On Black Holes and Cosmological Constant in Noncommutative Gauge Theory of Gravity

Deformed Reissner-Nordstr\"om, as well as Reissner-Nordstr\"om de Sitter, solutions are obtained in a noncommutative gauge theory of gravitation. The gauge potentials (tetrad fields) and the components of deformed metric are calculated to second order in the noncommutativity parameter. The solutions reduce to the deformed Schwarzschild ones when the electric charge of the gravitational source and the cosmological constant vanish. Corrections to the thermodynamical quantities of the corresponding black holes and to the radii of different horizons have been determined. All the independent invariants, such as the Ricci scalar and the so-called Kretschmann scalar, have the same singularity structure as the ones of the usual undeformed case and no smearing of singularities occurs. The possibility of such a smearing is discussed. In the noncommutative case we have a local disturbance of the geometry around the source, although asymptotically at large distances it becomes flat.Comment: Based on a talk given at the International Conference on Fundamental and Applied Research in Physics "Farphys 2007", 25-28 October 2007, Iasi, Romani

Order and Chaos in some Trigonometric Series: Curious Adventures of a Statistical Mechanic

This paper tells the story how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some "amateurs" to the discovery that the one-parameter family of deterministic trigonometric series \pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t), $p>1$, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. It is proved that \pzcS_p(t) = \alpha_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset, with explicitly computed constant $\alpha_p$. Experts' commentaries are reproduced stating the fluctuations of \pzcS_p(t) - \alpha_p{\rm{sign}}(t)|t|^{1/{p}} are presumably not Gaussian. Inspired by a central limit type theorem of Marc Kac, a well-motivated conjecture is formulated to the effect that the fluctuations of the $\lceil t^{1/(p+1)}\rceil$-th partial sum of \pzcS_p(t), when properly scaled, do converge in distribution to a standard Gaussian when $t\to\infty$, though --- provided that $p$ is chosen so that the frequencies \{n^{-p}\}_{n\in\Nset} are rationally linear independent; no conjecture has been forthcoming for rationally dependent \{n^{-p}\}_{n\in\Nset}. Moreover, following other experts' tip-offs, the interesting relationship of the asymptotics of \pzcS_p(t) to properties of the Riemann $\zeta$ function is exhibited using the Mellin transform.Comment: Based on the invited lecture with the same title delivered by the author on Dec.19, 2011 at the 106th Statistical Mechanics Meeting at Rutgers University in honor of Michael Fisher, Jerry Percus, and Ben Widom. (19 figures, colors online). Comments of three referees included. Conjecture 1 revised. Accepted for publication in J. Stat. Phy

Deriving the mass of particles from Extended Theories of Gravity in LHC era

We derive a geometrical approach to produce the mass of particles that could be suitably tested at LHC. Starting from a 5D unification scheme, we show that all the known interactions could be suitably deduced as an induced symmetry breaking of the non-unitary GL(4)-group of diffeomorphisms. The deformations inducing such a breaking act as vector bosons that, depending on the gravitational mass states, can assume the role of interaction bosons like gluons, electroweak bosons or photon. The further gravitational degrees of freedom, emerging from the reduction mechanism in 4D, eliminate the hierarchy problem since generate a cut-off comparable with electroweak one at TeV scales. In this "economic" scheme, gravity should induce the other interactions in a non-perturbative way.Comment: 30 pages, 1 figur