Distances in plane membranes

A flat membrane with given shape is displayed; two points in the membrane are randomly selected; the probability that the separation between the points have a specified value is sought. A simple method to evaluate the probability density is developed, and is easily extended to spaces with more dimensions.Comment: 8 pages, 6 figures, version 2 with corrected commands of figure

Cosmic crystallography: the hyperbolic isometries

All orientation preserving isometries of the hyperbolic three-space are studied, and the probability density of conjugate pair separations for each isometry is presented. The study is relevant for the cosmic crystallography, and is the theoretical counterpart of the mean histograms arising from computer simulations of the isometries.Comment: 18 pages, 16 figures include

Separations inside a cube

Two points are randomly selected inside a three-dimensional euclidian cube. The value l of their separation lies somewhere between zero and the length of a diagonal of the cube. The probability density P(l) of the separation is obtained analytically. Also a Monte Carlo computer simulation is performed, showing good agreement with the formulas obtained.Comment: 7 pages, 5 figure

Cosmic crystallography: the euclidean isometries

Exact expressions for probability densities of conjugate pair separation in euclidean isometries are obtained, for the cosmic crystallography.These are the theoretical counterparts of the mean histograms arising from computer simulation of the isometries. For completeness, also the isometries with fixed points are examined, as well as the orientation reversing isometries.Comment: 14 pages, Latex, 22 postscript figure

Cosmic crystallography: three multi-purpose functions

A solid sphere is considered, with a uniformly distributed infinity of points. Two points being pseudorandomly chosen, the analytical probability density that their separation have a given value is computed, for three types of the underlying geometry: $E^3, H^3$ and $S^3$. Figures, graphs and histograms to complement this short note are given.Comment: 6 pages, LaTe

Tempa voja^go kaj geodezioj en ^generala relativeco / Time travel and geodesics in general relativity

In the homogeneous metric of Som-Raychaudhuri, in general relativity, we study the three types of geodesics: timelike, null, and spacelike; in particular, the little known geodesics of simultaneities. We also study the non-geodetic circular motion with constant velocity, particularly closed timelike curves, and time travel of a voyager. ------------------- ^Ce la ^Generala Relativeco, en homogena metriko de Som-Raychaudhuri, ni studas geodeziojn de la tri tipoj: tempa, nula, kaj spaca, speciale la malmulte konatajn samtempajn geodeziojn. Ni anka^u studas ne-geodezian cirklan movadon kun konstanta rapido, speciale fermitajn kurbojn de tempa tipo, kaj movadon de voja^ganto al estinto.Comment: 16 pages, 6 figures, two columns Esperanto/Englis

Schwarz-Christoffel: piliero en rivero (a pillar on a river)

La transformoj de Schwarz-Christoffel mapas, konforme, la superan kompleksan duon-ebenon al regiono limigita per rektaj segmentoj. Cxi tie ni priskribas kiel konvene kunigi mapon de la suba duon-ebeno al mapo de la supera duon-ebeno. Ni emfazas la bezonon de klara difino de angulo de kompleksa nombro, por tiu kunigo. Ni diskutas kelkajn ekzemplojn kaj donas interesan aplikon pri movado de fluido. ------- Schwarz-Christoffel transformations map, conformally, the complex upper half plane into a region bounded by right segments. Here we describe how to couple conveniently a map of the lower half plane to the map of the upper half plane. We emphasize the need of a clear definition of angle of a complex, to that coupling. We discuss some examples and give an interesting application for motion of fluid.Comment: 19 pages, 18 figures, two columns, Esperanto/Englis

La relativeca tempo - I / The relativistic time - I

The relativistic time is different from the Newtonian one. We revisit some of these differences in Doppler effect, twin paradox, rotation, rigid rod, and constant proper acceleration. ------- La relativeca tempo estas malsama ol la Newtona. Ni revidos iujn el tiuj malsamoj en Dopplera efiko, gxemel-paradokso, rotacio, rigida stango, kaj konstanta propra akcelo.Comment: 19 pages, 2 columns Esperanto/English, eq.(5) corrected, ref.[15] adde

Geodesics of simultaneity in Schwarzschild

Geodesic of simultaneity is a spacelike geodesic in which every pair of neighbour events are simultaneous (g_{0\mu}\dd x^\mu=0). These geodesics are studied in the exterior region of \Sch's metric.Comment: 7 pages including 4 figures, Portuguese/Esperanto version at http://www.biblioteca.cbpf.br/pub/apub/nf/2009/nf02309.pd

Riemannian Space-times of G\"odel Type in Five Dimensions

The five-dimensional (5D) Riemannian G\"odel-type manifolds are examined in light of the equivalence problem techniques, as formulated by Cartan. The necessary and sufficient conditions for local homogeneity of these 5D manifolds are derived. The local equivalence of these homogeneous Riemannian manifolds is studied. It is found that they are characterized by two essential parameters $m^2$ and $\omega$: identical pairs $(m^2, \omega)$ correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian G\"odel-type metrics are exhibited. A classification of these manifolds based on the essential parameters is presented, and the Killing vector fields as well as the corresponding Lie algebra of each class are determined. It is shown that apart from the $(m^2= 4 \omega^2, \omega\not=0)$ and $(m^2\not=0, \omega = 0)$ classes the homogeneous Riemannian G\"odel-type manifolds admit a seven-parameter maximal group of isometry ($G_7$). The special class $(m^2= 4 \omega^2, \omega\not=0)$ and the degenerated G\"odel-type class $(m^2\not=0, \omega=0)$ are shown to have a $G_9$ as maximal group of motion. The breakdown of causality in these classes of homogeneous G\"odel-type manifolds are also examined.Comment: 26 pages. LaTeX file. To appear in J. Math. Phys. (1998