## MATH4GR Introduction to General Relativity

First Semester |

This module is an introduction to General Relativity, the theory of gravity by Albert Einstein. Building on the module Differential Geometry (MATH4DG), which takes place in the first half of semester 1, we develop Einstein’s idea that “space” and “time” form a continuum “spacetime” described by a 4-dimensional Lorentzian manifold. The module begins with Special Relativity, which applies to physical systems for which gravity is negligible in comparison to other forces. The spacetime of interest here is the *flat* (non-curved) Minkowski space. The idea is now to describe a *general* physical system in terms of general *curved* spacetimes. The curvature of the Lorentzian manifolds, which represent the spacetimes, is then interpreted as gravity.

Einstein found an analogue to the Poisson equation, which in Newtonian gravity describes the relation between the mass density of the matter fields and the gravitational field: Einstein’s field equations — the equations which govern the gravitational dynamics. In this module, we discuss the mathematical and physical ideas above in detail. Further topics are important classes of solutions of Einstein’s field equations, for example the Schwarzschild solution.

At Otago, we have a lively research group in this area, which includes Jörg Frauendiener, Jörg Hennig and Florian Beyer. Students should also consider the module MATH4AR Advanced Topics in General Relativity, which takes place in semester 2.

### Prerequisites

MATH4DG Differential Geometry, MATH306 is recommended

### Textbook

Hughston, Tod: An Introduction to General Relativity, (London Mathematical Society Student Texts),

Two copies of the book are on close reserve in the Science Library

### Lecturers

Jörg Frauendiener (phone 479-7770, email: joergf@maths.otago.ac.nz)

### Assessment

3 written assignments and a 30min oral exam

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = 0.5A + 0.5E**

where:

- E is the Exam mark
- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.