Module: Differential Geometry & General Relativity

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Differential Geometry & General Relativity
Module Code	MTH1004 (ITS: MS536)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	9	Credit Rating	7.5

Description

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.

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Lecture	24	Mix of live and online lectures on theory and problems in Differential Geometry and General Relativity.
Workshop	12	Students will work individually and in groups to solve problems in Differential Geometry and General Relativity.
Laboratory	6	Students will learn how to use computer algebra to complete calculations and solve problems.
Tutorial	6	Academics will provide additional support to students as they work through the module content.
Independent Study	140	No Description
Total Workload: 188

Section Breakdown
CRN	10067	Part of Term	Semester 1
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC1	Best Mark	N
Module Co-ordinator	Peter Taylor	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
Assignment	Take-home project on classical tests of GR.	10%	Week 12
In Class Test	In-class test on differential geometry.	10%	Week 8
Laboratory Portfolio	Lab exam on computer algebra.	5%	Week 6
Formal Examination	Terminal exam.	75%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

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.

Indicative Reading List

Books:

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

Articles:
None

Other Resources

None