Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds

Abstract

We study the Cauchy problem for the wave equation 2 g ψ = 0 on extreme Kerr backgrounds. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface Σ 0 which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of Σ 0 . We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T , and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N . Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H + . Published by Elsevier Inc.

Linear Waves in the Interior of Extremal Black Holes I

“…Quantitative decay estimates for axisymmetric solutions in extremal Kerr were proved in [8,9]. Conserved quantities along the event horizon can also arise for higher-spin equations on extremal Kerr [34].…”

Linear Waves in the Interior of Extremal Black Holes II

“…In [9] Aretakis extended the results of [8] to show non-decay and blow-up of higher-order derivatives of φ even in the case of data with vanishing Aretakis constants. There is still no proof of pointwise and energy boundedness or decay for non-axisymmetric φ in the exterior region of extremal Kerr (cf.…”

“…Aretakis considered in [8] axisymmetric solutions φ to (1.1) in the exterior region of extremal Kerr, arising from Cauchy data on a spacelike hypersurface Σ intersecting the event horizon H + ; see Fig. 1.…”

“…In [8], polynomial decay in affine time of axisymmetric φ and its tangential derivatives was shown to hold along the event horizon of extremal Kerr (a 2 = M 2 ) for suitably decaying Cauchy initial data. To obtain the analogue of (A) for axisymmetric φ in the extremal Kerr interior, we will assume the decay rates that follow from [8].…”

“…To obtain the analogue of (A) for axisymmetric φ in the extremal Kerr interior, we will assume the decay rates that follow from [8]. For the analogue of (A) for axisymmetric φ in extremal Kerr-Newman spacetimes with a 2 < M 2 and, moreover, for the analogues of (B) and (C) for axisymmetric φ in any extremal Kerr-Newman spacetime, Vol.…”