Black Hole Entropy: Certain Quantum Features (original) (raw)
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An elementary introduction is given to the problem of black hole entropy as formulated by Bekenstein and Hawking, based on the so-called Laws of Black Hole Mechanics. Wheeler's 'It from Bit' picture is presented as an explanation of plausibility of the Bekenstein-Hawking Area Law. A variant of this picture that takes better account of the symmetries of general relativity is shown to yield corrections to the Area Law that are logarithmic in the horizon area, with a finite, fixed coefficient. The Holographic hypothesis, tacitly assumed in the above considerations, is briefly described and the beginnings of a general proof of the hypothesis is sketched, within an approach to quantum gravitation which is non-perturbative in nature, namely Non-perturbative Quantum General Relativity (also known as Quantum Geometry). The holographic entropy bound is shown to be somewhat tightened due to the corrections obtained earlier. A brief summary of Quantum Geometry approach is included, with a sketch of a demonstration that precisely the log area corrections obtained from the variant of the It from Bit picture adopted earlier emerges for the entropy of generic black holes within this formalism.
Quantum aspects of black hole entropy
Pramana, 2000
This survey intends to cover recent approaches to black hole entropy which attempt to go beyond the standard semiclassical perspective. Quantum corrections to the semiclassical Bekenstein-Hawking area law for black hole entropy, obtained within the quantum geometry framework, are treated in some detail. Their ramification for the holographic entropy bound for bounded stationary spacetimes is discussed. Four dimensional supersymmetric extremal black holes in string-based N = 2 supergravity are also discussed, albeit more briefly.
Quantum geometry and black hole entropy
A 'black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4ℓ 2 P . With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal.
Entropy of Quantum Black Holes
Symmetry, Integrability and Geometry: Methods and Applications, 2012
In the Loop Quantum Gravity, black holes (or even more general Isolated Horizons) are described by a SU (2) Chern-Simons theory. There is an equivalent formulation of the horizon degrees of freedom in terms of a U (1) gauge theory which is just a gauged fixed version of the SU (2) theory. These developments will be surveyed here. Quantum theory based on either formulation can be used to count the horizon micro-states associated with quantum geometry fluctuations and from this the micro-canonical entropy can be obtained. We shall review the computation in SU (2) formulation. Leading term in the entropy is proportional to horizon area with a coefficient depending on the Barbero-Immirzi parameter which is fixed by matching this result with the Bekenstein-Hawking formula. Remarkably there are corrections beyond the area term, the leading one is logarithm of the horizon area with a definite coefficient −3/2, a result which is more than a decade old now. How the same results are obtained in the equivalent U (1) framework will also be indicated. Over years, this entropy formula has also been arrived at from a variety of other perspectives. In particular, entropy of BTZ black holes in three dimensional gravity exhibits the same logarithmic correction. Even in the String Theory, many black hole models are known to possess such properties. This suggests a possible universal nature of this logarithmic correction.
Black Hole Entropy and Quantum Gravity
Indian J Phys, 1998
An elementary introduction is given to the problem of black hole entropy as formulated by Bekenstein and Hawking. The information theoretic basis of Bekenstein's formulation is briefly reviewed and compared with Hawking's approach. The issue of calculating the entropy by actual counting of microstates is taken up next within two currently popular approaches to quantum gravity, viz., string theory and canonical quantum gravity. The treatment of the former assay is confined to a few remarks, mainly of a critical nature, while some of the computational techniques of the latter approach are elaborated. We conclude by trying to find commonalities between these two rather disparate directions of work.
Quantum geometry and microscopic black hole entropy
Classical and Quantum Gravity, 2007
Quantum black holes within the loop quantum gravity (LQG) framework are considered. The number of microscopic states that are consistent with a black hole of a given horizon area A0A_0A0 are counted and the statistical entropy, as a function of the area, is obtained for A0A_0A0 up to 550l2rmPl550 l^2_{\rm Pl}550l2rmPl. The results are consistent with an asymptotic linear relation and a logarithmic correction with a coefficient equal to -1/2. The Barbero-Immirzi parameter that yields the asymptotic linear relation compatible with the Bekenstein-Hawking entropy is shown to coincide with a value close to gamma=0.274\gamma=0.274gamma=0.274, which has been previously obtained analytically. However, a new and oscillatory functional form for the entropy is found for small, Planck size, black holes that calls for a physical interpretation.
Physics Letters B, 1998
We derive an exact formula for the dimensionality of the Hilbert space of the boundary states of SU (2) Chern-Simons theory, which, according to the recent work of Ashtekar et al, leads to the Bekenstein-Hawking entropy of a four dimensional Schwarzschild black hole. Our result stems from the relation between the (boundary) Hilbert space of the Chern-Simons theory with the space of conformal blocks of the Wess-Zumino model on the boundary 2sphere. The issue of the Bekenstein-Hawking (B-H) [1], [2] entropy of black holes has been under intensive scrutiny for the last couple of years, following the derivation of the entropy of certain extremal charged black hole solutions of toroidally compactified heterotic string and also type IIB superstring from the underlying string theories [3], [4]. In the former case of the heterotic string, the entropy was shown to be proportional to the area of the 'stretched' horizon of the corresponding extremal black hole, while in the latter case it turned out to be precisely the B-H result. The latter result was soon generalized to a large number of four and five dimensional black holes of type II string theory and M-theory (see [5] for a review), all of which could be realized as certain D-brane configurations and hence saturated the BPS bound. Unfortunately, the simplest black hole of all, the four *
Ten theses on black hole entropy
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2005
I present a viewpoint on black hole thermodynamics according to which the entropy: derives from horizon "degrees of freedom"; is finite because the deep structure of spacetime is discrete; is "objective" thanks to the distinguished coarse graining provided by the horizon; and obeys the second law of thermodynamics precisely because the effective dynamics of the exterior region is not unitary. Probably few people doubt that the twin phenomena of black hole entropy and evaporation hold important clues to the nature of quantum spacetime, but the agreement pretty much ends there. Starting from the same evidence, different workers have drawn very different, and partly contradictory, lessons. On one hand, there is perhaps broad agreement that the finiteness of the entropy points to an element of discreteness in the deep structure of spacetime. On the other hand there is sharp disagreement over whether the thermal nature of the Hawking radiation betokens an essential failure of unitarity in quantum gravity or whether it is instead betraying the need for a radical revision of the spacetime framework, as contemplated for instance in the "holographic principle". These alternatives are not necessarily in contradiction, of course, but in practice, the wish to retain unitarity has been one of the strongest motivations for taking seriously the latter type of possibility. My own belief is that non-unitarity is probably inevitable in connection with gravity and that, rather than shunning this prospect, we ought to welcome it because it offers a straightforward way to understand why the law of entropy increase continues to hold in the presence
Black-hole entropy from quantum geometry
Classical and Quantum Gravity, 2004
Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible. Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again.
Entropy of Black Holes: A Quantum Algebraic Approach
Entropy, 2003
In this paper we apply to a class of static and time-independent geometries the recently developed formalism of deformed algebras of quantum fields in curved backgrounds. In particular, we derive: i) some non-trivial features of the entanglement of the quantum vacuum, such as the robustness against interaction with the environment; ii) the thermal properties and the entropy of black holes for space-times with a unique event horizon, such as Schwarzschild, de Sitter and Rindler space-times.