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

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

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

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

Short Range Interactions in the Hydrogen Atom

In calculating the energy corrections to the hydrogen levels we can identify two different types of modifications of the Coulomb potential $V_{C}$, with one of them being the standard quantum electrodynamics corrections, $\delta V$, satisfying $\left|\delta V\right|\ll\left|V_{C}\right|$ over the whole range of the radial variable $r$. The other possible addition to $V_{C}$ is a potential arising due to the finite size of the atomic nucleus and as a matter of fact, can be larger than $V_{C}$ in a very short range. We focus here on the latter and show that the electric potential of the proton displays some undesirable features. Among others, the energy content of the electric field associated with this potential is very close to the threshold of $e^+e^-$ pair production. We contrast this large electric field of the Maxwell theory with one emerging from the non-linear Euler-Heisenberg theory and show how in this theory the short range electric field becomes smaller and is well below the pair production threshold

γγ\gamma \gamma Processes at High Energy pp Colliders

In this note we investigate the production of charged heavy particles via \gaga\ fusion at high energy pp colliders. We revise previous claims that the \gaga\ cross section is comparable to or larger than that for the corresponding Drell-Yan process at high energies. Indeed we find that the \gaga\ contribution to the total production cross section at pp is far below the Drell-Yan cross section. As far as the individual elastic, semi-elastic and inelastic contributions to the \gaga\ process are concerned we find that they are all of the same order of magnitude.Comment: REVTEX, 12 pages, two uuencoded figures appended at the end of the fil