Scattering for massive Dirac fields on the Kerr metric

Starting with the Dirac equation outside the event horizon of a non-extreme Kerr black hole, we develop a time-dependent scattering theory for massive Dirac particles. The explicit computation of the modified wave operators at infinity is done by implementing a time-dependent logarithmic phase shift from the free dynamics to offset the long range term in the full Hamiltonian due to the presence of the gravitational force. Analytical expressions for the wave operators are also given.Comment: 33 pages, 1 figure, new proof of Lemma III.1, minor changes in Section III, a few typos corrected in Thm. IV.1 and Thm. B.

Velocity and velocity bounds in static spherically symmetric metrics

We find simple expressions for velocity of massless particles in dependence of the distance $r$ in Schwarzschild coordinates. For massive particles these expressions put an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordstr\"om with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there exists always a region where the massless particle moves with a velocity bigger than the velocity of light in vacuum. In the case of Reissner-Nordstr\"om-de Sitter we completely characterize the radial velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.Comment: 20 pages, 5 figure

The Dirac propagator in the extreme Kerr metric

Starting with the Dirac equation in the extreme Kerr metric we derive an integral representation for the propagator of solutions of the Cauchy problem with initial data in the class of smooth compactly supported functions.Comment: 9 pages, one more reference adde

The Dirac propagator in the Kerr-Newman metric

We give an alternative proof of the completeness of the Chandrasekhar ansatz for the Dirac equation in the Kerr-Newman metric. Based on this, we derive an integral representation for smooth compactly supported functions which in turn we use to derive an integral representation for the propagator of solutions of the Cauchy problem with initial data in the above class of functions. As a by-product, we also obtain the propagator for the Dirac equation in the Minkowski space-time in oblate spheroidal coordinates.Comment: 29 pages, modifications in the abstract and in the introduction, small improvements in section 2.

Maximal extension of the Schwarzschild spacetime inspired by noncommutative geometry

We derive a transformation of the noncommutative geometry inspired Schwarzschild solution into new coordinates such that the apparent unphysical singularities of the metric are removed. Moreover, we give the maximal singularity-free atlas for the manifold with the metric under consideration. This atlas reveals many new features e.g. it turns out to describe an infinite lattice of asymptotically flat universes connected by black hole tunnels.Comment: 17 pages LaTex, 2 figure

The generalized Heun equation in QFT in curved space-times

In this article we give a brief outline of the applications of the generalized Heun equation (GHE) in the context of Quantum Field Theory in curved space-times. In particular, we relate the separated radial part of a massive Dirac equation in the Kerr-Newman metric and the static perturbations for the non-extremal Reissner-Nordstr\"{o}m solution to a GHE.Comment: 7 pages, some small improvements in section

Velocity and velocity bounds in static spherically symmetric metrics

We find simple expressions for velocity of massless particles in dependence of the distance $r$ in Schwarzschild coordinates. For massive particles these expressions put an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordstr\"om with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there exists always a region where the massless particle moves with a velocity bigger than the velocity of light in vacuum. In the case of Reissner-Nordstr\"om-de Sitter we completely characterize the radial velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.Comment: 20 pages, 5 figure

Quantum Mechanical Corrections to the Schwarzschild Black Hole Metric

Motivated by quantum mechanical corrections to the Newtonian potential, which can be translated into an $\hbar$-correction to the $g_{00}$ component of the Schwarzschild metric, we construct a quantum mechanically corrected metric assuming $-g_{00}=g^{rr}$. We show how the Bekenstein black hole entropy $S$ receives its logarithmic contribution provided the quantum mechanical corrections to the metric are negative. In this case the standard horizon at the Schwarzschild radius $r_S$ increases by small terms proportional to $\hbar$ and a remnant of the order of Planck mass emerges. We contrast these results with a positive correction to the metric which, apart from a corrected Schwarzschild horizon, leads to a new purely quantum mechanical horizon.Comment: 14 pages Latex, enlarged version as compared to the published on