Abstract: These lectures will begin by introducing the fundamental notion of "black hole" in general relativity, as exemplified by two elementary families of explicit solutions to the Einstein vacuum equations, the celebrated Schwarzschild and the more complicated Kerr metrics. The lectures will then turn to the problem of studying the propagation of waves on these backgrounds. A recurring theme will be the close connection between the analysis of hyperbolic equations and well known geometric/physical features of the underlying spacetime, for instance the red-shift effect, the photon sphere, and the ergo-region. These considerations are of crucial importance for the question of the nonlinear stability of the black hole spacetimes themselves as solutions to the Einstein vacuum equations, a great open question in the subject.
The lectures only assume basic differential geometry and analysis. Specifically Lorentzian geometric notions, as well as the most basic well-posedness statements for the wave equation, will be developed from the beginning.
Relevant references include the following, both written jointly with Igor Rodnianski: the lecture notes "Lecture notes on black holes and non-linear waves", in Evolution equations, Clay Mathematics Proceedings, Vol. 17. Amer. Math. Soc., Providence, RI, 2013, pp. 97-205, and the survey article "The black hole stability problem for linear scalar perturbations", in Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, 2011, pp. 132-189. The textbooks "The Cauchy problem in General Relativity", by Hans Ringstrom, published by the EMS, and "Partial Differential Equations in General Relativity", by Alan Rendall, published by Oxford University Press, may also be useful.
Time: Wednesdays 10:15-12
Auditorium: HG G43 (HWZ)
Begins: September 25