# MMath&Phys Mathematics and Physics

Year of entry: 2024

## Course unit details:Gravitation

Unit code PHYS40771 10 Level 4 Semester 1 No

Gravitation (M)

### Pre/co-requisites

Unit title Unit code Requirement type Description
Lagrangian Dynamics PHYS20401 Pre-Requisite Recommended
Cosmology PHYS30392 Pre-Requisite Optional
Complex Variables and Vector Spaces PHYS20672 Pre-Requisite Recommended
Electrodynamics (M) PHYS30441 Pre-Requisite Recommended

For recommneded theroty units following this module please see PHYS40722

### Aims

Development of the ideas of General Relativity within the framework of differential geometry on a curved manifold.

### Learning outcomes

On completion successful students will be able to:

1. apply the basic concepts of differential geometry on a curved manifold, specifically the concepts of metric, connection and curvature.

1. use the Einstein equations to describe the relation between mass-energy and curvature

1. understand the relation of General Relativity to Newtonian theory and post-Newtonian corrections, including gravitational waves.

1. describe spherical Black Holes.

1. derive the basic properties of the FLRW Universe.

### Syllabus

The weakest of all the fundamental forces, gravity has fascinated scientists throughout the ages. The great conceptual leap of Einstein in his 'General Theory of Relativity' was to realize that mass and energy curve the space in which they exist. In the first part of the course we will develop the necessary mathematics to study a curved manifold and relate the geometrical concept of curvature to the energy momentum tensor. In the second part of the course we solve the Einstein equations in a number of simple situations relevant to the solar system, black holes, and a homogeneous and isotropic universe.

1. Preliminaries (4 lectures)

Cartesian Tensors; Variational Calculus; Newtonian mechanics and gravity; Review of Special Relativity; Einstein's lift experiment; Einstein's vision of General Relativity, Rindler space.

1. Manifolds and differentiation (2lectures)

Manifolds, curves, surfaces; Tangent vectors; Coordinate transformations; Metric and line element; Vectors, co-vectors and tensors; Conformal metrics.

1. Connection and tensor calculus (4 lectures)

Covariant differentiation and Torsion; Affine Geodesics; Metric Geodesics and the Metric Connection; Locally Inertial Coordinates; Isometries and Killing's Equation; Computing Christoffel symbols and Geodesics.

1. Curvature (2 lectures)

Riemann Tens

### Assessment methods

Method Weight
Written exam 100%

### Feedback methods

Feedback will be available on students’ individual written solutions to selected examples, which will be marked when handed in, and model answers will be issued

The following texts are useful for revising the material for the course

Cheng, T. P., Relativity, Gravitation and Cosmology: A Basic Introduction (second edition, Cambridge University Press, 2010)

D'Inverno, R. Introducing Einstein's Relativity, (Oxford University Press, 1992)

Hartle, J. B. An Introduction to Einstein's General Relativity, (Addison Wesley, 2004)

Hobson, M. P., Efstathiou, G. & Lasenby, A. N. General Relativity: An Introduction for Physicists (Cambridge University Press, 2006)

Lambourne, R. J. A., Relativity, Gravitation and Cosmology (Cambridge University Press, 2010)

Misner, C.W. Thorne, K.S & Wheeler, J.A. Gravitation, (Freeman)

Wald, R.M. General Relativity (University of Chicago Press)

### Study hours

Scheduled activity hours
Assessment written exam 1.5
Lectures 24
Work based learning 6
Independent study hours
Independent study 68.5

### Teaching staff

Staff member Role
Michael Seymour Unit coordinator