Celebratio Mathematica

by Vincent Moncrief and Nishanth Gudapati

We consider the axisymmetric, linear perturbations of Kerr–Newman black holes, allowing for arbitrarily large (but subextremal) angular momentum and electric charge. By exploiting the famous Carter–Robinson identities, developed previously for the proofs of (stationary) black hole uniqueness results, we construct a positive definite energy functional for these perturbations and establish its conservation for a class of (coupled, gravitational and electromagnetic) solutions to the linearized field equations. Our analysis utilizes the familiar (Hamiltonian) reduction of the field equations (for axisymmetric geometries) to a system of wavemap fields coupled to a \( 2{+}1 \)-dimensional Lorentzian metric on the relevant quotient 3-manifold. The propagating “dynamical degrees of freedom” of this system are entirely captured by the wavemap fields, which take their values in a 4-dimensional, negatively curved (complex hyperbolic) Riemannian target space, whereas the base-space Lorentzian metric is entirely determined, in our setup, by elliptic constraints and gauge conditions.

The associated linearized equations are analyzed with insight derived from the so-called “linearization stability” program for such (generally covariant) systems. In particular this program provides a natural connection between the (conserved, positive definite) energy defined for first-order perturbations and the correction to the ADM mass induced therefrom at second order. A well-known technique allows one to generate, for sufficiently smooth perturbations, a sequence of higher-order (conserved, positive definite) energies that, in turn, bound certain higher-order (weighted) Sobolev norms of the linearized solutions. We anticipate that our results may prove useful in analyzing the dynamical stability of (arbitrarily rapidly rotating) Kerr–Newman black holes with respect to axisymmetric perturbations. Establishing such stability at the linearized level is expected to be an essential first step in dealing, ultimately, with the nonlinear problem.

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Yvonne Choquet-Bruhat

A positive definite energy functional for the axisymmetric perturbations of Kerr–Newman black holes

by Vincent Moncrief and Nishanth Gudapati