Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

MATH4AR Advanced Topics in General Relativity

Second Semester
10 points

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.

Further topics are cosmological models and gravitational waves.

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 indicate already that our universe could have arisen from a big bang. Gravitational waves are little ripples propagating through spacetime, which can result from the asymmetric acceleration of mass during massive astronomical events. They are a major topic of current research and their (still outstanding) direct observation would provide a “new window to the universe”.

Prerequisites

MATH4DG Differential Geometry, MATH4GR Introduction to General Relativity

Main topics

  • Black holes
  • Spherically symmetric stars
  • Cosmological models
  • Gravitational waves

Lectures

This module is a reading course in 2017 with weekly meetings

Tuesdays, 10-11 am, room 240 in weeks 1-7 of lectures.

Lecture notes will be available on the resources page.

Lecturer

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

Assessment

3 written assignments, 1 presentation

Final mark

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

F = 0.5A + 0.5P

where:

  • A is the Assignments mark
  • P is the Presentation mark

and all quantities are expressed as percentages.

Students must abide by the University’s Academic Integrity Policy

Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.

The two most common forms of academic misconduct are plagiarism and unauthorised collaboration.

Academic misconduct: Plagiarism

Plagiarism is defined as:

  • Copying or paraphrasing another person’s work and presenting it as your own.
  • Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
  • Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
  • Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

Academic misconduct: 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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.
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.
..., 1888-1925, 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.
Image credit: NASA
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 American LIGO detectors (Laser Interferometer Gravitational-Wave Observatory) have measured the first gravitational wave signal in September 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 eLISA, the evolved Laser Interferometer Space Antenna, which is a planned space mission to accurately measure gravitational waves.