Introduction to Mathematical Relativity
517 & 532 or permission of instructor
This is an introductory course on Einstein's theory of general relativity and gravitation, emphasizing the geometric-PDEs point of view of the equations and the recent advances in the field. Although welcomed and very much appreciated, no previous knowledge of physics is assumed. Some knowledge of differential geometry and PDEs is helpful. The following is an outline of the course:
I. The Geometry of Space-time:
Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit, The Penrose singularity theorem. Black holes and cosmic censorship conjectures. Homogeneous and isotropic solutions.
II. The Cauchy problem for Einstein Vacuum Equations (EVE):
The symbol and the characteristics of EVE. The local existence theorem in wave coordinates. Local existence using maximal hypersurfaces.
III. Conservation Laws and Noether's Theorem:
Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps, and for sections of vector bundles. Asymptotic flatness. The definition of global energy, momentum and angular momentum. The Positive Energy Theorem.
IV. Reduction under Symmetry:
Kerr and Kerr-Newman solutions. Cosmological solutions and asymptotically flat solutions. Spherically symmetric solutions of vacuum Einstein and Einstein-Maxwell equations.