General description

General Relativity describes gravity in geometric terms. Rather than thinking of gravity as a force, one is led to describe space and time together as a spacetime equipped with a metric, and to understand motion through the geometry of that spacetime. This viewpoint gives rise to many of the most striking predictions of modern physics, including black holes.

This project is intended as a physics-oriented introduction to some of the central ideas of curved spacetime. The emphasis will be on learning how to work with metrics, tensors, geodesics, and curvature in a concrete way, while keeping close contact with physical interpretation. Black holes will appear as a particularly important example through which these ideas can be explored.

The project duo is designed so that the group project develops a shared background in spacetime geometry and motion, while the individual project allows students to investigate a more advanced aspect of curved spacetime or black holes in greater depth.

Group project

The group project will follow a structured programme determined by the supervisor. It is designed for Year 3 students who may not have previously studied differential geometry or general relativity. The aim is to introduce the formal tools gradually, always in a way that is motivated by the physics of motion in spacetime.

Goals and expectations

By the end of the group project, students will have learned:

• how spacetime can be described using geometric objects like vectors, tensors, and metrics,
• how proper time, intervals, and causal structure arise from the metric,
• how free motion in spacetime is described by geodesics,

By the end of the group project, students will have practised:

• reading and discussing mathematical physics material collaboratively,
• carrying out coordinate-based calculations with metrics and geodesics,
• presenting technical material clearly to their peers as a group.

Indicative content

• Minkowski spacetime, spacetime intervals and worldlines.
• Vectors, tensors, and the role of coordinates.
• Metrics in flat and curved spacetimes.
• Geodesics and proper time as a description of free motion.
• Covariant derivatives and Christoffel symbols in concrete examples.
• Schwarzschild (Black Hole) spacetime as an example of the formalism.
• Radial motion, selected orbits, and the horizon at an introductory level.

Mode of Operation and Evidence of Learning

The group project will run through guided reading, regular meetings with the supervisor, structured discussion outside meetings, and a sequence of exercises designed to consolidate understanding. These exercises may include short derivations, coordinate calculations, conceptual questions, and explanatory tasks. The students will maintain a weekly group diary and will use their meetings to discuss material, ask questions, and work collaboratively on selected problems.

Evidence of learning will come from engagement in the group process, progress recorded in the diary, participation in the supervisor's meetings and group discussions, the group presentation, and the oral examination.

Individual project

In the individual project, students will investigate in more depth one aspect of curved spacetime, differential geometry, or black holes that builds on the background developed in the group project. The direction of study will be chosen with guidance from the supervisor.

Possible advanced topics

• Geodesics in Schwarzschild spacetime in greater detail.
• Effective potentials and circular orbits.
• Curvature tensors in selected examples.
• Gravitational redshift and observational effects.
• Black hole causal structure.

Mode of Operation and Evidence of Learning

The individual project will involve guided independent reading and problem solving, together with regular discussions with the supervisor. Students will be expected to develop ownership of a specific topic, identify and use relevant resources, and demonstrate understanding through the individual project work in the format specified for the module.

Prerequisites

• MATH2741 Methods of Mathematical Physics II

Co-requisites

• No formal co-requisite is required, though some prior exposure to special relativity would be helpful.

Resources

• S. Carroll, Spacetime and Geometry (selected introductory sections on flat spacetime, tensors, curvature, and Schwarzschild geometry).
• B. Schutz, A First Course in General Relativity (selected introductory sections).
• S. Carroll, Lecture Notes on General Relativity, for additional worked examples.
• R. Wald, General Relativity (for advanced individual-project directions).
• P. Townsend, Black Holes lecture notes (mainly for later stages of the project).

Associated research area

Mathematical/Theoretical Physics