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MA4K5 Content

* The wave equation and Special Relativity (Propagation of signals: the light-cone; finite speed of propagation; Transformations preserving the wave equation; the Lorentz group; Minkowski spacetime)

* Brief review of (pseudo-)Riemannian geometry (Vectors, one-forms and tensors; the metric tensor; the Levi-Civita connection and curvature; Stoke's theorem)

* Lorentzian geometry (Lorentzian metrics; causal classification of vectors and curves; global hyperbolicity; The d'Alembertian operator; Energy-momentum tensor for a scalar field; finite speed of propagation for a scalar field)

* General Relativity (Einstein's equations; discussion of local well posedness; Example: The Schwarzschild black hole; The Cauchy problem; discussion of open problems)

One of the crowning achievements of modern physics is Einstein's theory of general relativity, which describes the gravitational field to a very high degree of accuracy. As well as being an astonishingly accurate physical theory, the study of general relativity is also a fascinating area of mathematical research, bringing together aspects of differential geometry and PDE theory. In this course, I will introduce the basic objects and concepts of general relativity without assuming a knowledge of special relativity. The ultimate goal of the course will be a discussion of the Cauchy problem for the vacuum Einstein equations, including a statement of the relevant well-posedness theorems and a discussion of their relevance. We will take a 'field theory' approach to the subject, emphasising the deep connection between Lorentzian geometry and hyperbolic PDE. In contrast to the course PX436 General Relativity offered by the department of physics, we concentrate on the mathematical structure of the theory rather than its physical implications.

By the end of the module the student should be able to:

  • Understand how the Minkowski geometry and Lorentz group arise from considerations of signal propagation for the scalar wave equation.
  • Understand the basics of Lorentzian geometry: the metric; causal classification of vectors; connection and curvature; hypersurface geometry; conformal compacti cations; the d'Alembertian operator.
  • Be able to state the well-posedness theorems for the Cauchy problem for the Einstein equations and sketch the proof of local well posedness.

General Relativity and the Einstein Equations, Yvonne Choquet-Bruhat, Oxford University Press, 2009. (Available as an electronic resource.)
The large scale structure of spacetime, S.W. Hawking and G.F.R. Ellis, Cambridge University Press, 1973.
Gravitation, Charles W. Misner, Kip S. Thorne and John Archibald Wheeler.
General Relativity, Robert M. Wald, University of Chicago Press, c1984.