Relativity 2013-14

*Prof Mike Hobson*

## Synopsis

**Foundations of special relativity:** Inertial frames, spacetime geometry, Lorentz transformations, spacetime diagrams, 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, instrinsic and extrinsic geometry, the metric tensor, lengths areas and volumes, local Cartesian coordinates, tangent spaces, pseudo-Riemannian geometry, scalar, vector and tensor fields, basis vectors, raised and lowered indices, tangent vectors, the affine connection, covariant differentiation, intrinsic derivative, parallel transport, geodesics.

**Minkowski spacetime 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, accelerating observers, arbitrary coordinate systems.

**Electromagnetism: **the electromagnetic force, the 4-current density, the electromagnetic field equations, the electromagnetic field tensor, the Lorentz gauge, electric and magnetic fields, invariants, electromagnetism in arbitrary coordinates.

**The equivalence principle and spacetime curvature: **Newtonian gravity, the equivalence principle, gravity as spacetime curvature, local inertial coordinates, observers in a curved spacetime, weak gravitational fields, intrinsic curvature, the curvature tensor, the Ricci tensor, parallel transport, geodesic deviation, tidal forces, minimal coupling procedure.

**Gravitational field equations: **the energy-momentum tensor, perfect fluids, relativistic fluid dynamics, the Einstein equations, the weak field limit, the cosmological constant, particle motion from the field equations.

**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, Hawking radiation.

**Experimental tests of general relativity: **precession of planetary orbits, the bending of light, radar echoes, accretion discs around compact objects, gyroscope precession.

**Friedmann-Robertson-Walker spacetime: **the cosmological principle, comoving coordinates, the maximally-symmetric 3-space, the FRW metric, geodesics, cosmological redshift, the cosmological field equations.

*Kerr spacetime: **the general stationary axisymmetric metric, the dragging of inertial frames, stationary limit surfaces, event horizons, the Kerr metric, structure of a rotating black hole, trajectories of massive particles and photons, Penrose process. *

*Linearised gravity and gravitational waves: **weak field metric, linearised field equations, Lorenz gauge, wave solutions of linearised field equations. *

*Topics in italics are non-examinable, and might be omitted.*