Black-Hole Entropy as Causal Links

We model a black hole spacetime as a causal set and count, with a certain definition, the number of causal links crossing the horizon in proximity to a spacelike or null hypersurface . We find that this number is proportional to the horizon's area on , thus supporting the interpretation of the links as the horizon atoms that account for its entropy. The cases studied include not only equilibrium black holes but ones far from equilibrium.     

On the Origin of Black-Hole Entropy

A simple statistical interpretation of the origin of black hole entropy is presented. It is shown that this entropy can be understood as emerging as a result of missing information about the exact state of the matter from which the black hole was formed.     

Kaluza—Klein Black-Hole Entropy by Quantum Statistics

By using the method of quantum statistics, we directly derive the partition functions of bosonic and fermionic field in Kaluza—Klein black hole with axial symmetry. Then via the improved brick-wall method, membrane model, we obtain that the entropy of bosonic and fermionic field in black hole is proportional to the area of horizon. In our result, the stripped term and the divergent logarithmic term no longer exist. The problem that the state density is divergent around the horizon doesn't exist either. We also give the influence of the spining degeneracy of particles on the entropy of black hole. We offer a new, simple, and direct way of calculating the entropy of different complicated black holes.     

Duality Covariant Formulation of Extended Supergravity: Central Charges,BPS States and Black-Hole Entropy

In these lectures we give a geometrical formulation of N-extrended supergravities which generalizes N = 2 special geometry of N = 2 theories. In all these theories duality symmetries are related to the notion of “flat symplectic bundles” and central charges may be defined as “sections” over these bundles. Attractor points giving rise to “fixed scalars” of the horizon geometry and Bekenstein-Hawking entropy formula for extremal black-holes are discussed in some details.     

Regularity of SRB Entropy for Geometric Lorenz Attractors

Journal of Statistical Physics - We consider the classical geometric Lorenz attractors, showing that the SRB entropy admits $$\gamma $$ -Hölder continuity for any $$0<\gamma <1$$ .     

Black-Hole Thermodynamics and Electromagnetism

We show a strong parallel between the Hawking-Bekenstein black-hole thermodynamics and electromagnetism: When the gravitational coupling constant transforms into the electromagnetic coupling constant, the Schwarzchild radius, the Bekenstein temperature, the Bekenstein decay time and the Planck mass transform to respectively the Compton wavelength, the Hagedorn temperature, the Compton time and a typical elementary particle mass. The reasons underlying this parallalism are then discussed in detail.     

Extensible Embeddings of Black-Hole Geometries

Removing a black hole conic singularity by means of Kruskal representation is equivalent to imposing extensibility on the Kasner–Fronsdal local isometric embedding of the corresponding black hole geometry. Allowing for globally non-trivial embeddings, living in Kaluza–Klein-like M ₅ × S ₁ (rather than in standard Minkowski M ₆ ) and parametrized by some wave number k, extensibility can be achieved for apparently forbidden frequencies in the range ₁ (k) ₂ (k). As k 0, _{1, 2} (0) _H (e.g., _H = 1/4M in the Schwarzschild case) such that the Hawking–Gibbons limit is fully recovered. The various Kruskal sheets are then viewed as slices of the Kaluza–Klein background. Euclidean k discreteness, dictated by imaginary time periodicity, is correlated with flux quantization of the underlying embedding gauge field.     

Vaidya Space-Time in Black-Hole Evaporation

We take a boundary-value approach to quantum amplitudes arising in gravitational collapse to a black hole. Pose boundary data on initial and final space-like hypersurfaces Σ _F,I , separated at spatial infinity by a Lorentzian proper-time interval T. Quantum amplitudes are calculated following Feynman's approach; rotate: T→|T|exp (−iθ) into the complex, where 0< θ≤π/2, and solve the corresponding well-posed complex classical boundary-value problem. We compute the classical Lorentzian action S _class and corresponding semi-classical quantum amplitude, proportional to exp (iS _class). To recover the Lorentzian amplitude, take the limit θ→ 0₊ of the semi-classical amplitude. For the classical boundary-value problem with given perturbative boundary data, we compute an effective spherically-symmetric energy-momentum tensor 〉 T ^μν〈 _EFF , averaged over several wavelengths of the radiation, describing the averaged extra energy-momentum contribution in the Einstein field equations, due to the perturbations. This takes the form of a null fluid, describing the radiation (of quantum origin) streaming radially outwards. The classical space-time metric, in this region of the space time, is of Vaidya form, justifying the adiabatic radial mode equations, for spins s = 0 and s = 2.     

Geometric Partition Entropy: Coarse-Graining a Continuous State Space

Entropy is re-examined as a quantification of ignorance in the predictability of a one dimensional continuous phenomenon. Although traditional estimators for entropy have been widely utilized in this context, we show that both the thermodynamic and Shannon’s theory of entropy are fundamentally discrete, and that the limiting process used to define differential entropy suffers from similar problems to those encountered in thermodynamics. In contrast, we consider a sampled data set to be observations of microstates (unmeasurable in thermodynamics and nonexistent in Shannon’s discrete theory), meaning, in this context, it is the macrostates of the underlying phenomenon that are unknown. To obtain a particular coarse-grained model we define macrostates using quantiles of the sample and define an ignorance density distribution based on the distances between quantiles. The geometric partition entropy is then just the Shannon entropy of this finite distribution. Our measure is more consistent and informative than histogram-binning, especially when applied to complex distributions and those with extreme outliers or under limited sampling. Its computational efficiency and avoidance of negative values can also make it preferable to geometric estimators such as k-nearest neighbors. We suggest applications that are unique to this estimator and illustrate its general utility through an application to time series in the approximation of an ergodic symbolic dynamics from limited observations.     

Thermodynamic Properties of Radiation near the Black-Hole Horizon

With the approach of statistical physics we derive the thermodynamic functions for the scalar radiation near the horizon of a Schwarzschild black hole. The state equations are obtained, which are different from that obtained from the special relativity by the equivalence principle. The thermodynamic equilibrium can be achieved only if where T is the (local) temperature and is the red-shift factor, as the box containing the radiation approaches the horizon. The spectral distribution equation and the displacement law are obtained, which differ from the Planck distribution and the Wien's displacement law. The results can be easily extended to the electromagnetic radiation.     

Two Kinds of Magnetic Connection in Black-Hole Accretion Disc

We discuss two kinds of magnetic connection （MC） in the black hole （BH） accretion disc： the magnetic connection between the BH and the disc （MCHD） and that between the plunging region and the disc （MCPD）. The magnetic field configuration is produced by an electric current flowing at the inner edge of the disc. It turns out that the transfer direction of energy and angular momentum depends on the BH spin and a parameter λ for adjusting the angular velocities of the plunging matter, which corresponds to at most five regions in the disc. The effect of MCPD results in a much steeper emissivity than a standard accretion disc in the inner disc, however it fails to reach the observation range 4.3-5.5 in several objects, such as Seyfert 1 galaxy MCG-6-30-15, microquasars XTE J1650-500 and GX 399-4.     

Extracting Energy Magnetically from Plunging Region of Black-Hole Accretion Disk

An analytical expression for the jet power extracted from the plunging region between a black hole (BH)horizon and the inner edge of the disk (hereafter the PL power) is derived based on an improved equivalent circuit in BH magnetosphere with a mapping relation between the radial coordinate of the plunging region and that of the remote astrophysical load.It is shown that the PL power is of great importance in explaining jet power and dominates over the BZ and DL powers for a wide value range of the BH spin.In addition,we show that the PL power derived in our model can be fitted with the strong jet powers of several 3CR FR I radio galaxies,which cannot be explained by virtue of the BZ mechanism.Furthermore,the condition for negative energy of the accreting particles in the plunging region is discussed with the validity of the second law of BH thermodynamics.     

On ``Time-Periodic' Black-Hole Solutions to Certain Spherically Symmetric Einstein-Matter Systems

This paper explores black hole solutions of various Einstein-wave matter systems admitting a time-orientation preserving isometry of their domain of outer communications taking some point to its future. In the first two parts, it is shown that such solutions, assuming in addition that they are spherically symmetric and the matter has a certain structure, must be Schwarzschild or Reissner-Nordström. Non-trivial examples of matter for which the result applies are a wave map and a massive charged scalar field interacting with an electromagnetic field. The results thus generalize work of Bekenstein [1] and Heusler [13] from the static to the periodic case. In the third part, which is independent of the first two, it is shown that Dirac fields preserved by an isometry of a spherically symmetric domain of outer communications of the type described above must vanish. It can be applied in particular to the Einstein-Dirac-Maxwell equations or the Einstein-Dirac-Yang/Mills equations, generalizing work of Finster, Smoller and Yau [10, 8, 9 and also 7].     

The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics.     

Embedding Misner and Brill–Lindquist Initial Data for Black-Hole Collisions

In this article we consider the isometrical immersions into Euclidean three-space of two-dimensional slices of the Misner and the Brill–Lindquist initial data for black-hole collisions. We show negativity of curvature and deduce other geometric properties of the slices. Under the assumption that ends behave strongly like paraboloid of revolution, we prove that Misner and the Brill–Lindquist slices cannot be isometrically immersed into R ³. This condition on an end is natural in general relativity because it holds for each end of a slice of the Schwartzschild metric where it is embedded as a paraboloid of revolution.     

Geometric transitions

The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in mathematics and in physics.     

Geometric fermions

We study the Kähler-Dirac equation which linearizes the laplacian on the space of antisymmetric tensor fields. In flat space-time it is equivalent to the Dirac equation with internal symmetry and on the lattice it reproduces Susskind fermions. The KD equation in curved space-time differs from the Dirac equation by coupling the gravitational field to the internal symmetry generators. This new way of treating fermionic degrees of freedom may lead to a solution of the generation puzzle but is in conflict with the equivalence principle and with Lorentz invariance on the Planck-mass scale.     

Geometric hierarchy

We present one approach for solving the gauge hierarchy problem in a grand unified supersymmetric theory. Supersymmetry is broken at a scale of order 10¹² GeV. Both the grand scale (～10¹⁹ GeV) and the weak scale are generated via radiative corrections. The main phenomenological features of the model are: (i) the proton decays into $\text{K}{}^0\text{μ}{}^{\mathrm{+}}\text{and}\text{K}{}^{\mathrm{+}}\text{ν}{}_{\mathrm{\mu }}$ and the neutron decays into $\text{K}{}^0\text{ν}{}_{\mathrm{\mu }}$; (ii) the strong GP problem is solved with an invisible axion; (iii) the superpartners of quarks, leptons, gauge and Higgs bosons have masses ～ 50–100 GeV; and (iv) the lightest superpartner is stable.     

Geometric dequantization

Dequantization is a set of rules which turn quantum mechanics (QM) into classical mechanics (CM). It is not the WKB limit of QM. In this paper we show that, by extending time to a 3-dimensional “supertime,” we can dequantize the system in the sense of turning the Feynman path integral version of QM into the functional counterpart of the Koopman-von Neumann operatorial approach to CM. Somehow this procedure is the inverse of geometric quantization and we present it in three different polarizations: the Schrödinger, the momentum and the coherent states ones.