MPhys Physics

Gravitation (M)

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Follow Up Units

PHYS40772 - Early Universe

Development of the ideas of General Relativity within the framework of differential geometry on a curved manifold.

This course unit detail provides the framework for delivery in 21/22 and may be subject to change due to any additional Covid-19 impact.  Please see Blackboard / course unit related emails for any further updates.

On completion successful students will be able to:
1. apply the basic concepts of differential geometry on a curved manifold, specifically the concepts of metric, connection and curvature.
2. use the Einstein equations to describe the relation between mass-energy and curvature
3. understand the relation of General Relativity to Newtonian theory and post-Newtonian corrections.
4. describe spherical Black Holes.
5. derive the basic properties of the FRW Universe.

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.
 

1.   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.
2.   Manifolds and differentiation (2 lectures)
Manifolds, curves, surfaces; Tangent vectors; Coordinate transformations; Metric and line element; Vectors, co-vectors and tensors; Conformal metrics.
3.   Connection and tensor calculus (4 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.
4.   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.
5.   Einstein equations (2 lectures)
Energy-momentum tensor; Einstein tensor and the Einstein Equations; Newtonian limit; Gravitational radiation.
6.   Schwarzschild Solution (6 lectures)
Spherically symmetric vacuum solution; Birkhoff’s theorem; Dynamics in the Schwarzschild solution; Gravitational Redshift; Light deflection; Perihelion precession; Black holes.
7.   FRW universe (4 lectures)
Expansion, isotropy and homogeneity; FRW metric; Friedmann and Raychauduri equations; Solutions in radiation and matter eras; Cosmological constant; Cosmological redshift; Cosmological distance measures; Flatness and horizon puzzles.

Feedback will be available on students’ individual written solutions to selected examples, which will be marked when handed in, and model answers will be issued

The following texts are useful for revising the material for the course

Cheng, 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)