Maths page for handouts and examples sheets

[Part IB Methods and Variational Principles are very useful.]

**Brief review of Special Relativity**

Notion of proper time. Equation of motion for free point particle derivable from a variational principle. Noether’s Theorem. [1]

**Introduction and motivation for General Relativity**

Curved and Riemannian spaces. The Pound-Rebka experiment. Introduction to general relativity: interpretation of the metric, clock hypothesis, geodesics, equivalence principles. Static spacetimes. Newtonian limit. [4]

**Tensor calculus**

Covariant and contravariant tensors, tensor manipulation, partial derivatives of tensors. Metric tensor, magnitudes, angles, duration of curve, geodesics. Connection, Christofel symbols, covariant derivatives, parallel transport, autoparallels as geodesics. Curvature. Riemann and Ricci tensors, geodesic

deviation. [5]

**Vacuum field equations**

Spherically symmetric spacetimes, the Schwarzschild solution. Birkhoff’s Theorem. Rays and orbits, gravitational red-shift, light deflection, perihelion advance. Shapiro time delay. [2]

**Einstein Equations coupled to matter**

Concept of an energy momentum tensor. Maxwell stress tensor and perfect fluid as examples. Importance of Bianchi identities. The emergence of the cosmological term. Simple exact solutions: Friedmann- Lemaitre metrics, the Einstein Static Universe. Hubble expansion and redshift. De-Sitter spacetime, mention of Dark Energy and the problem of Dark matter. Notion of geodesic completeness and definition of a spacetime singularity. Schwarzschild and Friedmann-Lemaitre spacetimes as examples of spacetimes with singularities. [4]

**Linearized theory**

Linearized form of the vacuum equations. De-Donder gauge and reduction to wave equation. Comparison of linearized point mass solution with exact Schwarzschild solution and identification of the mass parameter. Gravitational waves in linearized theory. *The quadrupole formula for energy radiated.* Comparison of linearized gravitational waves with the exact pp-wave metric. [4]

**Gravitational collapse and black holes**

Non-singular nature of the surface r = 2*M *in the Schwarzschild solution using Finkelstein and Kruskal coordinates. The idea of an event horizon and the one-way passage of timelike geodesics through it. Qualitative account of idealized spherically symmetric collapse. The final state: statement of Israel’s Theorem. *Qualitative description of Hawking radiation.* [4]

Appropriate books

S.M. Carroll Spacetime and Geometry. Addison-Wesley 2004 (£53.99) J.B. Hartle Gravity: *An introduction to Einstein’s General Relativity*. AddisonWesley 2002 (£32.99 hardback)

L.P. Hughston and K.P. Tod *An Introduction to General Relativity*. Cambridge University Press 1990 (£16.95 paperback)

R. d’Inverno *Introducing Einstein’s Relativity*. Clarendon 1992 (£29.95 paperback)

†W. Rindler *Relativity: Special, General and Cosmological.* Oxford University Press 2001 (£24.95 paperback).

H. Stephani Relativity: An introduction to Special and General Relativity. Cambridge University Press, 2004 (£30.00 paperback, £80.00 hardback)

Page last updated: 7 February 2014 at 08:22