This course will emphasize the math of flat and curved spacetime with applications to Electromagnetism, Special Relativity and General Relativity.
Vectors and dual vectors in spacetime. General Lorentz transformations: boosts and rotations. The Lorentz and Poincaré groups: any element of the proper Lorentz group can be expressed as the product of a boost and a 3-D space rotation. Tensor analysis in flat spacetime. The electromagnetic field tensor and its dual form. Special Relativity and Maxwell's Equations expressed in tensor form. Transformations of electromagnetic fields under general Lorentz transformations. Applications. The energy-momentum tensor for matter and energy. The energy-momentum tensor in E&M.
Part 2 :
Curved spacetime. Vectors and tensors in curved spacetime. Parallel transport of vectors and tensors. Geodesics. Covariant derivatives. Maxwell equations in curved spacetime. The Riemann curvature tensor. Einstein's equations. The Schwarzchild solution. Static black holes. Lie Derivatives. Killing fields and conserved quantities.
SPECIAL INSTRUCTIONS AND/OR PREREQUISITES:
Knowledge of Special Relativity and Electricity and Magnetism (Physics CS 33, 34 and 35), and the subscript notation (Physics CS 31,32).
This course assumes a working knowledge of the subscript notation in vector calculus, as well as operations with matrices, and basic linear algebra.
BERNARD F. SCHUTZ A FIRST COURSE IN GENERAL RELATIVITY 2nd Ed CAMBRIDGE UNIVERSITY PRESS P.A.M. DIRAC, GENERAL THEORY OF RELATIVITY PRINCETON UNIVERSITY PRESS