Search for: All records

Black holes are important objects in our understanding of the universe, as they represent the extreme nature of General Relativity. The Kerr–Newman black hole is the most general asymptotically flat black hole solution and its stability properties have long been elusive due to the interaction between gravitational and electromagnetic radiations. We illustrate the main conjectures regarding the stability problem of known black hole solutions and present some recent theorems regarding the evolution of the Kerr–Newman black holes to coupled perturbations. 

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis exclusively in physical-space. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis [13] through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin [63] in the sub-extremal case, simplifying Aretakis’ derivation of Morawetz estimates through purely classical currents. 

The Carter tensor is a Killing tensor of the Kerr-Newman spacetime, and its existence implies the separability of the wave equation. Nevertheless, the Carter operator is known to commute with the D’Alembertian only in the case of a Ricci-flat metric. We show that, even though the Kerr-Newman spacetime satisfies the non-vacuum Einstein-Maxwell equations, its curvature and electromagnetic tensors satisfy peculiar properties which imply that the Carter operator still commutes with the wave equation. This feature allows to adapt to Kerr-Newman the physical-space analysis of the wave equation in Kerr by Andersson-Blue [4], which avoids frequency decomposition of the solution by precisely making use of the commutation with the Carter operator. We also extend the mathematical framework of physical-space analysis to the case of the Einstein-Maxwell equations on Kerr-Newman spacetime, representing coupled electromagnetic-gravitational perturbations of the rotating charged black hole. The physical-space analysis is crucial in this setting as the coupling of spin- 1 and spin-2 fields in the axially symmetric background prevents the separation in modes as observed by Chandrasekhar [19], and therefore represents an important step towards an analytical proof of the stability of the Kerr-Newman black hole.