Entropy of the Rindler Horizon

The entropy of the Rindler horizon to a nonuniformly accelerating observer is investigated. It is shown that result proportional to the area relies on a time-dependent cutoff, and the local energy determined by the cutoff is closer to the Planck scale than the brick-wall model. The method and result obtained in this paper can well be applied to the nonstationary black hole.     

Rindler solutions and their physical interpretation

We show that the singular behavior of Rindler solutions near horizon testifies to the currents of particles from a region arbitrarily close to the horizon. Besides, the Rindler solutions in right Rindler sector of Minkowski space can be represented as a superposition of only positive-or only negative-frequency plane waves; these states require infinite energy for their creation and possess infinite charge in a finite space interval, containing the horizon. The positive-or negative-frequency representations of Rindler solutions analytically continued to the whole Minkowski space make up a complete set of states in this space, which have, however, the aforementioned singularities. These positive (negative)-frequency states are characterized by positive (negative) total charge, the charge of the same sign in right (left) Rindler sector and by quantum number κ. But in other Lorentz invariant sectors they do not possess positive (negative)-definite charge density and have negative (positive) charge in left (right) Rindler sector. Therefore these states describe both the particle (antiparticle) and pairs, the mean number of which is given by Planck function of κ. These peculiarities make the Rindler set of solutions nonequivalent to the plane wave set and the inference on the existence of thermal currents for a Rindler observer moving in empty Minkowski space is unfounded. Zh. éksp. Teor. Fiz. 114, 777–785 (September 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Hawking Radiation of Dirac Particles on Rindler Horizon to a Uniformly Accelerating Observer

Following the method of Damour and Ruffini, the Hawking radiation of Dirac particles on Rindler horison to a uniformly accelerating observer is studied this paper. The temperature on Rindler horizon surface and the thermal spectrum formula of Dirac particles are obtained. The result is discussed.     

The effect of hadronic rindler horizon on hadronization process

In this study by extending the recent suggested mechanisms to hadronization processes, the information loss for QCD matter in hadronic Rindler horizon is found. We notice that for all finite values of quark and gluon energies, all information from all hadronization processes experiences some degree of loss. Then the effect of hedonic Rindler horizon on three jet cross section is explored. It is found that the three jet cross section is rising at y_cut =?0.0002 exhibits a turn-over at moderate value of y_cut =?0.01 and then rapidly decreases as y_cut increases. This model is consistent with OPAL data. Finally, different channels for producing Higgs boson near hadronic Rindler horizon are studied. It is shown that the cross section of Higgs boson produced via gluon fusion and quark interaction near a single Hadronic Rindler Horizon is much larger for higher center of mass energies. This is because an increase in the energy of hadronic Rindler horizon raises the temperature, thus intensifying the thermal radiation of QCD matter.     

Hawking Radiation of Dirac Particles on Rindler Horizon to a Uniformly Accelerating Observer

Following the method of Damour and Ruffini, the Hawking radiation of Dirac particles on Rindler horizonto a uniformly accelerating observer is studied in this paper. The temperature on Rindler horizon surface and the thermalspectrum formula of Dirac particles are obtained. The result is discussed.     

动态Rindler时空中Dirac旋量粒子的四分量波函数及Hawbing-Unruh热效应

在动态Rindler时空中对Dirac旋量粒子的动力学行为进行了研究,得到Dirac粒子四分量波函数的显式表示。同时还发现,对于一个作变加速直线运动的Rindler观察者存在一个随时间而变的视界,并将探测到随时间而变的热辐射,辐射温度正比于观察者的瞬时加速度,从而再一次证实了动态Rindler效应的存在。     

Rindler-type geometry inside a black hole

An anisotropic fluid with positive energy density and negative pressures is proposed in the black hole interior. The gravitational field is constant everywhere inside and is given by the horizon surface gravity. Even though the geometry is of Rindler type it is curved because of the spherical symmetry used and is singular at the origin. The Israel junction conditions are studied on the stretched horizon instead of using a matching process on the null surface r=2M$r=2M$. The entropy of any inner sphere is mass independent, maximally packed and unaffected by the outer layers.     

Boundary sources in the Doran-Lobo-Crawford spacetime

We take a null hypersurface (causal horizon) generated by a congruence of null geodesics as the boundary of the Doran-Lobo-Crawford spacetime to be the place where the Brown-York quasilocal energy is located. The components of the outer and inner stress tensors are computed and shown to depend on time and on the impact parameter b of the test-particle trajectory. The spacetime is a solution of Einstein’s equations with an anisotropic fluid as source. The surface energy density σ on the boundary is given by the same expression as that obtained previously for the energy stored on a Rindler horizon. For time intervals long compared to b (when the stretched horizon tends to the causal one), the components of the stress tensors become constant.      

Horizon Entropy

Although the laws of thermodynamics are well established for black hole horizons, much less has been said in the literature to support the extension of these laws to more general settings such as an asymptotic de Sitter horizon or a Rindler horizon (the event horizon of an asymptotic uniformly accelerated observer). In the present paper we review the results that have been previously established and argue that the laws of black hole thermodynamics, as well as their underlying statistical mechanical content, extend quite generally to what we call here causal horizons. The root of this generalization is the local notion of horizon entropy density.     

String quantization in accelerated frames and black holes

We quantize a closed bosonic string in a light-cone gauge in Rindler (uniformly accelerated) space-time and apply it to the Schwarzschild-Kruskal manifold. Inertial and accelerated particle states of the string associated to positive frequency modes with respect to the inertial and Rindler times respectively, are defined. There is a stretching effect of the string due to the presence of an event horizon. We explicitly solve the dynamical constraints leaving as physical degrees of freedom only those transverse to the acceleration. Different mass formulae are introduced depending on whether the centre of mass of the string has uniform speed or uniform acceleration. The expectation value of the Rindler (Schwarzschild) number-mode operator in the string around state (tachyon) results equal to a thermal spectrum at the Hawking-Unruh temperature T_s=α/2π (∼ M_Pl(M_Pl/M)^1/(D−3), where M is the black hole mass). We find T₀=M′/2π where M′ is the accelerated ground state string mass and T₀ the temperature T_s in dimensionless frequency units. Correlation functions of string coordinates and vertex operators and their Fourier transforms in accelerated time (string response functions) are computed and their thermal properties analyzed.     

Fermion entropy of non-uniformly rectilinearly accelerating black hole with electric and magnetic charges

Using the thin film brick-wall model, we calculate the fermion entropy on event horizon and the surface density of the entropy on the Rindler Horizon to a rectilinearly accelerating non-stationary black hole with electric and magnetic charges. The conclusion that black hole entropy is proportional to its area can still be applied by regulating the cut-off factor ?$?$ and the film's thickness δ$\delta$, which are time dependent.     

Does black-hole entropy make sense?

Bekenstein and Hawking saved the second law of thermodynamics near a black hole by assigning to the hole an entropyS _h proportional to the area of its event horizon. It is tempting to assume thatS _h possesses all the features commonly associated with the physical entropy. Kundt has shown, however, thatS _h violates several reasonable physical expectations. We review his criticism, augmenting it as follows: (a)S _h is a badly behaved state function requiring knowledge of the hole's future history; and (b) close analogs of event horizons in other space-times do not possess an “entropy.” We also discuss these questions: (c) IsS _h suitable for all regions of a black-hole space-time? And (b) shouldS _h be attributed to the exterior of a white hole? One can retainS _h for the interior (respectively, exterior) of a black (respectively, white) hole, but we reject this as contrary to the information-theoretic derivation of horizon entropy given by Bekenstein. The total entropy defined by Kundt (all ordinary entropy on space-section cutting through the hole, no horizon term) and that of Bekenstein-Hawking (ordinary entropy outside horizon plus horizon term) appear to be complementary concepts with separate domains of validity. In the most natural choice, an observer inside a black hole will use Kundt's entropy, and one remaining outside that of Bekenstein-Hawking.     

Essay: The Holography of Gravity Encoded in a Relation Between Entropy, Horizon Area, and Action for Gravity

I provide a general proof of the conjecture that one can attribute an entropy to the area of any horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature T = ^–1 and evaluating the exact partition function Z(). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z exp[S – E], where S = (1/4)4 a ² and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the black-hole spacetimes and provides a simple and consistent interpretation of entropy and energy for De Sitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. Further, I show that the relationship between entropy and area allows one to construct the action for the gravitational field on the bulk and thus the full theory. In this sense, gravity is intrinsically holographic.     

Rindler-like Horizon in Spherically Symmetric Spacetime

In this paper, the Rindler-like horizon in a spherically symmetric spacetime is proposed. It is showed that just like the Rindler horizon in Minkowski spacetimes, there is also a Rindler-like horizon to a family of special observers in general spherically symmetric spacetimes. The entropy of this type of horizon is calculated with the thin film brick-wall model. The significance of entropy is discussed. Our results imply some connection between Bekeinstein-Hawking entropy and entanglement entropy.     

Horizon thermodynamics in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) theory

We investigate whether the new horizon first law proposed recently still work in f(R) theory. We identify the entropy and the energy of black hole as quantities proportional to the corresponding value of integration, supported by the fact that the new horizon first law holds true as a consequence of equations of motion in f(R) theories. The formulas for the entropy and energy of black hole found here are in agreement with the results obtained in literatures. For applications, some nontrivial black hole solutions in f(R) theories have been considered, the entropies and the energies of black holes in these models are firstly computed, which may be useful for future researches.     

Radiation Energy Flux and Radiation Power of Schwarzschild Black Hole

Applying the entropy density near the event horizon, we obtained the result that the radiation energy flux of the black hole is always proportional to the quartic of the temperature of its event horizon. That is to say, the thermal radiation of the black hole always satisfies the generalized Stefan–Boltzmann law. The derived generalized Stefan–Boltzmann coefficient is no longer a constant. When the cut-off distance and the thin film thickness are both fixed, it is a proportional coefficient which is related to the black hole mass, the kinds of radiation particles and space–time metric near the event horizon. In this paper, we have put forward a thermal particle model in curved space–time. By this model, the result has been obtained that when the thin film thickness and the cut-off distance are both fixed, the radiation energy flux received by observer far away from the Schwarzschild black hole is proportional to the average radial effusion velocity of the radiation particles in the thin film, and inversely proportional to the square of the distance between the observer and the black hole.     

Entropy of a Rindler Observer

We compute the entropy of a Rindler particle-detector (observer) in the presence of a quantum field in the Minkowski vacuum state; due to the Unruh effect, the observer is immersed in a thermal bath at a temperature proportional to its proper acceleration.     

Entropy of a Rindler wedge

The entropy of a Rindler wedge is calculated from the action of its complexified section using the method of Gibbons and Hawking. It is equal to one quarter the area of the event horizon in fundamental units.     

Effective information loss outside the horizon

If a system falls through a black hole horizon, then its information is lost to an observer at infinity. But we argue that the accessible information is lost before the horizon is crossed. The temperature of the hole limits information carrying signals from a system that has fallen too close to the horizon. Extremal holes have T = 0, but there is a minimum energy required to emit a quantum in the short proper time left before the horizon is crossed. If we attempt to bring the system back to infinity for observation, then acceleration radiation destroys the information. All three considerations give a critical distance from the horizon \({d\sim \sqrt\frac{r_H}{\Delta E}}\), where r _H is the horizon radius and ΔE is the energy scale characterizing the system. For systems in string theory where we pack information as densely as possible, this acceleration constraint is found to have a geometric interpretation. These estimates suggest that in theories of gravity we should measure information not as a quantity contained inside a given system, but in terms of how much of that information can be reliably accessed by another observer.