An accurate formula for the period of a simple pendulum oscillating beyond the small-angle regime

A simple approximation formula is derived here for the dependence of the period of a simple pendulum on amplitude that only requires a pocket calculator and furnishes an error of less than 0.25% with respect to the exact period. It is shown that this formula describes the increase of the pendulum period with amplitude better than other simple formulas found in literature. A good agreement with experimental data for a low air-resistance pendulum is also verified and it suggests, together with the current availability/precision of timers and detectors, that the proposed formula is useful for extending the pendulum experiment beyond the usual small-angle oscillations.Comment: 15 pages and 4 figures. to appear in American Journal of Physic

Topology Change and Causal Continuity

The result that, for a scalar quantum field propagating on a ``trousers'' topology in 1+1 dimensions, the crotch singularity is a source for an infinite burst of energy has been used to argue against the occurrence of topology change in quantum gravity. We draw attention to a conjecture due to Sorkin that it may be the particular type of topology change involved in the trousers transition that is problematic and that other topology changes may not cause the same difficulties. The conjecture links the singular behaviour to the existence of ``causal discontinuities'' in the spacetime and relies on a classification of topology changes using Morse theory. We investigate various topology changing transitions, including the pair production of black holes and of topological geons, in the light of these ideas.Comment: Latex, 28 pages, 10 figures, small changes in text (one figure removed), conclusions remain unchanged. Accepted for publication in Physical Review

Notes on Euclidean Wilson loops and Riemann Theta functions

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5 space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3 space. Using known mathematical results for such minimal area surfaces we describe an infinite parameter family of analytic solutions for closed Wilson loops. The solutions are given in terms of Riemann theta functions and the validity of the equations of motion is proven based on the trisecant identity. The world-sheet has the topology of a disk and the renormalized area is written as a finite, one-dimensional contour integral over the world-sheet boundary. An example is discussed in detail with plots of the corresponding surfaces. Further, for each Wilson loops we explicitly construct a one parameter family of deformations that preserve the area. The parameter is the so called spectral parameter. Finally, for genus three we find a map between these Wilson loops and closed curves inside the Riemann surface.Comment: 35 pages, 7 figures, pdflatex. V2: References added. Typos corrected. Some points clarifie

On Charge-3 Cyclic Monopoles

We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of genus 2 data. The Richelot correspondence, a generalization of the arithmetic mean, is used to solve for this genus 2 curve. Results of other approaches are compared.Comment: 34 pages, 16 figures. Revision: Abstract added and a few small change

Nonadiabatic Pauli susceptibility in fullerene compounds

Pauli paramagnetic susceptibility $\chi$ is unaffected by the electron-phonon interaction in the Migdal-Eliashberg context. Fullerene compounds however do not fulfill the adiabatic assumption of Migdal's theorem and nonadiabatic effects are expected to be relevant in these materials. In this paper we investigate the Pauli spin susceptibility in nonadiabatic regime by following a conserving approach based on Ward's identity. We find that a sizable renormalization of $\chi$ due to electron-phonon coupling appears when nonadiabatic effects are taken into account. The intrinsic dependence of $\chi$ on the electron-phonon interaction gives rise to a finite and negative isotope effect which could be experimentally detected in fullerides. In addition, we find an enhancement of the spin susceptibility with temperature increasing, in agreement with the temperature dependence of $\chi$ observed in fullerene compounds. The role of electronic correlation is also discussed.Comment: Revtex, 10 pages, 8 figures include

G3-homogeneous gravitational instantons

We provide an exhaustive classification of self-dual four-dimensional gravitational instantons foliated with three-dimensional homogeneous spaces, i.e. homogeneous self-dual metrics on four-dimensional Euclidean spaces admitting a Bianchi simply transitive isometry group. The classification pattern is based on the algebra homomorphisms relating the Bianchi group and the duality group SO(3). New and general solutions are found for Bianchi III.Comment: 24 pages, few correction