Instability results for the wave equation in the interior of Kerr black holes

Abstract

We prove that a large class of smooth solutions ψ to the linear wave equation g ψ = 0 on subextremal rotating Kerr spacetimes which are regular and decaying along the event horizon become singular at the Cauchy horizon. More precisely, we show that assuming appropriate upper and lower bounds on the energy along the event horizon, the solution has infinite (non-degenerate) energy on any spacelike hypersurfaces intersecting the Cauchy horizon transversally. Extrapolating from known results in the Reissner-Nordström case, the assumed upper and lower bounds required for our theorem are conjectured to hold for solutions arising from generic smooth and compactly supported initial data on a Cauchy hypersurface. This result is motivated by the strong cosmic censorship conjecture in general relativity.

Linear Waves in the Interior of Extremal Black Holes I

“…See also upcoming results by Franzen [26]. The above inextendibility result (Theorem B) has been extended to subextremal Kerr by Luk-Sbierski [38], under suitable lower bound assumptions for solutions to (1.1) along the event horizon. A similar inextendibility result has also been obtained by Dafermos-Shlapentokh-Rothman via a scattering theory approach [23].…”

Boundedness of Massless Scalar Waves on Kerr Interior Backgrounds

“…Proof. As was already pointed out in [27] and also [47], in Kerr spacetime, by ϕ τ and ϕ φ we denote diffeomorphisms generated by the Killing fields ∂ t and ∂ φ , respectively. These contain the diffeomorphisms generated by T H + , defined in (C3).…”

“…In the following we will carry out the coordinate transformation (r, t, φ, θ) → (u, v,φ, θ ), since double null coordinates will turn out more convenient for our interior analysis. Using (46), (47) and dt = dv − du in (23) we obtain…”

“…Theorem 6.3. ("Poor man's" linear Kerr instability without symmetries, Luk and Sbierski, [47]). Let ψ be a solution of (1).…”

“…A brief overview on this topic as well as a mathematical formulation of the conjecture were already given in Section 1.2 of [30]. References discussing the instability behavior in similar settings are [44,47]. Also refer to [62] for a numerical analysis and [6,52,58] for early discussions of heuristic models.…”