## MATH4AR Advanced Topics in General Relativity

Second Semester |

General relativity (GR), Albert Einstein’s theory of gravitation, is one of the most elegant theories of mathematical physics. It gives a geometric description of gravitation in terms of the curvature of space and time, using mathematical methods like tensor algebra, differential geometry and the theory of ordinary and partial differential equations. GR is also one of the best verified theories of modern physics. In particular, it is regarded as the most satisfactory model of the large-scale universe that we have.

This course builds upon the previous modules Introduction to General Relativity (MATH4GR) and Differential Geometry (MATH4DG), where the mathematical prerequisites have been supplied. It consists of a selection of fundamental topics in GR, including discussions of black holes and of simple models for neutron stars, which are among the most extreme objects in our universe, and cosmology. The introduction to cosmological models will focus on the so-called Friedmann models, describing a homogeneous and isotropic universe. These models are mathematically relatively simple, but already indicate that our universe could have arisen from a big bang.

### Prerequisites

MATH4DG Differential Geometry, MATH4GR Introduction to General Relativity

### Main topics

- Black holes
- Spherically symmetric stars
- Cosmological models
- Gravitational waves (optional reading)

### Lectures

TBA

Lecture notes will be available on the resources page.

### Tutorials

There will be three tutorials, times TBA.

### Lecturer

Dr Jörg Hennig, room 215, email: jhennig@maths.otago.ac.nz

Office hours: by arrangement (or just pop in if I am in my office)

### Assessment

3 written assignments and a final exam

### Final mark

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

**F = 0.6E + 0.4A**

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.

### Albert Einstein

Where would modern physics be without the genius of**Albert Einstein**(1879-1955)? In 1905, he published four brilliant articles which contributed substantially to the foundation of modern physics and changed our views on space, time and matter. His general theory of relativity, developed between 1907 and 1915, is still regarded as the most satisfactory model of the large-scale universe that we have.

### Alexander Friedmann

**Alexander Friedmann**was a Russian physicist and mathematician. In 1922, he discovered cosmological models as solutions to Einstein's field equations which describe an expanding universe. Only 37 years old, Friedmann died in 1925 from typhoid fever — four years before the astronomer Edwin Hubble measured the redshifts of distant galaxies and showed that the universe is indeed expanding.

### Gravitational waves

**Gravitational waves**are waves of spacetime curvature (little ripples propagating through spacetime). They are the result of the asymmetric acceleration of mass that occurs during massive astronomical events, such as coalescing compact binary systems and supernovae, and they were predicted by Einstein himself in 1916. Since the amplitude of the waves is extremely small, they are very hard to detect. But finally, almost 100 years after Einstein's prediction, the LIGO detectors (Laser Interferometer Gravitational-Wave Observatory) observed the first gravitational wave in 2015. The signal was produced by two black holes that merged into a single black hole about 1.3 billion years ago. The above picture shows LISA, the Laser Interferometer Space Antenna, which is a planned space mission to accurately measure gravitational waves.