Module MATH4051: GENERAL RELATIVITY IV

Department: MATHEMATICAL SCIENCES

MATH4051: GENERAL RELATIVITY IV

Prerequisites

• Linear Algebra II (MATH2021) [2003/2004 and before: Linear Analysis and Applications II], Analysis in Many Variables II (MATH2031) and Mathematical Physics II (MATH2071); alternatively Linear Algebra II (MATH2021) [2003/2004 and before: Linear Analysis and Applications II], Analysis in Many Variables II (MATH2031) and Foundations of Physics II (PHYS2511).

Corequisites

• None.

Excluded Combination of Modules

• General Relativity III (MATH3201).

Aims

• To appreciate General Relativity, one of the fundamental physical theories.
• To develop and exercise mathematical methods.

Content

• (1) Differences between general and special relativity.
• Experiments showing up the differences.
• Gravity becomes geometry.
• (2) Differential manifold as model of spacetime.
• Coordinates and relations between different systems.
• Curves in spacetime.
• (3) Covariant derivative.
• Partial derivatives are inadequate.
• Properties (linearity, derivative of tensor product, commutation with contraction).
• Geodesic curves.
• Metric connection.
• (4) Distance relations, shape, units, light cones, locally inertial coordinate systems.
• Variational principles for geodesics.
• (5) Curvature tensor.
• Symmetries of curvature tensor.
• Einstein tensor.
• Geodesic deviation.
• (6) Newtonian gravity and Einstein's theory.
• Linear form of Einstein's theory.
• (7) Schwarzschild solution, blackholes with Kruskal-Szekeres coords.
• (8) Cosmology.
• Reading material on a topic related to: Global structures of spacetime; causal properties and topology .

Learning Outcomes

• By the end of the module students will: be able to solve complex, unpredictable and specialised problems in General Relativity.
• have an understanding of specialised and complex theoretical mathematics in the field of General Relativity.
• have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Special relativity.
• Differential manifolds.
• Metric, covariant derivative, curvature.
• General relativity.
• Black holes.
• Cosmology.
• Global structures of spacetime
• Knowledge and understanding of a topic in the following areas: global structures of spacetime, causal properties and topology.

• Students will have highly specialised and advanced mathematical skills which will be used with minimal guidance in the following areas: Spatial awareness, Modelling.
• Students will have the ability to read independently to acquire knowledge and understanding in the area of spacetime structure.

• Students will have enhanced problem solving skills.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University