Theory of Relativity

Aim

After completing the course, the student should be able to

• choose and apply methods to problems in Differential Geometry, Special Relativity, and General Relativity
• Present and justify solutions to tasks using relevant concepts and clear reasoning

Content

Part A: Differential Geometry

Manifolds and coordinate systems. Tangent vectors. Tensors. Metric tensors and line elements. Spacetimes. Commutators. Lie derivative. Covariant derivative. Connections. Christoffel symbols. Geodesic coordinates. Parallel transport. Geodesics. Curvature tensors of Riemann, Ricci, Weyl and Einstein. Symmetries. Integrability and flat spacetimes. Calculus of variations and geodesics. Isometries och Killing vectors.

Del B: Special Relativity

Inertial systems. k-calculus. Addition of velocities. Simultaneity. Lorentz transformations, length contraction och time dilation. Proper time. Transformation of velocity and acceleration. Comoving systems and uniform acceleration. Relativistic mechanics, relativistic mass and momentum. Energy. Doppler effect. Energymomentum tensors. Dust and perfect fluids. Electromagnetism.

Del C: General Relativity

Physical principles. Geodesic deviation. Einstein’s equations in vacuum. Weak fields. Particle motion. Linearized field equations. Einstein’s equations with matter. Structure of the field equations. The Cosmological constant. Stationary and static spacetimes. Spherical symmetry. The Schwarzschild solution. Planetary motion according to Newton and Einstein. Advance of the perihelion of Mercury. Bending of light. Gravitational redshift. Time delay of light. Singularities and black holes. Gravitational waves. Relativistic cosmology. Robertson-Walker’s line element. Friedmann’s equation.