# C7.5 General Relativity I - Material for the year 2020-2021

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*ASO: Special Relativity*, *B7.1 Classical Mechanics*, and *B7.2 Electromagnetism*.

16 lectures

### Assessment type:

- Written Examination

The course is intended as an elementary introduction to general relativity, the basic physical concepts of its observational implications, and the new insights that it provides into the nature of space time, and the structure of the universe. Familiarity with special relativity and electromagnetism as covered in the Part A and Part B courses will be assumed. The lectures will review Newtonian gravitation, tensor calculus and continuum physics in special relativity, physics in curved space time and the Einstein field equations. This will suffice for an account of simple applications to planetary motion, the bending of light and the existence of black holes.

This course starts by asking how the theory of gravitation can be made consistent with the special-relativistic framework. Physical considerations (the principle of equivalence, general covariance) are used to motivate and illustrate the mathematical machinery of tensor calculus. The technical development is kept as elementary as possible, emphasising the use of local inertial frames. A similar elementary motivation is given for Einstein's equations and the Schwarzschild solution. Cosmological solutions are discussed.

The learning outcomes are an understanding and appreciation of the ideas and concepts described above

Review of Newtonian gravitation theory and problems of constructing a relativistic generalisation. Review of Special Relativity. The equivalence principle. Tensor formulation of special relativity (including general particle motion, tensor form of Maxwell's equations and the energy momentum-tensor of dust). Curved space time. Local inertial coordinates. General coordinate transformations, elements of Riemannian geometry (including connections, curvature and geodesic deviation). Mathematical formulation of General Relativity, Einstein's equations (properties of the energy-momentum tensor will be needed in the case of dust only). Planetary motion, the bending of light, introduction to black hole solutions and the Schwarzschild solution. The introduction to cosmology including cosmological principles, homogeneity and isotropy, and the Friedman-Robertson-Walker solutions.

- S. Carroll,
*Space Time and Geometry: An Introduction to General Relativity*(Addison Welsey, 2003) - L.P. Hughston and K.P. Tod,
*An Introduction to General Relativity,*LMS Student Text 5 (London Mathematical Society, Cambridge University Press, 1990), Chs 1-18. - N.M.J. Woodhouse,
*Notes on Special Relativity*, Mathematical Institute Notes. Revised edition; published in a revised form as*Special Relativity, Lecture notes in Physics m6*(Springer-Verlag, 1992), Chs 1-7

- B. Schutz,
*A First Course in General Relativity*(Cambridge University Press, 1990). - R.M. Wald,
*General Relativity*(Chicago, 1984). - W. Rindler,
*Essential Relativity*(Springer-Verlag, 2nd edition, 1990).