Synopsis for C7.2a: General Relativity I

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit. OSS paper code 2B78.

Recommended Prerequisites

B7.2a Relativity and Electromagnetism.

Overview

The course is intended as an elementary introduction to general relativity, the basic physical concepts of its observational implications, and the new insights that it provides into the nature of space time, and the structure of the universe. Familiarity with special relativity and electromagnetism as covered in the B7 course will be assumed. The lectures will review Newtonian gravitation, tensor calculus and continuum physics in special relativity, physics in curved space time and the Einstein field equations. This will suffice for an account of simple applications to planetary motion, the bending of light and the existence of black holes.

Learning Outcomes

This course starts by asking how the theory of gravitation can be made consistent with the special-relativistic framework. Physical considerations (the principle of equivalence, general covariance) are used to motivate and illustrate the mathematical machinery of tensor calculus. The technical development is kept as elementary as possible, emphasising the use of local inertial frames. A similar elementary motivation is given for Einstein's equations and the Schwarzschild solution. Orbits in the Schwarzschild solution are given a unified treatment which allows a simple account of the three classical tests of Einstein's theory. Finally, the analysis of extensions of the Schwarzschild solution show how the theory of black holes emerges and exposes the radical consequences of Einstein's theory for space-time structure. Cosmological solutions are not discussed.

The learning outcomes are an understanding and appreciation of the ideas and concepts described above.

Synopsis

Review of Newtonian gravitation theory and problems of constructing a relativistic generalisation. Review of Special Relativity. The equivalence principle. Tensor formulation of special relativity (including general particle motion, tensor form of Maxwell's equations and the energy momentum-tensor of dust). Curved space time. Local inertial coordinates. General coordinate transformations, elements of Riemannian geometry (including connections, curvature and geodesic deviation). Mathematical formulation of General Relativity, Einstein's equations (properties of the energy-momentum tensor will be needed in the case of dust only). The Schwarzschild solution; planetary motion, the bending of light, and black holes.

1. L.P. Hughston and K.P. Tod, An Introduction to General Relativity, LMS Student Text 5 (London Mathematical Society, Cambridge University Press, 1990), Chs 1–18.
2. N.M.J. Woodhouse, Notes on Special Relativity, Mathematical Institute Notes. Revised edition; published in a revised form as Special Relativity, Lecture notes in Physics m6 (Springer-Verlag, 1992), Chs 1–7

Further Reading

1. B. Schutz, A First Course in General Relativity (Cambridge University Press, 1990).
2. R.M. Wald, General Relativity (Chicago, 1984).
3. W. Rindler, Essential Relativity (Springer-Verlag, 2nd edition, 1990).