Jonatan Lenells: Solutions of boundary value problems for the stationary axisymmetric Einstein equations

For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with Dirichlet boundary conditions imposed along a disk, Neugebauer and Meinel in the 1990s derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. I will discuss some other boundary value problems for the Ernst equation that can also be solved exactly. The exact solutions are obtained by using the integrable structure of the Ernst equation.