Sector Models - A Toolkit for Teaching General Relativity. Part 1: Curved Spaces and Spacetimes

Teaching the general theory of relativity to high school or undergraduate students must be based on an approach that is conceptual rather than mathematical. In this paper we present such an approach that requires no more than elementary mathematics. The central idea of this introduction to general relativity is the use of so-called sector models. Sector models describe curved spaces the Regge calculus way by subdivision into blocks with euclidean geometry. This procedure is similar to the approximation of a curved surface by flat triangles. We outline a workshop for high school and undergraduate students that introduces the notion of curved space by means of sector models of black holes. We further describe the extension to sector models of curved spacetimes. The spacetime models are suitable for learners with a basic knowledge of special relativity. For online teaching materials, see http://www.spacetimetravel.org . ----- F\"ur die Vermittlung der Allgemeinen Relativit\"atstheorie in der Schule, im Grund- oder im Nebenfachstudium besteht das Anliegen, eine fachlich befriedigende Darstellung zu geben, die nicht mehr als Schulmathematik voraussetzt. Wir stellen in diesem Beitrag einen solchen Zugang vor. Das zentrale Werkzeug unserer Einf\"uhrung sind sogenannte Sektormodelle, die gekr\"ummte R\"aume im Sinne des Regge-Kalk\"uls durch eine Zerlegung in kleine, ungekr\"ummte Sektoren beschreiben, \"ahnlich der Triangulierung einer gekr\"ummten Fl\"ache. Wir schildern einen Workshop f\"ur Sch\"uler/innen und Studierende, in dem gekr\"ummte R\"aume anhand von Sektormodellen Schwarzer L\"ocher eingef\"uhrt werden. Wir beschreiben ferner die Erweiterung auf Sektormodelle gekr\"ummter Raumzeiten. Raumzeitliche Sektormodelle setzen Grundkenntnisse der Speziellen Relativit\"atstheorie voraus. Online-Materialien unter http://www.tempolimit-lichtgeschwindigkeit.de .Comment: bilingual: English version 18 pages, 9 figures, for associated movies see http://www.spacetimetravel.org/sectormodels1/sectormodels1.html. German version 19 pages, 9 figures, for associated movies see http://www.tempolimit-lichtgeschwindigkeit.de/sectormodels1/sectormodels1.htm

Mapping vesicle shapes into the phase diagram: A comparison of experiment and theory

Phase-contrast microscopy is used to monitor the shapes of micron-scale fluid-phase phospholipid-bilayer vesicles in aqueous solution. At fixed temperature, each vesicle undergoes thermal shape fluctuations. We are able experimentally to characterize the thermal shape ensemble by digitizing the vesicle outline in real time and storing the time-sequence of images. Analysis of this ensemble using the area-difference-elasticity (ADE) model of vesicle shapes allows us to associate (map) each time-sequence to a point in the zero-temperature (shape) phase diagram. Changing the laboratory temperature modifies the control parameters (area, volume, etc.) of each vesicle, so it sweeps out a trajectory across the theoretical phase diagram. It is a nontrivial test of the ADE model to check that these trajectories remain confined to regions of the phase diagram where the corresponding shapes are locally stable. In particular, we study the thermal trajectories of three prolate vesicles which, upon heating, experienced a mechanical instability leading to budding. We verify that the position of the observed instability and the geometry of the budded shape are in reasonable accord with the theoretical predictions. The inability of previous experiments to detect the ``hidden'' control parameters (relaxed area difference and spontaneous curvature) make this the first direct quantitative confrontation between vesicle-shape theory and experiment.Comment: submitted to PRE, LaTeX, 26 pages, 11 ps-fi

Unbiased flux calibration methods for spectral-line radio observations

Position and frequency switching techniques used for the removal of the bandpass dependence of radio astronomical spectra are presented and discussed in detail. Both methods are widely used, although the frequency dependence of the system temperature and/or noise diode is often neglected. This leads to systematic errors in the calibration that potentially have a significant impact on scientific results, especially when using large-bandwidth receivers or performing statistical analyses. We present methods to derive an unbiased calibration using a noise diode, which is part of many heterodyne receivers. We compare the proposed methods and describe the advantages and bottlenecks of the various approaches. Monte Carlo simulations are used to qualitatively investigate both systematics and the error distribution of the reconstructed flux estimates about the correct flux values for the new methods but also the 'classical' case. Finally, the determination of the frequency-dependent noise temperature of the calibration diode using hot-cold measurements or observations of well-known continuum sources is also briefly discussed.Comment: 25 pages, 30 figures. Accepted for publication in A&

Analyzing X-Ray Pulsar Profiles: Geometry and Beam Pattern of EXO 2030+375

The pulse profiles of the transient Be/X-ray binary EXO 2030+375 show strong dependence on energy, as well as on its luminosity state, and are asymmetric in shape. We want to identify the emission components of the two magnetic poles in the pulsed emission to understand the geometry of the neutron star and its beam pattern. We utilize a pulse-profile decomposition method that enables us to find two symmetric pulse profiles from the magnetic poles of the neutron star. The symmetry characteristics of these single-pole pulse profiles give information about the position of the magnetic poles of the neutron star relative to its rotation axis. We find a possible geometry for the neutron star in EXO 2030+375 through the decomposition of the pulse profiles, which suggests that one pole gets closer to the line of sight than the other and that, during the revolution of the neutron star, both poles disappear behind the horizon for a short period of time. A considerable fraction of the emission arises from a halo while the pole is facing the observer and from the accretion stream of the other pole while it is behind the neutron star, but the gravitational line bending makes the emission visible to us.Comment: 8 pages, 9 figures, accepted for publication in A&

Gravity-Induced Shape Transformations of Vesicles

We theoretically study the behavior of vesicles filled with a liquid of higher density than the surrounding medium, a technique frequently used in experiments. In the presence of gravity, these vesicles sink to the bottom of the container, and eventually adhere even on non - attractive substrates. The strong size-dependence of the gravitational energy makes large parts of the phase diagram accessible to experiments even for small density differences. For relatively large volume, non-axisymmetric bound shapes are explicitly calculated and shown to be stable. Osmotic deflation of such a vesicle leads back to axisymmetric shapes, and, finally, to a collapsed state of the vesicle.Comment: 11 pages, RevTeX, 3 Postscript figures uuencode

Giant vesicles at the prolate-oblate transition: A macroscopic bistable system

Giant phospholipid vesicles are shown to exhibit thermally activated transitions between a prolate and an oblate shape on a time scale of several seconds. From the fluctuating contour of such a vesicle we extract ellipticity as an effective reaction coordinate whose temporal probability distribution is bimodal. We then reconstruct the effective potential from which we derive an activation energy of the order of $k_BT$ in agreement with theoretical calculations. The dynamics of this transition is well described within a Kramers model of overdamped diffusion in a bistable potential. Thus, this system can serve as a model for macroscopic bistability.Comment: 10 pages, LaTeX, epsfig, 4 eps figures included, to appear in Europhys. Let