The trapping effect on degenerate horizons

Understand

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-Nordstrom 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 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).