# Module MA4448: General Relativity

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2015-16
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof Andrei Parnachev
Learning Outcomes
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;
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)

Module Prerequisite

MA3429 Differential Geometry I
Assessment Detail
This module will be examined in a 2 hour examination at the annual session. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session. Supplemental exams if required will consist of 100% exam.