Aims:

To give an introduction to Einstein's general relativity theory of gravitation, and to explain how it provides a more accurate and satisfactory description of gravity than is available from Newtonian theory.  As time permits particular topics will be selected from tests of general relativity, black holes and cosmology.

Learning objectives:

At the end of the module you should be able to:

• appreciate the splendour of Einstein's achievement;
• understand the reasons for supposing that gravity may be modelled in terms of a curved space-time;
• appreciate how the differential geometry of surfaces in three dimensions may be generalised to give a theory of an n-dimensional curved space with metric, and to understand those parts of Riemannian geometry and the tensor calculus needed to follow the arguments leading to Einstein's equations;
• understand the conditions under which Einstein's theory reduces to the Newtonian Theory as a first approximation;
  
• solve the Einstein equations for a static and bounded spherically symmetric distribution of matter leading to the Schwarzchild exterior metric;
  
• appreciate (as time allows) formulas for the perihelion advance of planetary orbits, the deflection of light rays and the gravitational red shift, black holes, features of simple cosmological model.
  

Syllabus:

• A brief survey of the Newtonian theory of gravitation and the reasons for generalising the theory of special relativity in order to account for gravity;
  
• the idea that the paths of free particles or light rays are time-like or null geodesics, respectively, in a curved space-time;
  
• an introduction to Riemann geometry based on a metric as a generalisation of the differential geometry on a curved surface in three dimensions;
  
• tensors and the tensor calculus;
  
• the Einstein field equation;
  
• the Schwarzchild metric;
  
• a selection of - the advance of the perihelion of planetary orbits, the deflection of light rays and the gravitational red shift;
  black holes;
  the application of general relativity to cosmology.

Recommended texts:

• R d'Inverno, Introducing Einstein's Relativity, Oxford University Press (paperback) (U 0.11 DIN)
• L P Hughston and K P Tod, An Introduction to General Relativity, Cambridge University Press (U 0.11 TOD). 
• M Ludvigsen, General Relativity: a geometric approach, Cambridge University Press (S.82 LUD). 
• W Rindler, Essential Relativity, Springer-Verlag (U 0.11 RIN).
• S Weinberg, Gravitation and Cosmology, Wiley (U 0.11 WEI).

Teaching:

Two one-hour lectures per week and one one-hour seminar per week.

Assessment:

Two-hour unseen examination (90%)
Coursework (10%)

Elective information:

An introductory course on General Relativity.

Prerequisites

• Lorentz transformations and the space-time metric; properties of 4-vectors. Taught in: Special Relativity
• Newtonian gravity and planetary orbits. Taught in: Newtonian Mechanics
• Lagrangian formulation of mechanics Taught in: Analytical Mechanics