Module: Differential Geometry & General Relativity

Module Specifications.

Current Academic Year 2024 - 2025

All Module information is indicative, and this portal is an interim interface pending the full upgrade of Coursebuilder and subsequent integration to the new DCU Student Information System (DCU Key).

As such, this is a point in time view of data which will be refreshed periodically. Some fields/data may not yet be available pending the completion of the full Coursebuilder upgrade and integration project. We will post status updates as they become available. Thank you for your patience and understanding.

Date posted: September 2024

Module Title	Differential Geometry & General Relativity
Module Code	MS536 (ITS) / MTH1004 (Banner)
Faculty	Science & Health	School	Mathematical Sciences
Module Co-ordinator	Brien Nolan
Module Teachers	-
NFQ level	9	Credit Rating	7.5
Pre-requisite	Not Available
Co-requisite	Not Available
Compatibles	Not Available
Incompatibles	Not Available

None

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

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 Assessment	50%	Examination Weight	50%

Type	Description	% of total	Assessment Date
Course Work 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

Reassessment Requirement Type
Resit arrangements are explained by the following categories: Resit category 1: A resit is available for both* components of the module. Resit category 2: No resit is available for a 100% continuous assessment module. Resit category 3: No resit is available for the continuous assessment component where there is a continuous assessment and examination element.
* ‘Both’ is used in the context of the module having a Continuous Assessment/Examination split; where the module is 100% continuous assessment, there will also be a resit of the assessment
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

None