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The Carter tensor is a Killing tensor of the Kerr-Newman spacetime, and its existence implies the separability of the wave equation. Nevertheless, the Carter operator is known to commute with the D’Alembertian only in the case of a Ricci-flat metric. We show that, even though the Kerr-Newman spacetime satisfies the non-vacuum Einstein-Maxwell equations, its curvature and electromagnetic tensors satisfy peculiar properties which imply that the Carter operator still commutes with the wave equation. This feature allows to adapt to Kerr-Newman the physical-space analysis of the wave equation in Kerr by Andersson-Blue [4], which avoids frequency decomposition of the solution by precisely making use of the commutation with the Carter operator. We also extend the mathematical framework of physical-space analysis to the case of the Einstein-Maxwell equations on Kerr-Newman spacetime, representing coupled electromagnetic-gravitational perturbations of the rotating charged black hole. The physical-space analysis is crucial in this setting as the coupling of spin- 1 and spin-2 fields in the axially symmetric background prevents the separation in modes as observed by Chandrasekhar [19], and therefore represents an important step towards an analytical proof of the stability of the Kerr-Newman black hole.
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Electromagnetic-gravitational perturbations of Kerr–Newman spacetime: The Teukolsky and Regge–Wheeler equations
We derive the equations governing the linear stability of Kerr–Newman spacetime to coupled electromagnetic-gravitational perturbations. The equations generalize the celebrated Teukolsky equation for curvature perturbations of Kerr, and the Regge–Wheeler equation for metric perturbations of Reissner–Nordström. Because of the “apparent indissolubility of the coupling between the spin-1 and spin-2 fields”, as put by Chandrasekhar, the stability of Kerr–Newman spacetime cannot be obtained through standard decomposition in modes. Due to the impossibility to decouple the modes of the gravitational and electromagnetic fields, the equations governing the linear stability of Kerr–Newman have not been previously derived. Using a tensorial approach that was applied to Kerr, we produce a set of generalized Regge–Wheeler equations for perturbations of Kerr–Newman, which are suitable for the study of linearized stability by physical space methods. The physical space analysis overcomes the issue of coupling of spin-1 and spin-2 fields and represents the first step towards an analytical proof of the stability of the Kerr–Newman black hole.
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- PAR ID:
- 10327394
- Date Published:
- Journal Name:
- Journal of Hyperbolic Differential Equations
- Volume:
- 19
- Issue:
- 01
- ISSN:
- 0219-8916
- Page Range / eLocation ID:
- 1 to 139
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0219891622500011
