gr-qc/0003023v2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Adiabatic invariant Black Hole Continuation Description Eigenvalue Email Energy level Evaporation Gravitation Hawks (bird) Human body Mail (macOS) Observable Partition function (mathematics) Path integral formulation Perturbation theory (quantum mechanics) Physics and Astronomy Classification Scheme Quantization (signal processing) Quantum Theory Quantum mechanics Radiate Semiclassical physics Thermodynamics adrenomedullin |
| Content Type | Text |
| Resource Type | Article |
