# University of Geneva, Switzerland

## Teaching Relativity at Secondary Level II/High School

Activities and Course Material developed at the University of Geneva

### In brief:

Course material and students activities (accessible with conceptual and mathematical means at secondary level II/high school):

http://www.nccr-swissmap.ch/education/highschool/GRcourse

A textbook was published in 2018

https://www.ppur.org/produit/876/9782889152094/Cosmologie%20%20relativite%20generale%20

(Documents in French only at this time)

### Description

On the occasion of the centenary of Einstein’s General Theory of Relativity (GR), the National Centre of Competence in Research SwissMAP (The Mathematics of Physics) launched a pedagogical project aiming to introduce secondary school pupils to modern cosmology and GR. This project resulted in a course (in French) and a series of activities characterised by an intermediate level of difficulty: striking a balance between a “zero-equations” level suitable for popularising among the general public on the one hand, and tensor geometry exclusive to the academic experts on the other. The material is exclusively based on high-school level mathematics and physics; hence, these activities are *not* supposed to *replace* the basic tools taught in school, but to *complement* them.

The main purposes of this project are

- To consolidate and enlarge pupils’ existing knowledge in physics and mathematics while studying subjects they find fascinating;
- To improve links between high school and real research: students who learn physics up to the 19th century are left with distorted ideas about the main issues in modern physics. A more accurate awareness of what constitutes contemporary physics research (and hence what options are available to pupils after school) can have a strong impact on their choices for further study. This can counterbalance the general loss of interest toward hard sciences.

The course is divided into nine main chapters from an introduction to astrophysics through to gravitational waves, covering subjects such as gravitational lensing, black holes, cosmological ditances and the thermal history of the universe. Seven annexes complete the course integrating and/or broadening the complementary notions that pupils may need for a complete understanding of the main chapters. Each chapter is related to a series of exercises and/or activities including model answers (reserved for teachers).The level of difficulty increases with the chapters, allowing a gradual immersion into the subjects:

- The activities of the first chapters do not require any knowledge specific to high school physics or mathematics. Rather, they train some basic concepts from Secondary I including unit conversions, orders of magnitude and proportionality;
- Each chapter includes exercises and/or activities based on mechanics, waves, electricity, magnetism and thermodynamics of the general high school physics curriculum ;
- Some exercises in chapters 7-9 require mathematical tools generally acquired in the final years of high school (such as derivatives, integrals or function analysis);
- A few exercises in the final chapters rely on basic knowledge of programming language (such as Python or Mathematica).

Moreover, several common themes can be followed through the chapters of the course, using an overarching subject that can be developed while increasing the level of difficulty. The order and selection of the activities concerning a specific theme are flexible and can be adapted to teaching constraints and the pupils’ level.

For example, the thematic path “gravitational lensing” starts with chapters 3 and 4, and introduces pupils to the principle of equivalence and to the notion of curvature for a two-dimensional surface. Then, the formula of the deflection angle of a ray of light passing near a spherical mass can be explained in different ways: either by using a simple dimensional analysis (a basic notion in mathematics), or by using the Newtonian approach, which requires knowledge of vector formalism, as well as derivation and integration of functions. The analogy with the convergent lenses in optics and using the laws of refraction can easily explain the observed images (Einstein rings, crosses or gravitational arcs) and, thanks to some simple trigonometry (sine rule), pupils can find the formula for the Einstein radius and learn how this formula is used nowadays to estimate the mass of a gravitational lens, including dark matter. Indeed, treating the historical aspect of the question of the deflection angle and of gravitational lensing can be instructive for pupils, showing the interdisciplinarity of the scientific processes.

The thematic path on the parallels between electromagnetic and gravitational interactions starts in the first chapter, treating the order of magnitudes that at play. This parallel is then covered again in the chapter on the principle of equivalence and once more in chapter 8, on the Big Bang and the thermal history of the universe (by comparing the gravitational and the electromagnetic interactions with the strong nuclear interaction). A further development of this theme is achieved in the final chapter, dealing with the nature of gravitational waves.

A last example of theme path is “the expansion of the universe” that initially deals with observational discoveries related to the cosmic microwave background and the acceleration of the expansion of the universe. Moving forward through the chapters, mathematical modelling allows a description of universal dynamics, cosmological distances and theoretical implications of the most recent discoveries, opening a window on the current research in fundamental physics.

### Contact

Alice Gasparini, Alice.Gasparini@unige.ch

Andreas Müller, Andreas.Mueller@unige.ch

University of Geneva

Faculty of Science/Physics Section

Institut Universitaire de Formation des Enseignants (IUFE)

postal adress:

Pavillon d'Uni Mail (IUFE) // Boulevard du Pont d'Arve 40 // CH-1211 Geneva