This course provides the first part of an introduction to the theory of general relativity. Elementary tensor calculus on riemannian manifolds is introduced as a means of describing curved spacetimes and arriving at Einstein's field equations. Astrophysical applications will be discussed in Gravitation and Relativity II.

Course Content (10 lectures)

    1. The equivalence principle and its physical consequences
                Gravitational redshift and bending of light. Gravitation as curvature of spacetime. Illustration of curvature at the surface of the Earth.

    2. Covariance
                Component versus coordinate free notation

    3. Review of special relativity
                Invariant distance. Lorentz transformation. Spacelike, timelike, and null intervals. Proper time. Notion of covariance in special relativity.
                Four vectors, four velocity, four momentum and four acceleration. Tensors in special relativity.

    4. Spacetime as a manifold
                Manifolds and coordinate systems. Transformation of prototype contravariant vector. Covariant vector components with gradient as
                example. Tangent vector and one forms. The metric tensor and its inverse. Tensors of higher rank. Raising and lowering indices.
                Contraction.

    5. Covariant differentiation
                Parallel displacement of scalars, vectors and tensors. Christoffel symbols. Christoffel symbols expressed in terms of the metric tensor.
                Covariant differentiation of vectors and of tensors of higher rank.

    6. Geodesics
                Geodesic equation in Riemannian space as world line of test particle. Null geodesics and photons.

    7. Energy momentum tensor
                Physical interpretation, conservation of energy momentum

    8. Riemann-Christoffel tensor, and Ricci tensor
                Derivation of Riemann Christoffel tensor from equation of geodesic deviation. Bianchi identities.

    9. Einstein's equations

    10. Weak field limit and correspondence with Newtonian gravitation

Recommended Books

    Schutz, B. "A First Course in General Relativity", CUP (1985)

    This book is not essential for purchase, but is highly recommended

Books for consultation

    Misner, Thorne and Wheeler,"Gravitation", Freeman (1973)
    Wald, R.M. "General Relativity", University of Chicago (1984)
    Rindler, W. "Essential Relativity", Springer (1977).
    Berry, M. "Principles of Relativity and Cosmology", CUP (1976)
    Papapetrou, A. "Lectures in General Relativity", Reidel (1974)
    Ohanion, H.C. "Gravitation and Spacetime", Norton (1976)
    d'Inverno R. "Introducing Einstein's Relativity", Oxford University Press (1992)

    None of the above books is essential, or indeed worthwhile, for purchase but may be useful for occasional consultation and
        general background reading

Martin Hendry,
October 2000

Lecture Notes

Lecture notes for the course are provided below in pdf format. These notes are presented as a self-contained document (essentially like a short textbook) covering in detail all the material in the course syllabus.

These lecture notes are designed to avoid the need for students to copy down large amounts of text from the blackboard or OHP during the lectures.

Gravitation and Relativity is not an easy subject to master - in terms of both the abstract physical concepts and the mathematical notation. The widespread view among recent students, however, is that to understand the principles of general relativity is among the most intellectually satisfying achievements of their undergraduate degree.

In order to achieve this understanding most effectively - and painlessly - it is, therefore, very important that all students print out and read carefully in advance the material to be covered in each lecture. Setting aside some time to do this will greatly assist your appreciation of the lectures and help you to grasp quickly the key ideas of general relativity. Clear guidelines for advanced reading (and for tackling the example sheets, see below) will be given throughout the course.
 

(n.b. these .pdf files may not look very good on your computer screen, but they should print out OK.
    If you have any problems printing them please let me know)
 

Example Sheets
 
 

Examples 1

Examples 2

Examples 3

Examples 4

Examples 5

Examples 6

Example Sheets: Model Answers
 

Solutions to Examples in Lecture Notes
 

Please send any comments or questions on GR-I to Martin Hendry