Fourth Minho Meeting on
Mathematical Physics

Gustav Holzegel (Imperial College, London, U.K.)
Wave equations on Black Hole Spacetimes and the Stability Problem

I will start by motivating the study of the linear wave equation on black holes spacetimes and develop the necessary tools for its analysis, most prominently the vectorfield method (Lecture 1). I will then prove the uniform boundedness of waves on Schwarzschild, and at least state and explain the decay statements which have been proven for Schwarzschild and Kerr (Lectures 2+3). In the last lecture, I will explain our recent proof of the linear stability of the Schwarzschild spacetime under gravitational perturbations (Lecture 4). The latter is joint work with Dafermos and Rodnianski.

Carlos Herdeiro (Univ. Aveiro, Portugal)
Kerr black holes with scalar hair

Black holes are one of the most fascinating predictions of the General Theory of relativity. According to the conventional picture that emerged in the 1970s as a corollary of the uniqueness theorems, black holes are extremely constrained objects, determined only by a few global charges. For instance, two black holes with the same total mass and angular momentum must be precisely equal, in sharp contrast with stars. Such simplicity of black holes became immortalized in John Wheeler's mantra "Black holes have no hair". In this talk, I will discuss a novel mechanism that allows black holes to have 'hair' and challenges the standard paradigm. Some possible astrophysical consequences will be addressed.

Pedro Girão (IST, Lisbon, Portugal)
On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant

This talk is dedicated to the following problem: given spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant, with the data on the outgoing initial null hypersurface given by a subextremal Reissner-Nordstrom black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a ``suitably regular'' Lorentzian manifold. First, we establish the well posedness of the characteristic problem. Second, we study the stability of the radius function at the Cauchy horizon. Third, we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions which are classical solutions of the Einstein equations. These results comprise the trilogy arXiv:1406.7245, arXiv:1406.7253 and arXiv:1406.7261 by J. Costa, P. Girão, J. Natário and J. Silva.