Undergraduate Course: Geometry of General Relativity (MATH11138)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Year 5 Undergraduate)
||Availability||Available to all students
|Summary||Einstein's theory of General Relativity is a geometric theory of gravitation. This course is a modern introduction to this cornerstone of mathematical physics, formulated in the language of differential geometry.
There are two lectures a week and a workshop every two weeks. There are biweekly assignments and a closed-book exam.
This course assumes familiarity with the language of differentiable manifolds, but develops the theory of affine connections and enough pseudo-riemannian geometry (metric tensor, curvature) in order to describe the theory of General Relativity. This is done via the postulates of General Relativity and the Einstein field equations. The course then explores solutions of the Einstein field equations,including the famous Schwarzschild black hole and the cosmological solutions, which introduces the geometric notions of homogeneity andisotropy.
- Affine connections: covariant derivative, torsion, curvature, parallel transport, geodesics, geodesic deviation.
- Riemannian geometry: metric tensors, Lorentzian metrics, Levi-Civita connection, curvature tensors, moving frames, Cartan structure equations, isometries, Killing vector fields.
- General Relativity: special relativity and Minkowski spacetime, Maxwell's equations, postulates of General Relativity, spacetime, general covariance, energy-momentum tensor, Einstein equations.
- Causal structure and Penrose diagram for Minkowski spacetime.
- Schwarzschild solution: static and spherically symmetric spacetimes, black hole, Kruskal extension, causal structure and Penrose diagram.
- Cosmological models: homogeneity and isotropy, the Friedmann¿Lemaître¿Robertson¿Walker metric.
Students MUST have passed: Differentiable Geometry (MATH11235) . A pass in Differentiable Manifolds (MATH10088) in 2021/22 or earlier is an acceptable substitute for a pass in Differentiable Geometry (MATH11235).
Entry Requirements (not applicable to Visiting Students)
|| Students MUST have passed:
Differential Geometry (MATH11235)
||Other requirements|| A pass in Differentiable Manifolds (MATH10088) in 2021/22 or earlier is an acceptable substitute for a pass in Differentiable Geometry (MATH11235).
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling
|High Demand Course?
Course Delivery Information
|Academic year 2023/24, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Geometry of General Relativity (MATH11138) ||2:00|
On completion of this course, the student will be able to:
- Calculate efficiently with tensor fields in smooth (pseudoriemannian) manifold.
- Calculate the Riemann curvature tensor of spacetime via the Christoffel symbols or via Cartan's moving frames.
- Calculate the Ricci and Einstein tensors and find solutions of the Einstein field equations.
- Find Killing vectors for simple metrics and use them in the solutions of geodesic equation.
- Calculate the Ricci and Kretschmann scalars and infer something about the nature of singularities.
An Introduction to General Relativity, L.P Hughston and K.P. Tod (LMS, CUP, 1990)
General Relativity, R. M. Wald, University of Chicago Press (1984)
|Graduate Attributes and Skills
|Course organiser||Prof José Figueroa-O'Farrill
Tel: (0131 6)50 5066
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427