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Durham University

Department of Mathematical Sciences

MATH4051 General Relativity IV

A century ago, Einstein invented a geometric model of gravity, namely General Relativity. In addition to being mathematically beautiful, it provides an extraordinarily accurate description of gravitational phenomena. The key idea is that gravity is a manifestation of the geometry of space-time.

The first half of this course concentrates mainly on setting up the necessary geometrical structure, and introducing Einstein's equations which describe how matter curves space-time. In the second half, we shall see how all this is applied, for example to various solutions of Einstein's equations, and their interpretation; classical solar-system tests of GR, such as the bending of light around the sun and the perihelion shift of Mercury; black holes, event horizons and Penrose diagrams; and cosmology which studies the universe on a large scale.

Outline of Course

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

Term 1

  • Introduction to General Relativity: differences between GR & SR, gravity as geometry, equivalence principle.
  • Special Relativity: spacetime diagrams, line element, vectors and tensors, electromagnetism, stress-energy tensor.
  • Differential Manifolds: spacetime is a manifold, coordinates and coordinate transformations, tangent vectors, tensors revisited.
  • Metric: distance relationships, light cones, Riemann normal coordinates.
  • Covariant Derivative: inadequacy of partial derivatives, parallel transport, connection coefficients, differentiating tensors, metric connection, geodesics.
  • Curvature: Riemann tensor, characterisation of flat space, parallel transport around closed curves, commutation formulae, Bianchi identity, Einstein tensor, geodesic deviation.

Term 2

  • General Relativity: equivalence principle, physics in curved spacetime, Einstein's equations, linearized theory and Newtonian limit, Einstein-Hilbert action.
  • Black Holes: spherical symmetry, Schwarzschild solution, geodesics, solar-system applications, event horizon and Kruskal coordinates, black hole formation.
  • Cosmology: isotropy and homogeneity, FRW metric, examples of cosmologies, Hubble law, particle horizons.

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH4051 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.