In these lectures we give a general introduction to the theory of gravitational waves and the analytic approximation methods in general relativity. More precisely we focus on the theory which is necessary to accurately and reliably predict the gravitational waves generated by compact binary systems, made of black holes or neutron stars. The predictions are used in the form of gravitational-wave “templates” in the data analysis of the detectors LIGO, VIRGO, ... LISA. In particular we present the state-of-the-art on the post-Newtonian approximation in general relativity, which is the main tool for describing the famous gravitational wave “chirp” of compact binary systems. The outline of the lectures is :
1. Gravitational wave events
2. Methods to compute gravitational waves
3. Einstein quadrupole formalism
4. Post-Newtonian parameters
5. Finite size effects in compact binaries
6. Synergy with the effective field theory
7. Radiation reaction and balance equations. [-]

Conventional quantum field theory techniques do not work for extracting physical information from a background-independent quantum theory of gravity. A technique that works is Oeckl's boundary formalism, with semiclassical coherent states on the boundary. I illustrate how this technique has allowed us to compute the lifetime of a black hole in loop quantum gravity. This is an astrophysical relevant quantity that could have observational consequences.[-]

In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as (potentially) normally hyperbolic trapping, as well as the role of resonances.[-]

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.
I will discuss the geometry of these black holes, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In this talk, I will emphasize geometric aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.[-]