Course Content
General Relativity is the Geometrisation of gravitation. It is based on a small number of fairly simple ideas. It has, nonetheless, a reputation as a difficult subject for two reasons:
- Geometry is already complicated to describe, even before we try to include Physics.
- A lot of physical intuition from classical Physics, and even Special Relativity, turns out to be wrong.
Since you have just taken Differential Geometry, the first of these should not be too much of a problem. The second is, of course, a problem, but with enough practice you will gradually adjust.
For an indication of the topics covered in this course, see the table of contents of the text, given below. I will mostly follow that text, in a different order, except that I will cover somewhat more
- Examples (particularly Reissner-Norstrom)
- Differential forms and Stokes' Theorem
- Mechanics of point particles in curved spacetime
Exams
There will be single exam during the usual annual exam period, worth 90% of your course mark. There is a sample paper which you may find useful. We are not allowed to announce exam dates and locations on course web pages, but there is an official timetable here
Assignments
Assignments will count for 10% of your course mark. They will be posted here at roughly 2-3 week intervals, and will be due at the beginning of the 11:00am lecture. They will not be distributed in class. Solutions will be posted here once the assignment has been corrected.
| Assignment | Due | Version | Problems | Solutions |
|---|---|---|---|---|
| 1 | 2 February 2011 | Id: 4448-1011-1.m4,v 1.3 2011/02/14 23:49:59 john Exp john | PS PDF | PS PDF |
| 2 | 23 February 2011 | Id: 4448-1011-2.m4,v 1.2 2011/03/03 11:49:43 john Exp john | PS PDF | PS PDF |
| 3 | 9 March 2011 | Id: 4448-1011-3.m4,v 1.2 2011/03/25 12:42:24 john Exp john | PS PDF | PS PDF |
| 4 | 31 March 2011 | Id: 4448-1011-4.m4,v 1.6 2011/04/26 13:05:01 john Exp john | PS PDF | PS PDF |
Tutorials
There are no tutorials scheduled for this course.
Text
I will follow George Ellis' notes fairly closely. The only significant exception is in the Mechanics of point particles, where I have prepared some lecture notes. Ellis' notes develop Differential Geometry and General Relativity together. With one or two minor exceptions, you will have seen the Differential Geometry we need last term, although in a different notation. We'll go through those parts of the text very quickly. The table of contents is given below, to give you an idea of what topics we will be covering.
- 1 Introduction
- 2 Spacetime
- 2.1 Events and coordinates
- 2.1.1 Events
- 2.1.2 Curves
- 2.1.3 Surfaces
- 2.1.4 Curve-Surface relations
- 2.1.5 Coordinates, Curves and Surfaces
- 2.2 Distances, Angles and Times
- 2.2.1 Curved Spaces
- 2.2.2 Flat Spacetime
- 2.2.3 Curved Spacetimes
- 2.1 Events and coordinates
- 3 Tensors
- 3.1 Change of coordinates
- 3.1.1 Functions
- 3.1.2 Curves and vectors
- 3.1.3 Gradient of Function
- 3.1.4 Invariants
- 3.2 Tensors and Tensor transformations
- 3.2.1 Quotient Law: Tensor detection
- 3.3 Tensor Equations
- 3.3.1 Tensor operations
- 3.3.2 Symmetry properties
- 3.3.3 The volume element
- 3.1 Change of coordinates
- 4 General Bases
- 4.1 General basis
- 4.1.1 Coordinate bases
- 4.2 Tetrad bases
- 4.2.1 Orthonormal bases
- 4.2.2 Lorentz transformations
- 4.3 Other tetrads
- 4.4 Physics in an orthonormal basis
- 4.4.1 4-velocity of particle
- 4.4.2 The 4-momentum
- 4.4.3 Electromagnetism
- 4.4.4 Energy-momentum
- 4.5 Lorentz Transformations
- 4.1 General basis
- 5 Covariant Differentiation
- 5.1 Parallel transport
- 5.1.1 Flat space
- 5.1.2 The Parallel Transport Operator
- 5.2 Curved space
- 5.2.1 Curved space-time
- 5.3 Covariant Differentiation
- 5.3.1 Vector fields
- 5.3.2 General Tensors
- 5.4 Relation to the metric
- 5.4.1 The derivative of the metric
- 5.4.2 Vanishing torsion
- 5.4.3 The Christoffel relations
- 5.4.4 Euclidean space quantities
- 5.5 The Lie Derivative
- 5.1 Parallel transport
- 6 Physics in a curved space-time
- 6.1 Force law
- 6.2 Conservation Equations
- 6.2.1 Scalar conservation
- 6.3 Energy-momentum mass conservation
- 6.4 Maxwell's equations
- 6.5 Scalar Field
- 6.6 Geodesics
- 6.6.1 Parametrisation
- 6.6.2 Existence
- 6.6.3 Geodesic coordinates
- 6.6.4 Extremal properties
- 6.6.5 Calculation of Gammas
- 6.7 Gravity
- 6.8 Physical meaning of parallel transfer
- 6.8.1 Gravity as geometry
- 7 Curvature
- 7.1 The curvature tensor
- 7.1.1 Ricci identity for tensors
- 7.1.2 Symmetries
- 7.1.3 The contractions of the curvature tensor
- 7.1.4 Dimensionality of the curvature tensor
- 7.2 Geometry of curvature
- 7.2.1 Parallel transfer
- 7.2.2 Flat spacetime
- 7.2.3 Geodesic deviation
- 7.3 Integrability conditions
- 7.3.1 The Jacobi identities
- 7.3.2 The Bianchi identities
- 7.3.3 Contracted Bianchi identities
- 7.4 Spaces/spacetimes of constant curvature
- 7.5 Some consequences
- 7.5.1 Maxwell equations revisited
- 7.5.2 Killings equations revisited
- 7.5.3 Normal coordinates
- 7.1 The curvature tensor
- 8 The Einstein Field Equations
- 8.1 The Newtonian limit
- 8.1.1 The equations of motion
- 8.1.2 The gravitational equations
- 8.1.3 Implications
- 8.2 Properties of the field equations
- 8.2.1 PDE's for the metric
- 8.2.2 Evolution from initial values
- 8.2.3 The Free Gravitational field
- 8.2.4 Variational Principle
- 8.2.5 Obtaining solutions
- 8.1 The Newtonian limit