XXVI International Fall Workshop on Geometry and Physics

Braga, 4-7 September 2017

One of the most striking features of the Einstein equations is the prediction of gravitational waves. The presence of waves in any system of equations is naturally linked to the notion of hyperbolicity. This is to be compared with the Poisson equation of Newtonian gravity, an elliptic equation. Hyperbolicity is tied to propagation phenomena and hence the natural problem for the Einstein equations is the initial value problem. This raises several questions. What are the initial data? What is a solution corresponding to these data? How do we transform a geometric equation, like the Einstein equations, into a system of partial differential equations and how do we construct the PDE data starting from the geometric data? The course will answer these questions. If time permits, I will then continue with the global Cauchy problem in general relativity, that is to say the study of asymptotics and global properties of solutions. Prerequisites and references: I will assume basic knowledge of Lorentzian geometry and partial differential equations. Basic references for Lorentzian geometry can be found in the book of O'Neil "Semi-Riemannian geometry", and for partial differential geometry, in the book of Fritz John "Introduction to Partial differential equations" or that of Evans "Partial differential equations". Two good references concerning the Cauchy problem in relativity are Ringström's book "The Cauchy problem in relativity" and Christodoulou's book "Mathematical problem of general relativity I".