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Module Specifications..

Current Academic Year 2023 - 2024

Please note that this information is subject to change.

Module Title Differential Geometry & General Relativity
Module Code MS536
School School of Mathematical Sciences
Module Co-ordinatorSemester 1: Brien Nolan
Semester 2: Brien Nolan
Autumn: Brien Nolan
Module TeachersBrien Nolan
NFQ level 9 Credit Rating 7.5
Pre-requisite None
Co-requisite None
Compatibles None
Incompatibles None

This module provides students with the mathematical tools required to study General Relativity (GR), and introduces them to the area. The module includes the study of the conceptual foundations of GR and Einstein's equation, the study of the Schwarzschild solution, and the analysis of the classical tests of GR.

Learning Outcomes

1. Complete a range of calculations relevant to the mathematics of curved space-time (tensor algebra, metrics, connections, curvature, geodesics, isometries).
2. Use computer algebra to carry out a range of calculations relevant to the mathematics of curved space-time.
3. Solve a range of problems relating to the mathematics of curved space-time.
4. Apply their knowledge of differential geometry to describe and analyse different phenomena of gravitational fields.
5. Apply their knowledge of differential geometry to solve a range of problems relating to gravitational fields.
6. Analyse the conceptual foundations of General Relativity and discuss how these limit possible theories of gravitation.
7. Compare and contrast General Relativity with other theories of gravitation, both conceptually and in terms of observational evidence.

Workload Full-time hours per semester
Type Hours Description
Lecture24Mix of live and online lectures on theory and problems in Differential Geometry and General Relativity.
Workshop12Students will work individually and in groups to solve problems in Differential Geometry and General Relativity.
Laboratory6Students will learn how to use computer algebra to complete calculations and solve problems.
Tutorial6Academics will provide additional support to students as they work through the module content.
Independent Study140No Description
Total Workload: 188

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Differential Geometry
Tensor algebra. Manifolds, curves and tangent vectors. Connections and covariant derivatives. Metric tensors and the metric connection. Geodesics and geodesic deviation. Curvature. The Riemann, Ricci and Einstein tensors. Lie derivatives, isometries and Killing vectors.

Conceptual foundations of GR
The weak and strong equivalence principles. The energy-momentum tensor. Einstein's field equation. The cosmological constant. Alternative theories of gravity.

The Schwarzschild Solution.
Birkhoff's theorem and derivation of the Schwarzschild solution. Geodesics in the Schwarzschild exterior. Null geodesics and the Kruskal-Szekeres extension. Singularities in Schwarzschild space-time.

Classical test of GR
Gravitaional redshift. Bending of starlight. Precession of the perihelion of Mercury. Observational data and alternative theories.

Assessment Breakdown
Continuous Assessment50% Examination Weight50%
Course Work Breakdown
TypeDescription% of totalAssessment Date
AssignmentTake-home project on classical tests of GR.10%Week 12
In Class TestIn-class test on differential geometry.10%Week 8
Laboratory PortfolioLab exam on computer algebra.5%Week 6
Reassessment Requirement Type
Resit arrangements are explained by the following categories;
1 = A resit is available for all components of the module
2 = No resit is available for 100% continuous assessment module
3 = No resit is available for the continuous assessment component
This module is category 1
Indicative Reading List

  • James B. Hartle: 2002, Gravity: An Introduction to Einstein's General Relativity, Pearson, 0805386629
  • Eric Poisson and Clifford M. Will: 2014, Gravity: Newtonian, Post-Newtonian and Relativistic, Cambridge, 1107032865
Other Resources

Programme or List of Programmes
MSARMSc in Astrophysics and Relativity

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