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# Department of Mathematical Sciences

# MATH4051 General Relativity IV

A century ago, Einstein invented a geometric model of gravity, namely General Relativity. In addition to being mathematically beautiful, it provides an extraordinarily accurate description of gravitational phenomena. The key idea is that gravity is a manifestation of the geometry of space-time.

The first half of this course concentrates mainly on setting up the necessary geometrical structure, and introducing Einstein's equations which describe how matter curves space-time. In the second half, we shall see how all this is applied, for example to various solutions of Einstein's equations, and their interpretation; classical solar-system tests of GR, such as the bending of light around the sun and the perihelion shift of Mercury; black holes, event horizons and Penrose diagrams; and cosmology which studies the universe on a large scale.

## Outline of Course

Aim: To appreciate General Relativity, one of the fundamental physical theories. To develop and exercise mathematical methods.

### Term 1

• Introduction to General Relativity: differences between GR & SR, gravity as geometry, equivalence principle.
• Special Relativity: spacetime diagrams, line element, vectors and tensors, electromagnetism, stress-energy tensor.
• Differential Manifolds: spacetime is a manifold, coordinates and coordinate transformations, tangent vectors, tensors revisited.
• Metric: distance relationships, light cones, Riemann normal coordinates.
• Covariant Derivative: inadequacy of partial derivatives, parallel transport, connection coefficients, differentiating tensors, metric connection, geodesics.
• Curvature: Riemann tensor, characterisation of flat space, parallel transport around closed curves, commutation formulae, Bianchi identity, Einstein tensor, geodesic deviation.

### Term 2

• General Relativity: equivalence principle, physics in curved spacetime, Einstein's equations, linearized theory and Newtonian limit, Einstein-Hilbert action.
• Black Holes: spherical symmetry, Schwarzschild solution, geodesics, solar-system applications, event horizon and Kruskal coordinates, black hole formation.
• Cosmology: isotropy and homogeneity, FRW metric, examples of cosmologies, Hubble law, particle horizons.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.