Abstract
We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr–Newman black holes. These results are the analogues of results obtained in Gajic (Commun Math Phys 353(2), 717–770, 2017) for extremal Reissner–Nordström and stand in stark contrast to previously established results for the subextremal case, where the local energy was shown to generically blow up at the Cauchy horizon.
Highlights
In the precursor [23] of this paper, we established the following results for the linear wave equation, gφ = 0, (1.1)
Extremal Kerr–Newman spacetimes constitute a twoparameter subfamily of spacetimes, satisfying the constraint M 2 = a2 + e2; they can be viewed as a continuous family that connects the extremal Reissner– Nordstrom solutions (a2 = 0) to the extremal Kerr solutions (a2 = M 2)
For the analogue of (A) for axisymmetric φ in extremal Kerr–Newman spacetimes with a2 < M 2 and, for the analogues of (B) and (C) for axisymmetric φ in any extremal Kerr–Newman spacetime, Vol 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4007 we assume polynomial decay in time of φ along the event horizon that is conjectured, but has not yet been proved, to hold
Summary
For the analogue of (A) for axisymmetric φ in extremal Kerr–Newman spacetimes with a2 < M 2 and, for the analogues of (B) and (C) for axisymmetric φ in any extremal Kerr–Newman spacetime, Vol 18 (2017) Linear Waves in the Interior of Extremal Black Holes II 4007 we assume polynomial decay in time of φ along the event horizon that is conjectured, but has not yet been proved, to hold. Again, the decay for φ along the event horizon that is expected to hold for suitably decaying Cauchy initial data in this setting This assumption is necessary for the analogue of (A), as the required polynomial decay has not yet been proved to hold for φ (without axisymmetry) along the event horizon of slowly rotating extremal Kerr–Newman. See related conjectures in the introduction of [23]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
