Contracting rigid germs in higher dimensions

Following Favre, we define a holomorphic germ f:(C^d,0) -> (C^d,0) to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of f and its linear action on the fundamental group of the complement of the critical set.Comment: 30 pages, 0 figure

Sagnac Effect, Ring Lasers and Terrestrial Tests of Gravity

Light can be used as a probe to explore the structure of space-time: this is usual in astrophysical and cosmological tests, however it has been recently suggested that this can be done also in terrestrial laboratories. Namely, the GINGER project aims at measuring post-Newtonian effects, such as the gravito-magnetic ones, in an Earth based laboratory, by means of a ring lasers array. Here, we first review the theoretical foundations of the Sagnac Effect, on which ring lasers are based, and then we study the Sagnac Effect in a terrestrial laboratory, emphasizing the origin of the gravitational contributions that GINGER aims at measuring. Moreover, we show that accurate measurements allow to set constraints on theories of gravity different from General Relativity. Eventually, we describe the experimental setup of GINGER.Comment: 24 pages, 1 figure; accepted for publication in Galaxies, Special Issue "Advances in Gravitational Research

Gravito-electromagnetic Aharonov-Bohm effect: some rotation effects revised

By means of the description of the standard relative dynamics in terms of gravito-electromagnetic fields, in the context of natural splitting, we formally introduce the gravito-magnetic Aharonov-Bohm effect. Then, we interpret the Sagnac effect as a gravito-magnetic Aharonov-Bohm effect and we exploit this formalism for studying the General Relativistic corrections to the Sagnac effect in stationary and axially symmetric geometries.Comment: 19 pages, 2 figures, in in Proceedings of Analysis, Manifolds and Geometric Structures in Physics, International Conference in Honour of Y. Choquet-Bruhat, June 2004, Isola d'Elba, Ital

Gravitomagnetic Field of Rotating Rings

In the framework of the so-called gravitoelectromagnetic formalism, according to which the equations of the gravitational field can be written in analogy with classical electromagnetism, we study the gravitomagnetic field of a rotating ring, orbiting around a central body. We calculate the gravitomagnetic component of the field, both in the intermediate zone between the ring and the central body, and far away from the ring and central body. We evaluate the impact of the gravitomagnetic field on the motion of test particles and, as an application, we study the possibility of using these results, together with the Solar System ephemeris, to infer information on the spin of ring-like structures.Comment: 8 pages, 2 figures; revised to match the version accepted for publication in Astrophysics and Space Scienc

Gravito-electromagnetic Effects of Massive Rings

The Einstein field equations in linear post-Newtonian approximation can be written in analogy with electromagnetism, in the so-called gravito-electromagnetic formalism. We use this analogy to study the gravitational field of a massive ring: in particular, we consider a continuous mass distribution on Keplerian orbit around a central body, and we work out the gravitational field generated by this mass distribution in the intermediate zone between the central body and the ring, focusing on the gravito-magnetic component that originates from the rotation of the ring. In doing so, we generalize and complement some previous results that focused on the purely Newtonian effects of the ring (thus neglecting its rotation) or that were applied to the case of rotating spherical shells. Eventually, we study in some simple cases the effect of the the rotation of the ring, and suggest that, in principle, this approach could be used to infer information about the angular momentum of the ring.Comment: 13 pages, LaTeX, revised to match the version accepted for publication in the International Journal of Modern Physics

Light bending in f(T)f(T) gravity

In the framework of $f(T)$ gravity, we focus on a weak-field and spherically symmetric solution for the Lagrangian $f(T)=T+\alpha T^{2}$, where $\alpha$ is a small constant which parameterizes the departure from General Relativity. In particular, we study the propagation of light and obtain the correction to the general relativistic bending angle. Moreover, we discuss the impact of this correction on some gravitational lensing observables, and evaluate the possibility of constraining the theory parameter $\alpha$ by means of observations. In particular, on taking into account the astrometric accuracy in the Solar System, we obtain that $|\alpha| \leq 1.85 \times 10^{5}\, \mathrm{m^{2}}$; this bound is looser than those deriving from the analysis of Solar System dynamics, e.g. $|\alpha| \leq 5 \times 10^{-1}\, \mathrm{m^{2}}$, $|\alpha| \leq 1.8 \times 10^{4}\, \mathrm{m^{2}}$ or $|\alpha| \leq 1.2 \times 10^{2}\, \mathrm{m^{2}}$ . However we suggest that, since the effect only depends on the impact parameter, better constraints could be obtained by studying light bending from planetary objects.Comment: 14 pages, 1 figure; revised to match the version accepted for publication in IJMP

Rotation Effects and The Gravito-Magnetic Approach

Gravito-electromagnetism is somewhat ubiquitous in relativity. In fact, there are many situations where the effects of gravitation can be described by formally introducing "gravito-electric" and "gravito-magnetic" fields, starting from the corresponding potentials, in analogy with the electromagnetic theory (see also A. Tartaglia's contribution to these proceedings). The "many faces of gravito-electromagnetism" are related to rotation effects in both approximated and full theory approaches. Here we show that, by using a 1+3 splitting, relativistic dynamics can be described in terms of gravito-electromagnetic (GEM) fields in full theory. On the basis of this formalism, we introduce a "gravito-magnetic Aharonov-Bohm effect", which allows to interpret some rotation effects as gravito-magnetic effects. Finally, we suggest a way for measuring the angular momentum of celestial bodies by studying the gravito-magnetic effects on the propagation of electromagnetic signals.Comment: 3 pages, LaTeX, 1 EPS figure; to appear in the Proceedings for the ``XVI SIGRAV Conference'' in Vietri sul Mare (SA) 13-16 September 2004, References Changed, Misprints Correcte

A reversible allelic partition process and Pitman sampling formula

We introduce a continuous-time Markov chain describing dynamic allelic partitions which extends the branching process construction of the Pitman sampling formula in Pitman (2006) and the birth-and-death process with immigration studied in Karlin and McGregor (1967), in turn related to the celebrated Ewens sampling formula. A biological basis for the scheme is provided in terms of a population of individuals grouped into families, that evolves according to a sequence of births, deaths and immigrations. We investigate the asymptotic behaviour of the chain and show that, as opposed to the birth-and-death process with immigration, this construction maintains in the temporal limit the mutual dependence among the multiplicities. When the death rate exceeds the birth rate, the system is shown to have reversible distribution identified as a mixture of Pitman sampling formulae, with negative binomial mixing distribution on the population size. The population therefore converges to a stationary random configuration, characterised by a finite number of families and individuals.Comment: 17 pages, to appear in ALEA , Latin American Journal of Probability and Mathematical Statistic