CMM PDE Seminar “A Nonlinear Operator Approach to Black Hole Solution Classes in the Ernst Equation”

Abstract: The Ernst equation arises from the dimensional reduction of the vacuum Einstein equations in the presence of two spacelike Killing vector fields and plays an important role in the description of stationary axisymmetric spacetimes. Originally introduced as a simplified scheme for constructing the Kerr metric, it has become a fundamental tool in the study of stationary vacuum solutions. More generally, its integrability provides a powerful mechanism for constructing large families of physically relevant solutions, including the classical black hole spacetimes.

In this work, we introduce a profile–phase decomposition of the Ernst equation that naturally induces a system of nonlinear differential operators encoding its intrinsic analytic structure. Rather than relying on a prescribed solution-generating ansatz, the induced operator system determines a broad class of admissible solutions through nonlinear differential constraints, providing an intrinsic description of the corresponding solution space. Within this framework, we establish concrete characterizations of black hole solution classes in terms of the induced nonlinear differential system. In particular, the Schwarzschild, Kerr, and Kerr–NUT spacetimes arise naturally as particular instances of the resulting solution class. This is ongoing work carried out jointly with Felipe Poblete.

Speaker: Jessica Trespalacios (Universidad Austral de Chile)