# PX436 General Relativity

## Lecturer: Gareth Alexander

## Weighting: 15 CATS

Einstein's general theory of relativity is the basis for our understanding of black holes and the Universe on its largest scales. In general relativity the Newtonian concept of a gravitational force is abolished, to be replaced by a new notion, that of the curvature of space-time. This leads in turn to predictions of phenomena such as the bending of light and gravitational time dilation that are well tested, and others, such as gravitational waves, which are only now coming into the regime of direct detection.

The module starts with a recap of Special Relativity, emphasizing its geometrical significance. The formalism of curved coordinate systems is then developed. Einstein's equivalence principle is used to link the two to arrive at the field equations of GR. The remainder of the module looks at the application of general relativity to stellar collapse, neutron stars and black-holes, gravitational waves, including their detection, and finally to cosmology where the origin of the "cosmological constant" -- nowadays called "dark energy" - becomes apparent.

**Aims:**

To present the theory of General Relativity and its applications in modern astronomy, and to give an understanding of black-holes.

**Objectives:**

At the end of this module you should:

- understand the metric nature of special and general relativity, how the metric determines the motion of particles
- be able to undertake elementary calculations involving the Schwarzschild metric
- be able to describe the key features of black-holes
- be able to demonstrate knowledge of current attempts to detect gravitational waves

**Syllabus:**

- The geometry of space-time and the invariant “interval” in special relativity; the 4-vector formulation of special relativity; the metric of special relativity
- The equivalence principle and local inertial frames; the motivation for considering curved space-time; vectors and tensors in curved coordinate systems
- Geodesics: how the metric determines equations of motion; motion in almost-flat space-time: the Newtonian limit
- The curvature and stress-energy tensors; how the metric is determined: Einstein's field equations
- The Schwarzschild metric; observable consequences; black-holes; stability of orbits; extraction of energy
- Gravitational radiation and its detection; cosmology: the Robertson-Walker metric

**Commitment:** 25 Lectures (and 5 problems classes)

**Assessment:** 2 hour examination

The module has a website.

**Recommended Texts:** BF Schutz *A first course in general relativity*, Cambridge University Press,

M.P Hobson, G. Efstathiou, A.N. Lasenby, *General Relativity -- An Introduction for Physicists*, CUP.