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A challenge in teaching about special relativity is that a number of the theory's effects are at odds with the intuition of classical physics, as well as student's everyday experience. The relativity of simultaneity, time dilation and length contraction are prominent examples. This article describes two additional, less well-known counter-intuitive properties, both of which follow from the relativistic definition of relative motion, in situations with accelerated observers: (a) when two objects have a constant radar distance and are by that standard ``at relative rest,'' their relative speed is not necessarily zero, and (b) for two observers A and B, a situation is possible where A considers the two to be approaching each other, while B considers them to be moving away from each other. In general relativity, the generalisations of these two properties prove helpful for understanding static gravitational fields, and they also provide some insight into the nature of horizons., Comment: 7 pages, 2 figures

The "frozen star" picture of black holes, based on the fact that an outside observer will never see a collapsing sphere shrink sufficiently to form a black hole horizon, was a historical obstacle to the understanding of black holes, and continues to be a stumbling point for students trying to understand horizons. Promoting the discrete steps in Zeno's original story of Achilles and the tortoise to a "Zeno time coordinate" provides a quantitative toy model that allows students to understand why the "frozen star" phenomenon does not mean that objects cannot fall into a black hole. The toy model can be used for teaching about this particular feature of black holes in an introductory setting that does not introduce the Schwarzschild metric and its tortoise coordinates., Comment: Accepted for publication at Eur. J. Phys

In the context of the European Erasmus+ project Teaching ASTronomy at the Educational level (TASTE), we investigated the extent to which a learning module at school and a set of activities during a planetarium visit help students to gain insight in the Apparent Motion of the Sun and Stars. Therefore, we have set up a two treatment study with a pretest posttest design. In the four participating countries (Belgium, Germany, Greece and Italy), secondary school students studied the concept of the celestial globe at school using newly designed learning materials. By using a latent class analysis, we identified different classes of student answers on the AMoSS test. We show how students evolve from one class to another between pretest and posttest. Overall the results of the pretest and posttest show that a good understanding of the different aspects of the apparent motion of celestial bodies is difficult to achieve.

The Shapiro effect, also known as the gravitational time delay, is close kin to the gravitational deflection of light that was the central topic of our Summer School. It is also an interesting test bed for exploring a topic that provides the foundations for most of the calculations we have done in this school, yet is highly complex when treated more rigorously: the question of the Newtonian limit, and of the post-Newtonian corrections that must be applied to include the leading-order effects of general relativity. This contribution discusses simplified derivations for the gravitational redshift and the Shapiro effect, as well as astrophysical situations in which the Shapiro effect can be measured., Comment: 14 pages, 12 figures. Published in: K.-H. Lotze & Silvia Simionato (eds.), Proceedings of the Heraeus Summer School "Astronomy from 4 Perspectives: Thinking Gravitational Lensing for Teaching" (Jena, 2-7 Sep 2019), pp. 42-54

Teaching cosmology at the undergraduate or high school level requires simplifications and analogies, and inevitably brings the teacher into contact with at least one of the pedagogical interpretations of the expanding universe. The by far most popular interpretation holds that galaxies in an expanding universe are stationary, while space itself expands and thus causes the growing distances that characterize cosmic expansion. The alternative relativistic explosion interpretation regards cosmic expansion as a pattern of (relativistic) galaxy motion. The aim of this article is to discuss the two competing interpretations from the perspective of potential student preconceptions, taking into account both beneficial anchoring conceptions and potentially harmful preconceptions that can lead to misconceptions., Comment: 10 pages, 3 figures

The Schwarzschild metric is derived in a manner that does not require familiarity with the formalism of differential geometry beyond the ability to interpret a general spacetime metric. As such, the derivation is suitable for an undergraduate course on general relativity. The derivation uses infalling coordinates that are particularly well adapted to the situation, as well as Einstein's equation in the simple form introduced by Baez and Bunn. That version of the vacuum Einstein equations corresponds to requiring a particular local Newtonian limit: that, to first order, the deformation of a "test ball" of freely falling, initially-at-rest test particles is governed by the tidal forces of Newtonian gravity., Comment: 7 pages, 1 figure

The gravitational time delay of light, also called the Shapiro time delay, is one of the four classical tests of Einstein's theory of general relativity. This article derives the Newtonian version of the Shapiro time delay from Einstein's principle of equivalence and the Newtonian description of gravity, in a manner that is accessible to undergraduate students and advanced high-school students. The derivation can be used as a pedagogical tool, similar to the way that simplified derivations of the gravitational deflection of light are used in teaching about general relativity without making use of the more advanced mathematical concepts. Next, we compare different general-relativistic derivations of the Shapiro time delay from the Schwarzschild metric, which leads to an instructive example for the challenges of formulating the post-Newtonian limit of Einstein's theory. The article also describes simple applications of the time delay formula to observations within our solar system, as well as to binary pulsars., Comment: 14 pages, 9 figures

Cosmic event horizons separate spacetime into disjoint regions: those regions whose light signals can reach us, and more distant regions we cannot, even in principle, observe. For one type of cosmic horizon, associated with universes that keep expanding forever, there is a simple intuitive picture of where the cosmic horizon is located, in terms of the relative speed between a distant galaxy and our own approaching the speed of light: Where the light-speed limit is reached, light signals from that distant galaxy will not be able to catch up with our own galaxy; that galaxy and more distant galaxies are behind the cosmic horizon. Applied to the usual recession speeds of galaxies, that simple picture turns out to be wrong. But there is another relevant concept of speed, derived from the relativistic relative velocity of galaxies, which in turn is defined via the parallel transport of four-velocities. This article shows how, using this concept of relative velocity, key elements of the intuitive picture are valid, and can be used in a simplified explanation for cosmic horizons. When phrased as a statement about relative radial velocities, the notion of light not being able to catch up with our own galaxy is consistent with the calculations, and the cosmic event horizon is indeed the limit where the relative radial velocity of distant Hubble-flow galaxies reaches the speed of light. While the derivation itself requires advanced concepts of general relativity, we also discuss consequences of this result for teaching about cosmic event horizons in less advanced settings, where students are not introduced to the full formalism of general relativity., Comment: 15 pages, 4 figures

This elementary review covers the basics of working with astronomical data, notably with images, spectra and higher-level (catalog) data. The basic concepts and tools are presented using both application software (DS9 and TOPCAT) and Python. The level of presentation is suitable for undergraduate students, but should also be accessible to advanced high school students., Comment: 71 pages, 111 figures; revised version for publication in the Open Journal of Astrophysics

This review presents an overview of various kinds of models -- physical, abstract, mathematical, visual -- that can be used to present the concepts and applications of Einstein's general theory of relativity at the level of undergraduate and even high-school teaching. After a general introduction dealing with various kinds of models and their properties, specific areas of general relativity are addressed: the elastic sheet model and other models for the fundamental geometric properties of gravity, models for black holes including the river model, cosmological models for an expanding universe, and models for gravitational waves as well as for interferometric gravitational wave detectors., Comment: 97 pages, 46 figures. Based on an invited talk at the session of the section Gravitation and Relativity at the Spring Meeting 2017 of the German Physical Society (DPG) in Bremen, 16 March 2017

Noether's theorem is widely regarded as one of the most elegant results in theoretical physics. The article presents two simple examples that can be used to demonstrate the basic idea behind Noether's theorem, by deriving a relation between translational symmetry in the time coordinate and energy conservation in a way that is accessible to beginning students., Comment: 3 pages

We report on a project undertaken in Summer 2017 by pre-university student interns at Haus der Astronomie: point-particle simulations of galaxy collisions with the aim of reproducing observational data from such collisions. We succeeded in providing a visually similar representation of both NGC 5426/7 and the "Antennae" galaxies (NGC 4038/9), and were able to make deductions about the relative positions and orbits of these galaxies. The project is an example for how participants with little more than high-school level previous knowledge can successfully tackle, and understand, advanced topics from current astrophysical research. This report was written by the two participants (M. B.-C. and M. T.), on whose experiences it is based, in collaboration with their supervisor at Haus der Astronomie (M. P.)., Comment: 16 pages, 8 figures. Project presented at the WE Heraeus Summer School "Astronomy From Four Perspectives: The Dark Universe" in Heidelberg on September 1, 2017, and at the German national teacher training in astronomy at the University of Jena on July 3, 2018

This article presents a simple model that reproduces key concepts of modern cosmology within the framework of special relativity, at a level that is suitable for an undergraduate or high school setting. The model includes cosmic expansion governed by a universal scale factor, the Hubble relation, proper distances between galaxies and the associated recession speeds, comoving coordinates, angular and luminosity distances, and the relation between the cosmic scale factor and the cosmological redshift. It allows for a pedagogical discussion of the role of superluminal recession speeds of galaxies, the applicability of the special-relativistic Doppler formula to describe the cosmological redshift, and the validity of energy conservation for the redshifted photons. The model can be viewed as a toy version of the Milne universe., Comment: 20 pages, 4 figures

The Heraeus Summer School series "Astronomy from four perspectives", funded by the WE Heraeus Foundation, draws together teachers and teacher students, astronomers, physicists and astronomy students from Germany and Italy. For each summer school, participants gather at one of the four participating nodes: Heidelberg, Padua, Jena, and Florence. The main goal of the series is to bring astronomy into schools, which is achieved by educating and training the teachers and teacher students. In this e-print, we present the exercises, tutorials, and high-school classroom materials developed during the fifth summer school of the series, which took place at Haus der Astronomie in Heidelberg August 26 -- September 2, 2017. "The tutorials" were prepared beforehand for the participants of the Summer schools, and are suitable for use in teacher training. "Classroom materials" were developed mainly during the summer school itself, and are suitable for high-school level teaching. They include question sheets for pupils, and some pointers on where to use the material in the German high school curriculum. Both sets of materials address the summer school's four main topics: Supernova cosmology, the virial theorem, rotation curves of galaxies, and the temperature of the cosmic microwave background (CMB)., Comment: 39 pages, over 50 exercises, questions and computer simulations as well as solution sheets; Updated affiliations of authors

An introduction to the physics and mathematics of the expanding universe, using no more than high-school level / undergraduate mathematics. Covered are the basics of scale factor expansion, the dynamics of the expanding universe, various distance concepts and the generalized redshift-luminosity relation, among other topics., Comment: 107 pages, 33 figures. Extended notes for a lecture held as part of the WE Heraeus Summer School "Astronomy from Four Perspectives: The Dark Universe" at Haus der Astronomie, Heidelberg, on 28 August 2017

Parallax is the most important astronomical method for determining the distances from the Earth to nearby stars. We present two hands-on activities that explore the concepts and practice of parallax measurement. The first activity is a simple geometrical set-up using simple theodolites, and is accessible for students grade 7 and above. The second activity uses digital photography to recreate astronomical parallax measurements in a classroom setting, and is suitable for students grade 10 and above., Comment: 7 pages, 12 figures. Published in two parts as "Parallax: reaching the stars with geometry" in Science in School 39, pp. 40-44 (2017) and "Finding the scale of space" in Science in School 40, pp. 40-45 (2017)

Gravitational waves and cosmic expansion are both described in terms of Einstein's general relativity. This article explores the similarities between the two phenomena, as well as some differences, using the fundamental concept of the metric of spacetime, expressed via a line element. In both contexts, the focus is on the metric description of space, as opposed to the components of the metric determining the properties of time. This allows for a simplified, more accessible discussion., Comment: 12 pages, 6 figures

Astronomers measure cosmic distances to objects beyond our own galaxy using standard candles: objects of known intrinsic brightness, whose apparent brightnesses in the sky are then taken as an indication of their distances from the observer. In this activity, we use street lights and a digital camera to explore the method of standard candles as well as some of its limitations and possible sources of error., Comment: 7 pages, 7 figures

We study the hidden symmetries arising in the dimensional reduction of d=5, N=2 supergravity to three dimensions. Extending previous partial results for the bosonic part, we give a derivation that includes fermionic terms, shedding light on the appearance of the local hidden symmetry SO(4) in the reduction., Comment: 11 pages LaTeX, uses elsart and amssymb

We show that the Goedel solution of five-dimensional gauged supergravity contains either closed time-like curves through every space-time point or none at all, dependent on the rotational parameter. In addition, we present a deformation of that solution with a parameter kappa which characterizes the symmetry of four-dimensional base space: for kappa = 1, 0, -1 it has spherical, flat and hyperbolic symmetry, respectively. Also investigated are the causal properties of the lifted solution in 10 dimensions., Comment: 10 pages, LaTeX (v2: added new references)