 On successful completion of the module, students should be able to:  Describe the geometry of curved spaces and curved spacetimes in terms of a metric; derive the LeviCivita connection associated with a given metric; obtain the Riemann curvature tensor, the Ricci tensor, the Ricci scalar and the Einstein tensor associated with such a metric
 Obtain the geodesic equations in arbitrary spacetimes
 Derive the Schwarzschild solution to the vacuum Einstein field equations under the assumption of time translational and spherical symmetry
 Derive expressions for the gravitational redshift, the precession of the perihelion of planetary orbits, and the deflection of light rays in the Schwarzschild geometry
 Show that there exist planewave solutions to the vacuum Einstein equations in the linear approximation, corresponding to gravational waves, and derive some of their physical properties
 Obtain the energymomentum tensor for a fluid, discuss the consequences of spatial isotropy for a Universe with a cosmological constant filled with a fluid, and hence construct the FriedmannRobertsonWalker metric
 Obtain the Friedmann equations from the Einstein field equations, solve these equations for the scale factor and discuss the implications of these solutions
 Describe the evolution of the Universe from the Big Bang, from the first three minutes to the era of star and galaxy formation; explain the origin of low mass chemical elements in the early Universe.
