The Trapping Effect on Degenerate Horizons

Abstract

We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner–Nordström up to and including the future event horizon. We show that such an estimate requires (1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and (2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a new $L^{2}$ concentration phenomenon (namely, a global trapping effect, in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces. Our methods and results play a crucial role in our upcoming works on linear and nonlinear wave equations on extremal black hole backgrounds.