Instability of the massive Klein-Gordon field on the Kerr spacetime

2007/05/20 by Sam R. Dolan · 1 voice · 7 citations
Physics and Astronomy · #Astrophysical Phenomena and Observations #Black Holes and Theoretical Physics #Field (mathematics) #Instability #Klein–Gordon equation #Mathematical physics #Nonlinear system #Physics #Quantum Electrodynamics and Casimir Effect #Quantum mechanics #Scalar field #Spacetime #Superradiance #Upper and lower bounds #gr-qc #hep-th

paper · pdf · doi:10.1103/physrevd.76.084001

published as Phys.Rev.D76:084001,2007 · Added references. 27 pages, 7 figures

arxiv published 2007/05/20 · arxiv created 2007/06/05 · arxiv updated 2007/06/05 · openalex publication_date 2007/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We investigate the instability of the massive scalar field in the vicinity of a rotating black hole. The instability arises from amplification caused by the classical superradiance effect. The instability affects bound states: solutions to the massive Klein-Gordon equation which tend to zero at infinity. We calculate the spectrum of bound state frequencies on the Kerr background using a continued fraction method, adapted from studies of quasinormal modes. We demonstrate that the instability is most significant for the l = 1, m = 1 state, for M μ\lesssim 0.5. For a fast rotating hole (a = 0.99) we find a maximum growth rate of τ^-1 ≈ 1.5 × 10^-7 (GM/c³)^-1, at M μ≈ 0.42. The physical implications are discussed.

Citations

• Bound states and decay times of fermions in a Schwarzschild black hole background
• Black strings andp-branes are unstable
• Stability and quasinormal modes of the massive scalar field around Kerr black holes
• Large Mass Hierarchy from a Small Extra Dimension
• Regular coordinate systems for Schwarzschild and other spherical spacetimes
• Klein-Gordon equation and rotating black holes
• Stability of a Schwarzschild Singularity

Cited by

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• Superradiant dark matter production from primordial black holes: Impact of multiple modes and gravitational wave emission
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• Testing General Relativity with Present and Future Astrophysical Observations
• Fundamental fields around compact objects: Massive spin-2 fields,\n Superradiant instabilities and Stars with dark matter cores
• BLACK-HOLE BOMBS AT THE LHC
• Superradiant instability of charged scalar field in stringy black hole mirror system

Discussions

• Effectively the OP is asking if black holes in general relativity are classically stable. This is a notoriously difficult question answer. Early on some partial results were obtained. For example, Ber [stackexchange, 0 points]

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