# 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**

**Assessment Detail**

**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.