Axially symmetric spacetimes: numerical and analytical perspectives

Sergio Dain

In this talk I will discuss some new aspects of axially symmetric spacetimes which open the door for future interplay between analytical and numerical studies. These new developments are based on the role of the total mass in axial symmetry.

The mass plays two important roles in axial symmetry. First, there exists a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution and also it allows to impose outer boundary conditions in a natural way. Second, extreme Kerr black holes are minimizer of the mass. I will discuss recent developments on both issues. At the end, I will present relevant open problems that can be hopefully solved with an interaction between numerical and analytical insights.