On successful completion of this module students will be able to;
Define the Einstein-Hilbert action and derive Einstein's equations from an action principle;
Define the stress-energy-momentum tensor, obtain its components in an orthonormal tetrad, and obtain explicit expressions for the stress-energy-momentum tensor describing a perfect fluid matter distribution;
Derive the canonical form of the Schwarzschild solution to the vacuum field equations under the sole assumption of spherical symmetry, and hence state Birkhoff's Theorem;
Derive expressions for the gravitational redshift, perihelion advance of the planets, and light deflection in the Schwarzschild space-time and hence discuss solar system tests of General Relativity;
Obtain the geodesic equations in arbitrary space-times and hence describe various trajectories such as radially in-falling particles or circular geodesics etc.;
Obtain the maximal extension of the Schwarzschild solution in Kruskal coordinates and hence discuss the Schwarzschild black hole;
Define spatial isotropy with respect to a universe filled with a congruence of time-like world-lines, discuss the consequences of global isotropy on the shear, vorticity and expansion of the congruence and hence construct the Friedmann-Robertson-Walker metric;
Obtain the Friedmann and Raychaudhuri equations from the Einstein field equations, solve these equations for the scale factor and discuss the cosmogonical and eschatological consequences of the solutions;
Derive the Einstein equations in the linear approximation and discuss the Newtonian limit in the weak-field, slow-moving approximation;
Use the gauge freedom to show that, in the Einstein-deDonder gauge, the perturbations satisfy an inhomogeneous wave-equation, to solve in terms of plane-waves, and to use the residual gauge freedom to show that for waves propagating in the positive z-direction there are only two linearly independent non-zero components;
Derive the Reissner-Nordstrom solution of the Einstein-Maxwell field equations, obtain its maximal extension and discuss the Reissner-Nordstrom black hole solution;
Module Content
Module Prerequisite
Recommended Reading
Each of these texts are useful, but beware of the many different signatures and notations.
Introducing Einstein's Relativity, R. D'Inverno, (Oxford University Press 1992)
A First Course in General Relativity, B.Schutz, (Cambridge University Press 2011)
General Relativity, H.Stephani, (Cambridge University Press 1982)
General Relativity, R.M. Wald, (Chicago University Press 1984)
Assessment Detail
This module will be examined jointly with MA3429
in a 3-hour examination in Trinity term,
except that those taking just one of the
two modules will have a 2 hour examination.
Continuous assessment
will contribute 15% to the final grade for the module at the annual
examination session.