# General Relativity

**In brief: **Course material and students activities on General Relativity and Cosmology, accessible with conceptual and mathematical means at secondary level II/high school (Alice Gasparini, Andreas Müller)

The University of Geneva proposes a course and several series of learning activities (in French) introducing high school pupils to General Relativity and Modern Cosmology. Subjects like cosmological expansion, cosmological equations, dark matter and dark energy, black holes, and gravitational waves are approached with a medium level of difficulty, i.e. based on a high school level in mathematics and physics.

**Description**

On the occasion of the centenary of Einstein’s General Theory of Relativity (GR), a educational project was launched, with the aim to introduce secondary school pupils to modern cosmology and GR. This project resulted in a course (in French) and a series of activities characterized by an intermediate level of difficulty: striking a balance between a “zero-equations” level suitable for popularizing 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 at 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 designed for an annual special physics option course (2h/week) and can be used as a «toolbox» of occasional activities to be inserted into a standard course of physics or mathematics. It 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 an equivalent two 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 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 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 comes into 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.

**Contents (in French only at this time):**

__Theory:__

"Cosmologie et Relativité Générale: une première approche", PPUR (2018)

__Exercises: __

- Série 1 : Grandeurs: enoncé (pdf)
- Série 2 : Expansion: enoncé (pdf)
- Série 3 : Principe d’équivalence: enoncé (pdf)
- Série 4 : Courbure: enoncé (pdf)
- Série 5 : Lentilles gravitationnelles: enoncé (pdf)
- Série 6 : Trous noirs: enoncé (pdf)
- Série 7 : Equations cosmologiques: enoncé (pdf)
- Série 8 : Chronologie du Big Bang: enoncé (pdf)
- Série 9 : Ondes gravitationnelles: enoncé (pdf)

__Activities:__

- Activité 1: Loi de Hubble enoncé (pdf)
- Activité 2: Courbure enoncé (pdf)

__Related documents: __

Tous les énoncés des activités expérimentales (1 et 2)

Tous les énoncés des séries d'exercices (1-9)

The corrections for each series of exercises and for activities are available for teachers only, contacting Alice Gasparini.

__Posters:__

__Publications: __

- “La cosmologie et la relativité générale au secondaire II au service de la motivation des élèves pour la physique”, M. A. Gasparini, A. Müller et L. Weiss, Revue de Mathématiques pour l’école n. 230 (2018).

- “La cosmologie et la relativité générale au gymnase : Exemple d’une séquence sur l’effet de lentille gravitationnelle”, M. A. Gasparini, Société Suisse des Professeurs de Physique et Mathématiques (SSPMP), Bulletin n. 137 (2018).

- “La cosmologie et la relativité générale par les mathématiques et la physique du lycée”, M. A. Gasparini, Communications de la Société Suisse de Physique n. 55 (2018).

- “La relativité générale et ses applications en cosmologie : un cours pour le secondaire”, M. A. Gasparini, Expériment@l-Tremplin (2018).