Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

Abstract

We prove that there are no non-stationary (with respect to the Hawking vectorfield) real mode solutions to the Teukolsky equations on all $(3+1)$-dimensional subextremal Kerr-anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking-Reall bound and $\left|a\sqrt{-\Lambda }\right|<\frac{\sqrt{3}}{20}$. We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky-Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking-Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime.

30 pages, 4 figures. A Mathematica notebook containing all the computations is attached as ancillary file. v2 includes Teukolsky-Starobinsky conservation laws and a comparison with the asymptotically flat case. v3 agrees with the journal version and fixes in addition a typo in equation (1.4a) and a small error in the proof of Lemma 2.5 (statement unchanged)

References in corpus (6)

• Kerr-CFT and gravitational perturbations
• Hidden spectral symmetries and mode stability of subextremal Kerr(-dS) black holes
• A proof of the instability of AdS for the Einstein--massless Vlasov system
• Kerr stability for small angular momentum
• The Teukolsky--Starobinsky constants: facts and fictions
• Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes away from extremality

Cited by in corpus (2)

• Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes
• Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes