The Scalar Field Equation in Schwarzschild–de Sitter Space

The scalar wave equation between the inner and the outer horizon in the Schwarzschild–de Sitter geometry is solved numerically, and the spatial variations of the field amplitude, as well as of the potential, are shown graphically. By generalizing the "tortoise" coordinate x known from Schwarzschild theory to the SdS system we first transfer the wave equation to a convenient form in which the potential V is written as a function of x. We then show how a useful "tangent" approximation can be introduced which leads to a simple, analytically invertible, relation between x and the radius r. We concentrate on two limiting cases. The first case is when the two horizons are close to each other, the so-called Nariai black hole, and the second case is when the horizons are far apart. Reflection and transmission coefficients are worked out on the basis of a replacement of the real barrier V(x) by a square barrier.     

Black Hole and Cosmic Entropy for Schwarzschild–de Sitter Space–Time

We calculate the free energy and the entropy of a scalar field in terms of the brick-wall method in the background of the Schwarzschild–de Sitter space–time. We obtain the entropy of a black hole and the cosmic entropy at nonasymptotic flat space. When the cut-off satisfies the proper condition, the entropy of a black hole is proportional to the area of a black hole horizon, and the cosmic entropy is proportional to the cosmic horizon area.     

Nonequatorial circular orbits of spinning particles in the Schwarzschild–de Sitter background

In this paper analytical solutions of the Mathisson–Papapetrou equations that describe nonequatorial circular orbits of a spinning particle in the Schwarzschild–de Sitter background are studied, and the role of the cosmological constant is emphasized. It is shown that generally speaking a highly relativistic velocity of the particle is a necessary condition of motion along this type of orbits, with an exception of orbits locating close to the position of the static equilibrium, where low velocities are possible as well. Depending on the correlation between the spin orientation of the particle and its orbital velocity some of the possible nonequatorial circular orbits exist due to the repulsive action on the particle caused by the spin–gravity coupling and the others are caused by the attractive action. Here values of the energy of the particle on the corresponding orbits are also analyzed.     

Classical tests of general relativity in the Newtonian limit of the Schwarzschild–de Sitter spacetime

Recently it has been shown that despite previous claims the cosmological constant affects light bending. In the present article we study light bending and the advance of Mercury’s perihelion in the context of the Newtonian limit of the Schwarzschild–de Sitter spacetime employing the special relativistic equivalence of mass and energy. In both cases, up to a constant factor, we find the same results as in the full general relativistic treatment of the same phenomena. These approximate and intuitive arguments demonstrate clearly what effects should have been expected from the presence of Λ in the general relativistic treatment of these phenomena.     

Dirac Equation and Its Solution Around Schwarzschild–de Sitter Black Hole

This paper aims to solve the radial parts of Dirac equation between the inner and the outer horizon in Schwarzschild–de Sitter geometry numerically. A logarithm approximation which almost has the same nature with the modified tortoise coordinate r* even close to the two horizons is found. It is used to replace the modified tortoise coordinate r*, this leads to a simple analytically invertible relation between r* and the radius r. Then, the potential V (r*) is replaced by a collection of step functions. By a quantum mechanical method, the solution of the wave equation as well as the reflection and transmission coefficients are computed. The resulting wave turns out to be not close to that of a harmonic wave globally.     

Spinning particles in Schwarzschild–de Sitter space–time

After considering the reference case of the motion of spinning test bodies in the equatorial plane of the Schwarzschild space–time, we generalize the results to the case of the motion of a spinning particle in the equatorial plane of the Schwarzschild–de Sitter space–time. Specifically, we obtain the loci of turning points of the particle in this plane. We show that the cosmological constant affect the particle motion when the particle distance from the black hole is of the order of the inverse square root of the cosmological constant.     

Entropy of Neutrino Field in Kerr—Newman—de Sitter Space—time

By using the improved brick wall model,we regard the two null horizons in kerr-Newman-de Sitter space-time as two independent thermal equilibrium systems and we calculate the entropies of both horizons due to neutrino field.We propose that the so-called“superradiant”mode cannot be considered because fermion fields does not display supperradiance.In fact,the nonsuperradiant part does exactly show the expected area-law entropy.Moreover,our cut-offε,which does not require an angular cut-off,is independent of angle θ.     

Foliation of the Kottler–Schwarzschild–de Sitter spacetime by flat spacelike hypersurfaces

There exist Kruskal like coordinates for the Reissner–Nordstrom (RN) black hole spacetime which are regular at coordinate singularities. Non-existence of such coordinates for the extreme RN black hole spacetime has already been shown. Also the Carter coordinates available for the extreme case are not manifestly regular at the coordinate singularity, therefore, a numerical procedure was developed to obtain free fall geodesics and flat foliation for the extreme RN black hole spacetime. The Kottler–Schwarzschild–de Sitter (KSSdS) spacetime geometry is similar to the RN geometry in the sense that, like the RN case, there exist non-singular coordinates when there are two distinct coordinate singularities. There are no manifestly regular coordinates for the extreme KSSdS case. In this paper foliation of all the cases of the KSSdS spacetime by flat spacelike hypersurfaces is obtained by introducing a non-singular time coordinate.     

Entropy of higher-dimensional charged Gauss–Bonnet black hole in de Sitter space

The fundamental equation of the thermodynamic system gives the relation between the internal energy, entropy and volume of two adjacent equilibrium states. Taking a higher-dimensional charged Gauss–Bonnet black hole in de Sitter space as a thermodynamic system, the state parameters have to meet the fundamental equation of thermodynamics. We introduce the effective thermodynamic quantities to describe the black hole in de Sitter space. Considering that in the lukewarm case the temperature of the black hole horizon is equal to that of the cosmological horizon, we conjecture that the effective temperature has the same value. In this way, we can obtain the entropy formula of spacetime by solving the differential equation. We find that the total entropy contains an extra term besides the sum of the entropies of the two horizons. The corrected term of the entropy is a function of the ratio of the black hole horizon radius to the cosmological horizon radius, and is independent of the charge of the spacetime.     

Klein–Gordon and Dirac Equations in de Sitter Space–Time

We present and discuss the Klein–Gordonand Dirac wave equations in the de Sitter universe. Toobtain the Dirac wave equation we use the factorizationof the second-order invariant Casimir operatorassociated to the Fantappie–de Sitter group. Boththe Klein–Gordon and Dirac wave equations arediscussed in terms of the spherical harmonics with spinweight. A particular case of Dirac wave equation issolved in terms of a new class of polynomials.     

Entropy of Dirac Field in Non-stationary and Slowly Changing Reissner-Nordstrom Black Hole

Using the thin film brick-wall model and WKB approximation, the entropy of the Dirac field in the nonstationary and slowly changing Reissner-Nordstrom （R-N） black hole is calculated. The result shows that the entropy of the R-N black hole is still proportional to its surface area if we choose proper cut-off.     

Quantum Radiation of a Non-stationary Kerr-Newman Black Hole in de Sitter Space-Time

Hawking radiation of Klein-Gordon and Dirac particles in a non-stationary Kerr-Newman-de-Sitter black hole is studied by introducing a new tortoise coordinate transformation. The result shows that the Fermi-Dirac radiant spectrum displays a new term that represents the interaction between the spin of spinor particles and the rotation of black holes, which is absent in the Bose-Einstein distribution of Klein-Gordon particles.     

Entropy and topology of the Kerr—de Sitter black hole

By using the path integral method of Gibbons and Hawking, the entropy of the Kerr-de Sitter black hole is investigated under the microcanonical ensemble. We find that the entropy is one eighth the sum of the products of the Euler number of its cosmological horizon and event horizon with their respective areas. It is shown that the origin of the entropy of the black hole is related to the topology of its instanton.     

Emergent universe in brane world scenario with Schwarzschild–de Sitter bulk

A model of an emergent universe is obtained in brane world. Here the bulk energy is in the form of cosmological constant, while the brane consists of a fluid satisfying an equation of state of the form , which is effectively a radiation equation of state at high energies. It is shown that with the positive bulk cosmological constant, one of our models represents an emergent universe.     

Entropy in the NUT—Kerr—Newman Black Holes in the Background of de Sitter Spacetime

We calculate the entropy of the fermion field in the NUT-Kerr-Newman black holes in the background of the de Sitter spacetime by using the improved brick-wall method and the membrane model.Here the Euler characteristic of the black holes is over two.The results show that,as the cut-off is properly chosen,the entropy in the black hole satisfies the Bekenstein-Hawking area law.     

Real scalar field scattering in the nearly extremal Schwarzschild de Sitter space

Reasonable approximations are introduced to investigate the real scalar field scattering in the nearly extremal Schwarzschild-de Sitter (SdS) space.The approximations naturally lead to the invertible x(r) and the global replacement of the true potential by a Pshl-Teller one.Meanwhile,the Schrdinger-like wave equation is transformed into a solvable form.Our numerical solutions to the wave equation show that the wave is characteristically similar to the harmonic under the tortoise coordinate x,while the wave piles up near the two horizons and the wavelength tends to its maximum as the potential approaches to the peak under the radial coordinate r.     

Asymmetry of Hawking Radiation of Dirac Particles in a Charged Vaidya–de Sitter Black Hole

The Hawking radiation of Dirac particles in a charged Vaidya–de Sitter black hole is investigated by using the method of generalized tortoise coordinate transformation. It is shown that the Hawking radiation of Dirac particles does not exist for P₁, Q₂ components, but for P₂, Q₁ components it does. Both the location and the temperature of the event horizon change with time. The thermal radiation spectrum of Dirac particles is the same as that of Klein-Gordon particles.     

Entropy quantization of Reissner-Nordström de Sitter black hole via adiabatic covariant action

Based on the ideas of adiabatic invariant quantity, we attempt to quantize the entropy of a charged black hole in de Sitter spacetime in two different coordinates. The entropy spectrum is obtained by imposing Bohr-Sommerfeld quantization rule and the laws of black hole thermodynamics to the modified adiabatic covariant action of the charged black hole. The result shows that the spacing of entropy spectrum is equidistant, and the corresponding horizon area quantum is identical to Bekenstein’s result. Interestingly, in contrast to the quasinormal mode analysis, we note that there is no need to impose the small charge limit for the obtained entropy spectrum of the charged black hole. We also note that the modified adiabatic covariant action gives the same value for the black hole entropy spectrum in different coordinate frames. This is a physically desired result since the entropy spectrum should be invariant under the coordinate transformations.     

A Remark on the Residual Entropy of the Antiferromagnetic Ising Model in the Maximal Critical Field

By means of a transfer matrix method, we show that the residual entropy S of the two-dimensional square lattice antiferromagnetic Ising model in the maximal critical field satisfies (ln λ _n )/(n+1)≤S≤(ln λ _n )/n, where λ _n is the largest eigenvalue of the transfer matrix F _n on a strip of width n. Using these bounds, we numerically calculate the value of S, with precise estimates on the errors, namely, S=0.394198±0.020747.     

Phase Diagram of the BCS–Hubbard Model in a Magnetic Field

We propose an extended BCS–Hubbard model and investigate its ground state phase diagram in an external magnetic field. By mapping the model onto a model of spinless fermions coupled with conserving Z₂ variables which are mimicked by pseudospins, the model is shown to be exactly solvable along the symmetric lines for an arbitrary on-site Hubbard interaction on the bipartite lattice. In the zero field limit, the ground states exhibit an antiferromagnetic order of pseudospins. In the lar...