Introduction: problems with Newtonian gravity, the equivalence principle, gravity as spacetime curvature.
Foundations of special relativity: Inertial frames, spacetime geometry, Lorentz transformations, length contraction and time dilation, Minkowski line element, particle worldlines and proper time, Doppler effect, addition of velocities, acceleration and event horizons in special relativity.
Manifolds, coordinates and tensors: Concept of a manifold, curves and surfaces, coordinate transformations, Riemannian geometry, intrinsic and extrinsic geometry, the metric tensor, lengths and volumes, local Cartesian coordinates, pseudo- Riemannian geometry, scalar fields, vectors and dual vectors, tensor fields, tensor algebra, covariant differentiation and the metric connection, intrinsic derivative, parallel transport, geodesics.
Minkowski space and particle dynamics: Cartesian inertial coordinates, Lorentz transformations, 4-tensors and inertial bases, 4-vectors and the lightcone, 4-velocity, 4-acceleration, 4-momentum of massive and massless particles, relativistic mechanics, arbitrary coordinate systems.
Electromagnetism: Lorentz force, the current 4-vector, the electromagnetic field tensor and field equations, the electromagnetic 4-potential.
Spacetime curvature: Locally-inertial coordinates, weak gravitational fields, intrinsic curvature, the curvature tensor, the Ricci tensor, parallel transport, geodesic deviation and tidal effects, physical laws in curved spacetime.
Gravitational field equations: the energy-momentum tensor, perfect fluids, relativistic fluid dynamics, the Einstein equations, the weak field limit, the cosmological constant.
Schwarzschild spacetime: static isotropic metrics, solution of empty-space field equations, Birkhoff’s theorem, gravitational redshift, trajectories of massive particles and photons. Singularities, radially-infalling particles, event horizons, Eddington- Finkelstein coordinates, gravitational collapse, tidal forces.
Experimental tests of general relativity: precession of planetary orbits, the bending of light.
Friedmann–Robertson–Walker spacetime: the cosmological principle, comoving coordinates, the maximally-symmetric 3-space, the FRW metric, geodesics, cosmological redshift, the cosmological field equations.
Linearised gravity and gravitational waves: weak field metric, linearised field equations, Lorenz gauge, wave solutions of linearised field equations.*
*The topics starred in the schedules may be lectured, but questions will not be set on them in examinations.