Undergraduate Course: Geometry of General Relativity (MATH11138)
Course Outline
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 
SCQF Credits  10 
ECTS Credits  5 
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 two assignments and a closedbook exam. 
Course description 
This course assumes familiarity with the language of differentiable manifolds, but develops the theory of affine connections and enough pseudoriemannian 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.
Syllabus:
 Affine connections: covariant derivative, torsion, curvature, parallel transport, geodesics, geodesic deviation.
 Riemannian geometry: metric tensors, Lorentzian metrics, LeviCivita 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, energymomentum 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.
Prerequisites

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)
Prerequisites 
Students MUST have passed:
Differential Geometry (MATH11235)

Corequisites  
Prohibited Combinations  
Other requirements  A pass in Differentiable Manifolds (MATH10088) in 2021/22 or earlier is an acceptable substitute for a pass in Differentiable Geometry (MATH11235).
Note that PGT students on School of Mathematics MSc programmes are not required to have taken prerequisite courses, but they are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling. 
Information for Visiting Students
Prerequisites  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? 
Yes 
Course Delivery Information

Academic year 2024/25, Available to all students (SV1)

Quota: None 
Course Start 
Semester 2 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)  Geometry of General Relativity (MATH11138)  2:00  
Learning Outcomes
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.

Reading List
Recommended:
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) 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  GGR 
Contacts
Course organiser  Prof José FigueroaO'Farrill
Tel: (0131 6)50 5066
Email: j.m.figueroa@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

