Universal Property of Quantum Gravity implied by Bekenstein-Hawking Entropy and Boltzmann formula

We search for a universal property of quantum gravity. By "universal", we mean the independence from any existing model of quantum gravity (such as the super string theory, loop quantum gravity, causal dynamical triangulation, and so on). To do so, we try to put the basis of our discussion on theories established by some experiments. Thus, we focus our attention on thermodynamical and statistical-mechanical basis of the black hole thermodynamics: Let us assume that the Bekenstein-Hawking entropy is given by the Boltzmann formula applied to the underlying theory of quantum gravity. Under this assumption, the conditions justifying Boltzmann formula together with uniqueness of Bekenstein-Hawking entropy imply a reasonable universal property of quantum gravity. The universal property indicates a repulsive gravity at Planck length scale, otherwise stationary black holes can not be regarded as thermal equilibrium states of gravity. Further, in semi-classical level, we discuss a possible correction of Einstein equation which generates repulsive gravity at Planck length scale. (This article is bease on ref.[4] (Entropy 13(2011)1611-1647), but the conclusion in sec.4 is modified from ref.[4].)Comment: This article is bease on ref.[4] (Entropy 13(2011)1611-1647), but the conclusion in sec.4 is modified from ref.[4]. This is a refereed proceeding article of International Conference on Mathematical Modeling in Physical Sciences (IC-Msquare 2012

Limit of Universality of Entropy-Area Law for Multi-Horizon Spacetimes

It may be a common understanding at present that, once event horizons are in thermal equilibrium, the entropy-area law holds inevitably. However, no rigorous verification is given to such a very strong universality of the law in multi-horizon spacetimes. In this article, based on thermodynamically consistent and rigorous discussion, we investigate thermodynamics of Schwarzschild-deSitter spacetime in which the temperatures of two horizons are different. We recognize that three independent state variables exist in thermodynamics of the horizons. One of the three variables represents the effect of "external gravity" acting on one horizon due to another one. Then we find that thermodynamic formalism with three independent variables suggests the breakdown of entropy-area law, and clarifies the necessary and sufficient condition for the entropy-area law. As a by-product, the special role of cosmological constant in thermodynamics of horizons is also revealed.Comment: 44 pages, 4 figures. Invited as a chapter contribution to an edited book, "Classical and Quantum Gravity: Theory, Analysis and Application", 2011, Nova Pub

Black Hole Evaporation and Nonequilibrium Thermodynamics for a Radiation Field

When a black hole is put in an "empty" space (zero temperature space) on which there is no matter except the matter of the Hawking radiation (Hawking field), then an outgoing energy flow from the black hole into the empty space exists. By the way, an equilibrium between two arbitrary systems can not allow the existence of an energy (heat) flow from one system to another. Consequently, in the case of a black hole evaporation in the empty space, the Hawking field should be in a nonequilibrium state. Hence the total behaviour of the evaporation, for example the time evolution of the total entropy, should be analysed with a nonequilibrium thermodynamics for the Hawking field. This manuscript explains briefly the way of constructing a nonequilibrium thermodynamic theory for a radiation field, and apply it to a simplified model of a black hole evaporation to calculate the time evolution of the total entropy.Comment: Corrected version of the manuscript in "Proceedings of 14th Workshop on General Rel. and Grav. in Japan", 4 pages, 2 figure

How to Measure Black Hole's Mass, Spin and Direction of Spin Axis in Kerr Lens Effect 1: test case with simple source emission near BH

We propose a theoretical principle to measure the mass, spin and direction of spin axis of Kerr black holes (BHs) through observing 2 quantities of the spinning strong gravitational lens effect of BHs. Those observable quantities are generated by 2 light rays emitted at the same time by a source near the BH: the primary and secondary rays that reach a distant observer, respectively, the earliest and secondary temporally. The time delay between detection times and the ratio of observed specific fluxes of those rays are the observable quantities. Rigorously, our proposal is applicable to a single burst-like (short duration) isotropic emission by the source. An extension of our principle to cases of complicated emissions may be constructed by summing up appropriately the result of this paper, which will be treated in future works.Comment: Accepted for publication, Prog.Theor.Exp.Phys. in 2017, 42 pages(26 pages for body + 16 pages for appendices), Typos are correcte

An Axiomatic Review of Israel-Stewart Hydrodynamics and Extended Irreversible Thermodynamics

The causality of dissipative phenomena can not be treated in traditional theories of dissipations, Fourier laws and Navier-Stokes equations. This is the reason why the dissipative phenomena have not been studies well in relativistic situations. Furthermore, the interactions among dissipations, e.g. the heating of fluid due to viscous flow and the occurrence of viscous flow due to heat flux, are not explicitly described in those traditional laws. One of the phenomenologies which describe the causality and interaction of dissipations is the Extended Irreversible Thermodynamics (EIT). (Israel-Stewart theory of dissipative hydrodynamics is one approximate form of EIT.) This manuscript reviews an axiomatic construction of EIT and Israel-Stewart hydrodynamic theory. Also, we point out that the EIT is also applicable to radiative transfer in optically \emph{thick} matters. However, radiative transfer in optically \emph{thin} matters can not be described by EIT, because the non-self-interacting nature of photons is incompatible with a basic requirement of EIT, "the bilinear form of entropy production rate". The break down of EIT in optically thin situation is not explicitly recognized in standard references of EIT and Israel-Stewart theory. Some detail of how EIT fails to describe a radiative transfer in optically thin situations is also explained. (This manuscript is a revision of the contribution to a book Ref.[26] published in 2011. So, recent developments made after 2011 may not be cited.)Comment: 19 pages, 1 figur

The generalised second law and the black hole evaporation in an empty space as a nonequilibrium process

When a black hole is in an empty space on which there is no matter field except that of the Hawking radiation (Hawking field), then the black hole evaporates and the entropy of the black hole decreases. The generalised second law guarantees the increase of the total entropy of the whole system which consists of the black hole and the Hawking field. That is, the increase of the entropy of the Hawking field is faster than the decrease of the black hole entropy. In naive sense, one may expect that the entropy increase of the Hawking field is due to the self-interaction among the composite particles of the Hawking field, and that the "self"-relaxation of the Hawking field results in the entropy increase. Then, when one consider a non-self-interacting matter field as the Hawking field, it is obvious that the self-relaxation does not take place, and one may think that the total entropy does not increase. However, using nonequilibrium thermodynamics which has been developed recently, we find for the non-self-interacting Hawking field that the rate of entropy increase of the Hawking field (the entropy emission rate by the black hole) grows faster than the rate of entropy decrease of the black hole along the black hole evaporation in the empty space. The origin of the entropy increase of the Hawking field is the increase of the black hole temperature. Hence an understanding of the generalised second law in the context of the nonequilibrium thermodynamics is suggested; even if the self-relaxation of the Hawking field does not take place, the temperature increase of the black hole during the evaporation process causes the entropy increase of the Hawking field to result in the increase of the total entropy.Comment: 13 pages, 6 figures. Accepted for publication in "Classical and Quantum Gravity" at 1 September 2006, mis-typo is correcte

Monomer-Dimer Mixture on a Honeycomb Lattice

We study a monomer-dimer mixture defined on a honeycomb lattice as a toy model for the spin ice system in a magnetic field. In a low-doping region of monomers, the effective description of this system is given by the dual sine-Gordon model. In intermediate- and strong-doping regions, the Potts lattice gas theory can be employed. Synthesizing these results, we construct a renormalization-group flow diagram, which includes the stable and unstable fixed points corresponding to ${\cal M}_5$ and ${\cal M}_6$ in the minimal models of the conformal field theory. We perform numerical transfer-matrix calculations to determine a global phase diagram and also to proffer evidence to check our prediction.Comment: 5 pages, 3 figure

de Sitter thermodynamics in the canonical ensemble

The existing thermodynamics of the cosmological horizon in de-Sitter spacetime is established in the micro-canonical ensemble, while thermodynamics of black hole horizons are established in the canonical ensemble. Generally in the ordinary thermodynamics and statistical mechanics, both of the micro-canonical and canonical ensembles yield the same equation of state for any thermodynamic system. This implies the existence of a formulation of de-Sitter thermodynamics based on the canonical ensemble. This paper reproduces the de-Sitter thermodynamics in the canonical ensemble. The procedure is as follows: We put a spherical wall at the center of de-Sitter spacetime, whose mass is negligible and perfectly reflects the Hawking radiation coming from the cosmological horizon. Then the region enclosed by the wall and horizon settles down to a thermal equilibrium state, for which the Euclidean action is evaluated and the partition function is obtained. The integration constant (subtraction term) of Euclidean action is determined to reproduce the equation of state (e.g. entropy-area law) verified already in the micro-canonical ensemble. Our de-Sitter canonical ensemble is well-defined to preserve the "thermodynamic consistency", which means that the state variables satisfy not only the four laws of thermodynamics but also the appropriate differential relations with thermodynamic functions; e.g. partial derivatives of the free energy give the entropy, pressure, and so on. The special role of cosmological constant in de-Sitter thermodynamics is also revealed.Comment: 28 pages, 2 figures, Accepted for publication in Prog.Theor.Phys, Typos are correcte

Black Hole Evaporation as a Nonequilibrium Process

When a black hole evaporates, there arises a net energy flow from the black hole into its outside environment due to the Hawking radiation and the energy accretion onto black hole. Exactly speaking, due to the net energy flow, the black hole evaporation is a nonequilibrium process. To study details of evaporation process, nonequilibrium effects of the net energy flow should be taken into account. In this article we simplify the situation so that the Hawking radiation consists of non-self-interacting massless matter fields and also the energy accretion onto the black hole consists of the same fields. Then we find that the nonequilibrium nature of black hole evaporation is described by a nonequilibrium state of that field, and we formulate nonequilibrium thermodynamics of non-self-interacting massless fields. By applying it to black hole evaporation, followings are shown: (1) Nonequilibrium effects of the energy flow tends to accelerate the black hole evaporation, and, consequently, a specific nonequilibrium phenomenon of semi-classical black hole evaporation is suggested. Furthermore a suggestion about the end state of quantum size black hole evaporation is proposed in the context of information loss paradox. (2) Negative heat capacity of black hole is the physical essence of the generalized second law of black hole thermodynamics, and self-entropy production inside the matter around black hole is not necessary to ensure the generalized second law. Furthermore a lower bound for total entropy at the end of black hole evaporation is given. A relation of the lower bound with the so-called covariant entropy bound conjecture is interesting but left as an open issue.Comment: Modified version of contribution (Chap.8) to an edited book by M.N.Christiansen and T.K.Rasmussen, Classical and Quantum Gravity Research (Nova Science Publishers, 2008). 36 pages and 12 figure

Black Hole Evaporation and Generalized 2nd Law with Nonequilibrium Thermodynamics

In general, when a black hole evaporates, there arises a net energy flow from black hole into its outside environment due to Hawking radiation and energy accretion onto black hole. The existence of energy flow means that the thermodynamic state of the whole system, which consists of a black hole and its environment, is in a nonequilibrium state. To know the detail of evaporation process, the nonequilibrium effects of energy flow should be taken into account. The nonequilibrium nature of black hole evaporation is a challenging topic including issues of not only black hole physics but also nonequilibrium physics. Using the nonequilibrium thermodynamics which has been formulated recently, this report shows: (1) the self-gravitational effect of black hole which appears as its negative heat capacity guarantees the validity of generalized 2nd law without entropy production inside the outside environment, (2) the nonequilibrium effect of energy flow tends to shorten the evaporation time (life time) of black hole, and consequently specific nonequilibrium phenomena are suggested. Finally a future direction of this study is commented.Comment: Typo is corrected. 10 pages, 2 figures. Based on proceedings and talks given at: APCTP Jeju Meeting on Gravitation and Cosmology (Jeju, Korea, 2007), Dynamics and Thermodynamics of Black Holes and Naked Singularities II, (Politecnico di Milano, Italy, 2007) and 16th General Relativity and Gravitation (Niigata, Japan, 2006