Horizon Entropy Research Articles

Black hole interior mass formula

AbstractWe argue by explicit computations that, although the area product, horizon radii product, entropy product, and irreducible mass product of the event horizon and Cauchy horizon are universal, the surface gravity product, the surface temperature product and the Komar energy product of the said horizons do not seem to be universal for Kerr–Newman black hole spacetimes. We show the black hole mass formula on the Cauchy horizon following the seminal work by Smarr [Phys Rev Lett 30:71 (1973), Phys Rev D 7:289 (1973)] for the outer horizon. We also prescribe the four laws of black hole mechanics for the inner horizon. A new definition of the extremal limit of a black hole is discussed.

Near horizon symmetry and entropy of black holes in the presence of a conformally coupled scalar

We analyze the near horizon conformal symmetry for black hole solutions in gravity with a conformally coupled scalar field using the method proposed by Majhi and Padmanabhan recently. It is shown that the entropy of the black holes of the form ds2 = −f(r) dt2 + dr2/f(r) + ⋅⋅⋅ agrees with the Wald entropy. This result is different from the previous result obtained by M Natsuume, T Okamura and M Sato using the canonical Hamiltonian formalism, which claims a discrepancy from the Wald entropy.

Horizon thermodynamics and gravitational field equations in quasi-topological gravity

In this paper we show that the gravitational field equations of $(n+1)$% -dimensional topological black holes with constant horizon curvature, in cubic and quartic quasi-topological gravity, can be recast in the form of the first law of thermodynamics, $dE=TdS-PdV$, at the black hole horizon. This procedure leads to extract an expression for the horizon entropy as well as the energy (mass) in terms of the horizon radius, which coincide exactly with those obtained in quasi-topological gravity by solving the field equations and using the Wald's method. We also argue that this approach is powerful and can be extended to all higher order quasi-topological gravity for extracting the corresponding entropy and energy in terms of horizon radius.

3D flat holography: entropy and logarithmic corrections

We compute the leading corrections to the Bekenstein-Hawking entropy of the Flat Space Cosmological (FSC) solutions in 3D flat spacetimes, which are the flat analogues of the BTZ black holes in AdS3. The analysis is done by a computation of density of states in the dual 2D Galilean Conformal Field Theory and the answer obtained by this matches with the limiting value of the expected result for the BTZ inner horizon entropy as well as what is expected for a generic thermodynamic system. Along the way, we also develop other aspects of holography of 3D flat spacetimes.

Holographic dark energy and generalized second law

We explore the validity of the generalized second law (GSL) of thermodynamics in flat FRW universe with apparent horizon and event horizon as the boundary. We found that in a universe with holographic dark energy and dark matter, interacting with each other, the GSL is satisfied at the apparent horizon and partially satisfied at the event horizon under thermal equilibrium conditions. We also analyzed the GSL under non-equilibrium conditions and shows that the fulfillment of GSL at the apparent horizon implies that the temperature of the dark energy is greater than that of the horizon. Thus, there occurs a flow of dark energy towards the apparent horizon. As a result, the entropy of the dark energy decreases and that of the horizon increases. This is verified by finding the evolution of the dark energy entropy and horizon entropy in a dark energy dominated universe under non-equilibrium conditions.

Quantization of Horizon Entropy and the Thermodynamics of Spacetime

This is a review of my work published in the papers of Skakala (JHEP 1201:144, 2012; JHEP 1206:094, 2012) and Chirenti et al. (Phys. Rev. D 86:124008, 2012; Phys. Rev. D 87:044034, 2013). It offers a more detailed discussion of the results than the accounts in those papers, and it links my results to some conclusions recently reached by other authors. It also offers some new arguments supporting the conclusions in the cited articles. The fundamental idea of this work is that the semiclassical quantization of the black hole entropy, as suggested by Bekenstein (Phys. Rev. D 7:2333–2346, 1973), holds (at least) generically for the spacetime horizons. We support this conclusion by two separate arguments: (1) we generalize Bekenstein’s lower bound on the horizon area transition to a much wider class of horizons than only the black-hole horizon, and (2) we obtain the same entropy spectra via the asymptotic quasi-normal frequencies of some particular spherically symmetric multi-horizon spacetimes (in the way proposed by Maggiore (Phys. Rev. Lett. 100:141301, 2008)). The main result of this paper supports the conclusions derived by Kothawalla et al. (Phys. Rev. D 78:104018, 2008) and Kwon and Nam (Class. Quant. Grav. 28:035007, 2011), on the basis of different arguments.

Horizon spectroscopy in and beyond general relativity

In this work we generalize the results for the entropy spectra typically derived for black holes in general relativity to a generic horizon within the spherically symmetric (asymptotically flat and nonflat) space-times of more general theories of gravity. We use all the standard approaches---Bekenstein's universal lower bound on the entropy transition, the highly damped quasinormal modes and reduced phase-space quantization---to derive the spectra. In particular, the three approaches show that the Bekenstein-like spectra for the horizon entropy is a robust result. Our results confirm the suggestion made relatively recently by an independent fourth argument by Kothawala et al. [Phys. Rev. D 78, 104018 (2008)].

Why the expansion of the Universe is accelerated

There is compelling evidence, nowadays, that the present expansion of the Universe is accelerated. In Einstein gravity this acceleration is explained by invoking some exotic energy component with a huge negative pressure. More general gravity theories simply modify the Einstein–Hilbert action in ad hoc manner. Here, we show that whatever its origin, this acceleration is consistent with the second law of thermodynamics in the sense that the entropy of the causal cosmic horizon plus that of matter and fields inside it must never decrease and it should approach a finite maximum in the long run. In this light, the acceleration could have been anticipated on thermodynamic grounds well before its actual discovery.

Cardy-Verlinde Formula of Noncommutative Schwarzschild Black Hole

Few years ago, Setare (2006) has investigated the Cardy-Verlinde formula of noncommutative black hole obtained by noncommutativity of coordinates. In this paper, we apply the same procedure to a noncommutative black hole obtained by the coordinate coherent approach. The Cardy-Verlinde formula is entropy formula of conformal field theory in an arbitrary dimension. It relates the entropy of conformal field theory to its total energy and Casimir energy. In this paper, we have calculated the total energy and Casimir energy of noncommutative Schwarzschild black hole and have shown that entropy of noncommutative Schwarzschild black hole horizon can be expressed in terms of Cardy-Verlinde formula.

Initiating the Effective Unification of Black Hole Horizon Area and Entropy Quantization with Quasi-Normal Modes

Black hole (BH) area quantization may be the key to unlocking a unifying theory of quantum gravity (QG). Surmounting evidence in the field of BH research continues to support a horizon (surface) area with a discrete and uniformly spaced spectrum, but there is still no general agreement on the level spacing. In the specialized and important BH case study, our objective is to report and examine the pertinent groundbreaking work of thestrictly thermal and nonstrictly thermalspectrum level spacing of the BH horizon area quantization with included entropy calculations, which aims to tackle this gigantic problem. In particular, such work exemplifies a series of imperative corrections that eventually permits a BH’s horizon area spectrum to begeneralizedfrom strictly thermal to nonstrictly thermal with entropy results, thereby capturing multiple preceding developments by launching an effective unification between them. Moreover, the results are significant because quasi-normal modes (QNM) and “effective states” characterize the transitions between the established levels of the nonstrictly thermal spectrum.

Horizon area spectrum and entropy spectrum of a noncommutative geometry inspired regular black hole in three dimensions

By employing an adiabatic invariant and implementing the Bohr-Sommerfield quantization rule, I study the quantization of a regular black hole inspired by noncommutative geometry in AdS3 spacetime. The entropy spectrum as well as the horizon area spectrum of the black hole is obtained. It is shown that the spectra are discrete, and the spacing of the entropy spectrum is equidistant; in the limit , the area spectrum depends on the noncommutative parameter and the cosmological constant, but the spacing of the area spectrum is equidistant up to leading order in θ, and is independent of the noncommutative parameter and the cosmological constant.

Thermality and heat content of horizons from infinitesimal coordinate transformations

Thermal properties of a static horizon, (like the entropy S, heat content TS etc.) can be obtained either from the surface term of the Einstein-Hilbert action or by evaluating the Noether charge, corresponding to the diffeomorphisms generated by the timelike Killing vector field. We show that, for a wide class of geometries, the same results can be obtained using the vector field which produces an infinitesimal coordinate transformation between two physically relevant reference frames, viz. the freely falling frame near the horizon and the static, accelerated, frame. In particular, the infinitesimal coordinate transformation from inertial coordinates to uniformly accelerated frame can be used to obtain the heat content and entropy of the Rindler horizon. This result offers insight into understanding the observer dependent degrees of freedom which contribute to the entropy of null surfaces.

Study of thermodynamic laws in f(R,T,RμνTμν) gravity

We study first and second laws of black hole thermodynamics at theapparent horizon of FRW spacetime in f(R,T,RμνTμν)gravity, where R, Rμν are the Ricci scalar and Riemanntensor and T is the trace of the energy-momentum tensorTμν. We develop the Friedmann equations for any spatialcurvature in this modified theory and show that these equations canbe transformed to the form of Clausius relationThSeff = δ\U0001d4ac. Here Th is the horizontemperature, Seff is the entropy which contains contributionsboth from horizon entropy and additional entropy term introduced dueto the non-equilibrating description and δ\U0001d4ac isthe energy flux across the horizon. The generalized second law ofthermodynamics is also established in a more comprehensive form andone can recover the corresponding results in Einstein, f(R) andf(R,T) gravities. We discuss GSLT in the locality of assumptionthat temperature of matter inside the horizon is similar to that ofhorizon. Finally, we consider particular models in this theory andgenerate constraints on the coupling parameter for the validity ofGSLT.

Cosmological horizon entropy and generalized second law for flat Friedmann universe

We discuss the generalized second law (GSL) and the constraints imposed by it for two types of Friedmann universes. The first one is the Friedmann universe with radiation and a positive cosmological constant, and the second one consists of non-relativistic matter and a positive cosmological constant. The time evolution of the event horizon entropy and the entropy of the contents within the horizon are analyses in an analytical way by obtaining the Hubble parameter. It is shown that the GSL constraint the temperature of both the radiation and matter of the Friedmann universe. It is also shown that, even though the net entropy of the radiation (or matter) is decreasing at sufficiently large times as the universe expand, it exhibit an increase during the early times when universe is decelerating. That is the entropy of the radiation within the comoving volume is decreasing only when the universe has got an event horizon.

A q-parameter bound for particle spectra based on black hole thermodynamics with Rényi entropy

By regarding the Hawking–Bekenstein entropy of Schwarzschild black hole horizons as a non-extensive Tsallis entropy, its formal logarithm, the Rényi entropy, is considered. The resulting temperature – horizon radius relation has the same form as the one obtained from a (3+1)-dimensional black hole in anti-de Sitter space using the original entropy formula. In both cases the temperature has a minimum. A semi-classical estimate of the horizon radius at this minimum leads to a Bekenstein bound for the q-parameter in the Rényi entropy of micro black holes (q⩾1+2/π2), which is surprisingly close to fitted q-parameters of cosmic ray spectra and power-law distribution of quarks coalescing to hadrons in high energy accelerator experiments.

Entropy of viscous Universe models

The cosmological event horizon entropy and the apparent horizon entropy of the ΛCDM and the Bianchi type I Universe model with viscosity has been calculated numerically, and analytically in the large time limit. It is shown that for these Universe models the cosmological event horizon entropy increases with time and for large times it approaches a finite maximum value. The effect of viscosity upon the entropy is also studied and we have found that its role is to decrease the entropy. The bigger the viscosity coefficient is the less the entropy will be. Furthermore, the radiation entropy for the ΛCDM Universe model with and without viscosity is investigated, and together with the cosmological event horizon entropy are used to examine the validity of the generalized second law of thermodynamics, which states that the total rate of change of entropy of the Universe is never negative, in this Universe model.

Charged Dirac Particles’ Hawking Radiation via Tunneling of Both Horizons and Thermodynamics Properties of Kerr–Newman–Kasuya–Taub–NUT–AdS Black Holes

We investigate Hawking radiation of electrically and magnetically charged Dirac particles from a dyonic Kerr–Newman–Kasuya–Taub–NUT–Anti-de Sitter (KNKTN–AdS) black hole by considering thermal characters of both the outer and inner horizons. We apply Damour–Ruffini method and membrane method to calculate the temperature and the entropy of the inner horizon of the KNKTN–AdS black hole. The inner horizon admits thermal character with positive temperature and entropy proportional to its area. The inner horizon entropy contributes to the total entropy of the black hole in the context of Nernst theorem. Considering conservation of energy, charges, angular momentum, and the back-reaction of emitting particles to the spacetime, we obtain the emission spectra for both the inner and outer horizons. The total emission rate is obtained as the product of the emission rates of the inner and outer horizons. It deviates from the purely thermal spectrum with the leading term exactly the Boltzman factor and can bring some information out. The result thus can be treated as an explanation to the information loss paradox.

Thermodynamics in f (T) gravity and corrected entropies

This paper is devoted to studying the generalized second law of thermodynamics in f (T) gravity. We use quantum corrections such as power law and logarithmic corrected entropies to the horizon entropy along with Gibbs’ equation in the thermal equilibrium state. We derive the f (T) model by taking into account a power law scale factor through the first modified Friedmann equation which obeys the condition for a realistic model. Two types of horizons, i.e. Hubble and event horizons, are used to check the validity of the generalized second law of thermodynamics with corrected entropies. We conclude that this law holds with a specific range of entropy parameter on both horizons in the case of power law corrected entropy, while it violates for all values of entropy parameter on both horizons for logarithmic corrected entropy.

On the universality of inner black hole mechanics and higher curvature gravity

Black holes are famous for their universal behavior. New thermodynamic relations have been found recently for the product of gravitational entropies over all the horizons of a given stationary black hole. This product has been found to be independent of the mass for all such solutions of Einstein-Maxwell theory in d=4,5. We study the universality of this mass independence by introducing a number of possible higher curvature corrections to the gravitational action. We consider finite temperature black holes with both asymptotically flat and (A)dS boundary conditions. Although we find examples for which mass independence of the horizon entropy product continues to hold, we show that the universality of this property fails in general. We also derive further thermodynamic properties of inner horizons, such as the first law and Smarr relation, in the higher curvature theories under consideration, as well as a set of relations between thermodynamic potentials on the inner and outer horizons that follow from the horizon entropy product, whether or not it is mass independent.

Emergence of spaces and the dynamic equations of FRW universes inthe f(R) theory and deformed Hořava-Lifshitztheory

It has been shown that Friedmann equation of FRW universe can be derivedfrom the idea which says cosmic space is emergent as cosmic time progressesand our universe is expanding towards the state with the holographicequipartition by Padmanabhan. In this note, we give a general relationshipbetween the horizon entropy and the number of the degrees of freedomon the surface, which can be applied to quantum gravity.we also obtain the corresponding dynamic equationsby using the idea of emergence of spaces in the f(R) theory anddeformed Hořava-Lifshitz(HL)theory.