Gravitation (M,A)

PHYS40771 Gravitation (M,A) 10 Semester 1 Undergraduate Level 4 Richard Battye Unit coordinator Peter Millington Unit coordinator Department of Physics & Astronomy 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 5.0 Gravitation (M) Gravitation (M) Development of the ideas of General Relativity within the framework of differential geometry on a curved manifold. On completion successful students will be able to:<ol><li>apply the basic concepts of differential geometry on a curved manifold, specifically the concepts of metric, connection and curvature.</li><li>use the Einstein equations to describe the relation between mass-energy and curvature</li><li>explain the relation of General Relativity to Newtonian theory and post-Newtonian corrections, including gravitational waves.</li><li>describe spherical black holes and their gravitational effects.</li><li>derive the basic properties of the FLRW Universe. </li></ol> The weakest of all the fundamental forces, gravity has fascinated scientists throughout the ages. The great conceptual leap of Einstein in his 'General Theory of Relativity' was to realize that mass and energy curve the space in which they exist. In the first part of the course we will develop the necessary mathematics to study a curved manifold and relate the geometrical concept of curvature to the energy momentum tensor. In the second part of the course we solve the Einstein equations in a number of simple situations relevant to the solar system, black holes, and a homogeneous and isotropic universe.Preliminaries (4 lectures) Cartesian tensors; Variational calculus; Newtonian mechanics and gravity; Review of Special Relativity; Einstein's lift experiment; Einstein's vision of General Relativity, Rindler space.Manifolds and differentiation (2 lectures) Manifolds, curves, surfaces; Tangent vectors; Coordinate transformations; Metric and line element; Vectors, co-vectors and tensors; Conformal metrics.Connection and tensor calculus (2 lectures) Covariant differentiation and torsion; Affine geodesics; Metric geodesics and the metric connection; Locally inertial coordinates; Isometries and Killing's equation; Computing Christoffel symbols and geodesics.Curvature (2 lectures) Riemann tensor; Ricci tensor and scalar; Symmetries of the Riemann tensor and the Bianchi identities; Round trips by parallel transport; Geodesic deviation.Einstein equations (2 lectures) Energy-momentum tensor; Einstein tensor and the Einstein equations; Newtonian limit; Gravitational radiation.Schwarzschild solution (6 lectures) Spherically symmetric vacuum solution; Birkhoff’s theorem; Dynamics in the Schwarzschild solution; Gravitational redshift; Light deflection; Perihelion precession; Black holes.FLRW Universe (4 lectures) Expansion, isotropy and homogeneity; FRW metric; Friedmann and Raychaudhuri equations; Solutions in radiation and matter eras; Cosmological constant; Cosmological redshift; Cosmological distance measures; Flatness and horizon puzzles. 1 Written exam 100% PHYS10672 Advanced Dynamics Pre-Requisite Compulsory PHYS30392 Cosmology Pre-Requisite Optional PHYS20672 Complex Variables and Vector Spaces Pre-Requisite Recommended PHYS30441 Electrodynamics (M) Pre-Requisite Recommended For recommended theroty units following this module please see PHYS40722. N  The following texts are useful for revising the material for the courseCheng, T. P., Relativity, Gravitation and Cosmology: A Basic Introduction (second edition, Cambridge University Press, 2010)D'Inverno, R. Introducing Einstein's Relativity, (Oxford University Press, 1992)Hartle, J. B. An Introduction to Einstein's General Relativity, (Addison Wesley, 2004)Hobson, M. P., Efstathiou, G. & Lasenby, A. N. General Relativity: An Introduction for Physicists (Cambridge University Press, 2006)Lambourne, R. J. A., Relativity, Gravitation and Cosmology (Cambridge University Press, 2010)  More advanced texts Misner, C.W. Thorne, K.S & Wheeler, J.A. Gravitation, (Freeman)Wald, R.M. General Relativity (University of Chicago Press)  Assessment written exam 2 Lectures 22 Work based learning 6 0 70