Relativistic quantum bouncing particles in a homogeneous gravitational field

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein–Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schrödinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein–Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.

Entropy of Reissner–Nordström 3D Black Hole in Roegenian Economics

The subject of this paper is to analyse the Mathematical Principia of Economic 3D Black Holes in Roegenian economics. In detail, we study two main problems: (i) mathematical origin of economic 3D black holes; and (ii) entropy and internal political stability depending on national income and the total investment, for economic Reissner–Nordström (RN) 3D black hole. To solve these problems, it was necessary to jump from macroeconomic side to microeconomic side (a substantial approach as they are so different), to complete the thermodynamics–economics dictionary with new entities, and to introduce the flow between two macroeconomic systems. The main contribution is about introducing and studying the Schwarzschild-type metric on an economic 4D system, together with Rindler coordinates, Einstein 4D partial differential equations (PDEs), and economic RN 3D black holes. In addition, we introduce some economic Ricci type flows or waves, for further research.

Entropy of Reissner-Nordström 3D Hole in Roegenian Economics

The subject of this paper is to analyse the Math Principia of Economic 3D Black Holes in Roegenian economics. This idea is totally new in the related literature, excepting our papers. In details, we study two special problems: (i) math origin of economic 3D black holes, (ii) entropy and internal political stability depending on national income and the total investment, for economic RN 3D black hole. To solve these problems, it was necessary to jump from macroeconomic side to microeconomic side (a substantial approach so different), to complete the thermodynamics-economics dictionary with new entities, to introduce the flow between two macroeconomic systems, to study the Schwarzschild type metric properties on an economic 4D system, together with Rindler coordinates, Einstein 4D PDEs, and economic RN 3D black hole. In addition, we introduce some economic Ricci type flows or waves, for further research.

DISPERSION RELATIONS FOR COLD PLASMA AROUND THE HORIZON OF SCHWARZSCHILD–DE SITTER BLACK HOLE

In this paper, we investigate the wave properties of cold plasma in the vicinity of Schawarzchild–de Sitter black hole horizon using 3 + 1 formalism. The general relativistic magnetohydrodynamical equations are formulated for this space–time with the use of Rindler coordinates. We consider both the rotating and nonrotating surroundings with magnetized and nonmagnetized plasmas. Linear perturbation and Fourier analysis techniques are applied by introducing simple harmonic waves. We derive complex dispersion relation from the determinant of Fourier analyzed equations for each case which provides real and complex values of the wave number. From the wave number we determine the phase and group velocities, the refractive index etc., which are used to discuss the characteristics of the waves around the event horizon.

On the Gravitational Energy of the Hawking Wormhole

The surface energy for a conformally flat space–time which represents the Hawking wormhole in spherical (static) Rindler coordinates is computed using the Hawking–Hunter formalism for nonasymptotically flat space–times. The physical gravitational Hamiltonian is proportional to the Rindler acceleration g of the hyperbolic observer and is finite on the event horizon ξ=b (b — the Planck length, ξ — the Minkowski interval). The corresponding temperature of the system of particles associated to the massless scalar field Ψ=1-b2/ξ2, coupled conformally to Einstein's equations, is given by the Davies–Unruh temperature up to a constant factor of order unity.

STATIC, MASSIVE FIELDS AND VACUUM POLARIZATION POTENTIAL IN RINDLER SPACE

In Rindler space, we determine in terms of special functions the expression of the static, massive scalar or vector field generated by a point source. We find also an explicit integral expression of the induced electrostatic potential resulting from the vacuum polarization due to an electric charge at rest in the Rindler coordinates. For a weak acceleration, we give then an approximate expression in the Fermi coordinates associated with the uniformly accelerated observer.

Alternative Fock quantization of neutrinos in flat space-time

The neutrino equations in flat space-time are studied in static Rindler accelerated coordinates and in the Penrose conformal coordinates. It is found that the definition of a positive frequency solution to the Dirac equation in the Rindler coordinates is not equivalent to that in Minkowski coordinates, implying inequivalent Fock representations in the two coordinate systems. The definition of positive frequency in the Penrose conformal metric is found to be exactly equivalent to that of Minkowski spacetime. A complete set of solutions is found, and the ‘energy' spectrum is discrete ( ω = 2 n + 1, n > j , where j is the total angular momentum of the eigenfunction) and excludes zero frequency.