Problems covering gravity as a central force, motion in polar coordinates, effective potentials, Kepler's problem, and perihelion precession using a perturbation term to the Newtonian gravitational potential.

Derivation of Maxwell's equations in covariant form from the action principle, the continuity equation, and charge conservation in flat and curved spacetimes.

Properties of the electromagnetic 2-form tensor and showing that a mass term in the Proca Lagrangian breaks gauge invariance under U(1) transformation — concluding the photon must be massless.

Rindler spacetime for uniformly accelerating observers, Rindler wedges, observer-dependent quantum states, inner products, field quantisation, and Bogoliubov coefficients.

Classical experimental tests: gravitational frequency shift, the Shapiro delay, bending of light, and precession of planetary orbits.

Geodesic equations for cylindrical surfaces, 2-spheres, and the Schwarzschild metric. The geodesic deviation equation and its relation to the Riemann tensor.

Ricci tensor components in FRW spacetime, the energy-momentum tensor for a perfect fluid in a homogeneous and isotropic universe.

Killing vector fields of the Schwarzschild metric, conserved quantities, and first-order differential equations for radial trajectories.

Demonstrating the Unruh effect using Bogoliubov transformations and field quantisation in non-inertial frames.

The Kerr metric — ergosphere, frame dragging, Killing horizons, and the Penrose process for extracting energy from rotating black holes.

Near-horizon geometries of Schwarzschild and Reissner-Nordström black holes. Field quantisation and observer-dependent vacuum states in relation to Hawking radiation.