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MMath&Phys Mathematics and Physics / Course details
Year of entry: 2023
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Course unit details:
|Unit level||Level 4|
|Teaching period(s)||Semester 1|
|Available as a free choice unit?||No|
|Unit title||Unit code||Requirement type||Description|
|Complex Variables and Vector Spaces||PHYS20672||Pre-Requisite||Recommended|
For recommneded theroty units following this module please see PHYS40722
Development of the ideas of General Relativity within the framework of differential geometry on a curved manifold.
On completion successful students will be able to:
apply the basic concepts of differential geometry on a curved manifold, specifically the concepts of metric, connection and curvature.
use the Einstein equations to describe the relation between mass-energy and curvature
understand the relation of General Relativity to Newtonian theory and post-Newtonian corrections, including gravitational waves.
describe spherical Black Holes.
derive the basic properties of the FLRW Universe.
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.
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.
Manifolds and differentiation (2lectures)
Manifolds, curves, surfaces; Tangent vectors; Coordinate transformations; Metric and line element; Vectors, co-vectors and tensors; Conformal metrics.
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.
Curvature (2 lectures)
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)
More advanced texts
Misner, C.W. Thorne, K.S & Wheeler, J.A. Gravitation, (Freeman)
Wald, R.M. General Relativity (University of Chicago Press)
|Scheduled activity hours|
|Assessment written exam||1.5|
|Work based learning||6|
|Independent study hours|
|Michael Seymour||Unit coordinator|