Victor Berezin

Quantum black hole model and Hawking’s radiation

The black hole model with a self-gravitating charged spherical symmetric dust thin shell as a source is considered. The Schroedinger-type equation for such a model is derived. This equation appeared to be a finite differences equation. A theory of such an equation is developed and general solution is found and investigated in details. The discrete spectrum of the bound state energy levels is obtained. All the eigenvalues appeared to be infinitely degenerate. The ground state wave functions are evaluated explicitly. The quantum black hole states are selected and investigated. It is shown that the obtained black hole mass spectrum is compatible with the existence of Hawkings radiation in the limit of low temperatures both for large and nearly extreme Reissner-Nordstrom black holes. The above mentioned infinite degeneracy of the mass (energy) eigenvalues may appeared helpful in resolving the well known information paradox in the black hole physics.

Black hole mass spectrum vs spectrum of Hawking radiation

Abstract We consider a massive self-gravitating shell as a model for collapsing body and a null self-gravitating shell as a model for quanta of Hawking radiation. It is shown that the mass-energy spectra for the body and the radiation do not match. The way out of this difficulty is to consider not only out-going radiation but also the ingoing one. It means that the structure of black hole is changing during its evaporation resulting in the Bekenstein-Mukhanov spectrum for large masses.

Square-root quantization: Application to quantum black holes

Two different ways of quantizing the relativistic Hamiltonian for radial motion in the field of Coulomb-like potential are compared. The results depend slightly on choice of time. In the case of Lorentzian time a Sommerfeld spectrum is recovered. Application to quantum black holes gives a sqrt{n} mass spectrum with about the same numerical factors.

The phantom shell around a black hole and global geometry

We describe the possible scenarios for the evolution of a thin spherically symmetric self-gravitating phantom shell around the Schwarzschild black hole. The general equations describing the motion of the shell with a general form of the equation of state are derived and analysed. The different types of spacetime R± and T± regions and shell motion are classified depending on the parameters of the problem. It is shown that in the case of a positive shell mass there exist three scenarios for the shell evolution with an infinite motion and two distinctive types of collapse. Analogous scenarios were classified for the case of a negative shell mass. In particular, this classification shows that it is impossible for a physical observer to detect the phantom energy flow. We shortly discuss the importance of our results for astrophysical applications.

Towards a theory of quantum black hole

We describe some specific quantum black hole model. It is pointed out that the origin of a black hole entropy is the very process of quantum gravitational collapse. The quantum black hole mass spectrum is extracted from the mass spectrum of the gravitating source. The classical analog of quantum black hole is constructed.

Could a real (not virtual) static observer exist outside a Schwarzschild black hole

We considered the static spherically symmetric ensemble of observers, having finite bare mass and trying to measure geometrical and physical properties of the environmental static (Schwarzschild) space–time. The word “virtual” in the title means the test particle serving as an observer, and the “real” is the observer whose mass and its influences on the space–time metric cannot be neglected. It is shown that, using the photon rockets (which the mass together with the mass of their fuel is also taken into account) they can managed to keep themselves on the fixed value of radius. The process of diminishing the total bare mass up to zero lasts infinitely long time. It is important that the problem is solved self-consistently, i.e., with full account for the back reaction of both bare mass and radiation from rockets on the space–time geometry.

On maximal analytical extension of the Vaidya metric

The classical Vaidya metric is transformed to the special diagonal coordinates in the case of the linear mass function allowing rather easy treatment. We find the exact analytical expressions for metric functions in these diagonal coordinates. Using these coordinates, we elaborate the maximum analytic extension of the Vaidya metric with a linear growth of the black hole mass and construct the corresponding Carter-Penrose diagrams for different specific cases. The derived global geometry seemingly is valid also for a more general behavior of the black hole mass in the Vaidya metric.

ON THE THEORY OF SPHERICALLY SYMMETRIC THIN SHELLS IN CONFORMAL GRAVITY

The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy-momentum tensor of the thin shell is proportional to Dirac delta-function. We constructed the theory of the spherically symmetric thin shells for three types of gravitational theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl) gravity where the gravitational part of the total Lagrangian is just the square of the Weyl tensor; (3) Weyl+Einstein gravity. The results are compared with these in General Relativity (Israel equations). We considered in details the shells immersed in the vacuum. Some peculiar properties of such shells are found. In particular, for the traceless (= massless) shells it is shown that their dynamics can not be derived from the matching conditions and, thus, is completely arbitrary. On the contrary, in the case of the Weyl+Einstein gravity the trajectory of the same type of shell is completely restored even without knowledge of the outside solution.

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g00 = 0 or g00 = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.

Effect of the variation rate of an external magnetic field on the high-frequency response of surface superconductivity in a critical state

The effect of the magnetic field variation rate on the high-frequency absorption of surface superconductivity of cylindrical samples has been studied. It has been shown that the magnitude of additional highfrequency absorption attributed to Kulik vortices does not demonstrate saturation and other critical changes up to a magnetic field variation rate of 350 kOe/s, which indicates that the velocity of motion of Kulik vortices is higher than that of Abrikosov vortices. The effect of the normal core of the sample on the dynamics of a magnetic flux has been discussed.