Kinematic relative velocity with respect to stationary observers in Schwarzschild spacetime

We study the kinematic relative velocity of general test particles with respect to stationary observers (using spherical coordinates) in Schwarzschild spacetime, obtaining that its modulus does not depend on the observer, unlike Fermi, spectroscopic and astrometric relative velocities. We study some fundamental particular cases, generalizing some results given in other work about stationary and radial free-falling test particles. Moreover, we give a new result about test particles with circular geodesic orbits: the modulus of their kinematic relative velocity with respect to any stationary observer depends only on the radius of the circular orbit, and so, it remains constant.Comment: 8 pages, 2 figure

Geometric description of lightlike foliations by an observer in general relativity

We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $\Omega$ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $\Omega$ for this observer. A new distribution $\Omega_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega$ and $\Omega _U^-$ are integrable, the pair \Omega ,\Omega_U^- $represents the wave fronts of a stationary wave for the observer$U$. However, we show in an example that the integrability of$\Omega $does not imply the integrability of$\Omega_U^-$.Comment: 14 pages, 4 figure

Lightlike simultaneity, comoving observers and distances in general relativity

We state a condition for an observer to be comoving with another observer in general relativity, based on the concept of lightlike simultaneity. Taking into account this condition, we study relative velocities, Doppler effect and light aberration. We obtain that comoving observers observe the same light ray with the same frequency and direction, and so gravitational redshift effect is a particular case of Doppler effect. We also define a distance between an observer and the events that it observes, that coincides with the known affine distance. We show that affine distance is a particular case of radar distance in the Minkowski space-time and generalizes the proper radial distance in the Schwarzschild space-time. Finally, we show that affine distance gives us a new concept of distance in Robertson-Walker space-times, according to Hubble law.Comment: 17 pages, 5 figures. Since "lightlike distance" is in fact the known "affine distance", the notation has been change

Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations

We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and we give conditions for such existence and uniqueness in some cases. Finally we illustrate these methods with an example of a collocation problem, and we give some examples of collocation problems that do not fit in the cases studied previously.Comment: 18 pages, 4 figure

Invariant manifolds of the Bonhoeffer-van der Pol oscillator

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.Comment: 14 pages, 15 figure

Intrinsic definitions of "relative velocity" in general relativity

Given two observers, we define the "relative velocity" of one observer with respect to the other in four different ways. All four definitions are given intrinsically, i.e. independently of any coordinate system. Two of them are given in the framework of spacelike simultaneity and, analogously, the other two are given in the framework of observed (lightlike) simultaneity. Properties and physical interpretations are discussed. Finally, we study relations between them in special relativity, and we give some examples in Schwarzschild and Robertson-Walker spacetimes.Comment: 29 pages, 12 figures. New proofs in special relativity and a new open problem in general relativity (see Remark 5.2). An Appendix has been added, studying the relative velocities in Schwarzschild, with new figures. Some spelling erros fixe

A note on the computation of geometrically defined relative velocities

We discuss some aspects about the computation of kinematic, spectroscopic, Fermi and astrometric relative velocities that are geometrically defined in general relativity. Mainly, we state that kinematic and spectroscopic relative velocities only depend on the 4-velocities of the observer and the test particle, unlike Fermi and astrometric relative velocities, that also depend on the acceleration of the observer and the corresponding relative position of the test particle, but only at the event of observation and not around it, as it would be deduced, in principle, from the definition of these velocities. Finally, we propose an open problem in general relativity that consists on finding intrinsic expressions for Fermi and astrometric relative velocities avoiding terms that involve the evolution of the relative position of the test particle. For this purpose, the proofs given in this paper can serve as inspiration.Comment: 8 pages, 2 figure

Aplicació de criteris de sostenibilitat en el disseny de polígons industrials

La finalitat d’aquest projecte és crear una metodologia per facilitar el disseny sostenible de polígons industrials. El model conceptual d’activitat sostenible s’eleva sobre la base de tres pilars fonamentals on tots tenen la mateixa importància: cohesió i progrés social, eficiència econòmica i conservació ambiental. L’estudi que s’ha dut a terme en aquest projecte ha deixat al marge la part de la viabilitat econòmica del triangle de la sostenibilitat centrant-se principalment en els equilibris socials i ambientals derivats de la nova planificació territorial. El mètode utilitzat per a cada una de les parts estudiades ha estat en primer lloc la recerca d’informació i normativa vigent i la posterior definició d’uns criteris de sostenibilitat per tal de ser aplicats. Aquests criteris pretenen ser una eina per ajudar a planificar adequadament un polígon industrial de manera que l’usuari que la utilitzi ho faci d’una forma simple i visual. A més a més, no només serveix per avaluar un cert polígon, sinó que té el valor afegit de donar un punt de vista aproximat dels diferents paràmetres avaluats i permet visualitzar els possibles punts conflictius i fer actuacions per resoldre’ls. A causa de la complexitat de dissenyar una metodologia que donés resposta a la projecció d’un polígon industrial a qualsevol part del món, l’eina creada serveix per donar suport a la fase de disseny i planificació de polígons només a l’Estat Espanyol, i, en particular, a Catalunya. No obstant això, recollint les dades necessàries també és possible aplicar-la en altres parts del planeta