Gibbs paradox, black hole entropy, and the thermodynamics of isolated horizons

Abstract

This letter presents a new, solely thermodynamical argument for considering the states of the quantum isolated horizon of a black hole as distinguishable. We claim that only if the states are distinguishable, the thermodynamic entropy is an extensive quantity and can be well-defined. To show this, we make a comparison with a classical ideal gas system whose statistical description makes only sense if an additional 1/N !-factor is included in the state counting in order to cure the Gibbs paradox. The case of the statistical description of a quantum isolated horizon is elaborated, to make the claim evident. As solutions of the gravitational field equations teach us, a star having a sufficiently large mass collapses beyond its Schwarzschild radius and will continue to collapse completely into a gravitational singularity at the end of its lifetime. This renders the spacetime geodesically incomplete. The star disappears and leaves a black hole behind, whose size is given by the radius of the event horizon, i.e. the Schwarzschild radius, which covers the singularity. The event horizon is the frontier of all events, which in principle could be observed by an external observer -all inner events are concealed from the latter. The existence of these tantalizing objects is supported by an ever-increasing number of astrophysical observations. However, the motivation to attend to a more fundamental account of the gravitational field is driven firstly by the fact, that these singularities are unphysical divergences of the gravitational field and hence the predictability of classical general relativity for the description of the interior structure of the black hole breaks down as implied by the singularity theorems [1]. Secondly, dimensional arguments suggest, that one can't neglect quantum effects near these singularities.Furthermore, based on its gravitational properties, it is possible to assign thermodynamic quantities to black holes. Concretely, this is accomplished through relating the horizon area to entropy, the black hole surface gravity to temperature and the black hole mass to energy, whose nomological relation is expressed by means of the four laws of black hole mechanics [2]. Hence, these laws suggest a close analogy to the laws of thermody- † Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités AMI, AMII, et du Sud Toulon-Var * pithis@cpt.univ-mrs.fr, andreas.pithis@campus.lmu.de namics. Additionally, calculations by means of semiclassical quantum field theory on curved spacetime predicting that quantum mechanically a black hole radiates like a black body with a temperature proportional to the surface gravity of the black hole, clarified the right proportionalities between the above related quantities. This gives rise to the Bekenstein-Hawking formula for the relation between entropy and area,expressing a remarkable and intriguing relation between the geometry of spacetime, gravity, quantum theory and thermodynamics [3,4]. Taking the microscopic underpinning of thermodynamics through statistical mechani...

Statistics, holography, and black hole entropy in loop quantum gravity

“…We believe that even though the statements made in [23] are correct in their context, their validity is only a peculiarity of the effective models where such results were derived. Moreover, in such models where no holographic feature (5) is present among other things because matter degrees of freedom are completely ignored, BH entropy will not grow linearly with A/ 2 p if punctures are taken to be indistinguishable [14,24]. It is only when one includes the effect of matter through (5) that indistinguishability leads to the correct leading-order entropy and the correct leading order temperature.…”

“…There has been some discussions in the past about the issue of statistics of the punctures that define the quantum black hole states [13,14,24]. In this section, we explore the consequences of assuming that punctures are indistinguishable excitations of the quantum geometry of the horizon.…”

Thermodynamics in Curved Space-Time and Its Application to Holography

“…Moreover, in loop quantum gravity, the holographic entropy can be derived by counting the DOFs on the isolated horizon of a black hole, which consists of horizon punctures pierced by the edges of spin network. In [19] these punctures are confirmed to be distinguishable and the corresponding Gibbs paradox is analyzed. Moreover, it is proved in [20,21] that infinite statistics (quantum Boltzmann statistics) saturates the holographic entropy bound.…”

Anyonic statistics and large horizon diffeomorphisms for loop quantum gravity black holes

“…This choice drastically influences the quantum statistics of the model and is well motivated in the LQG literature [6,[30][31][32][33]. We want to revise the supporting arguments here.…”

“…The total number of punctures is denoted by N ¼ P k=2 j n j and the combinatorial prefactor indicates that in the purely gravitational case the punctures are considered as distinguishable [6,[31][32][33]. Since the level of k the theory is proportional to a H =l 2 p it is convenient to assume the limit k → ∞ of (26) giving…”