Kerr–Newman
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| Kerr–Newman | |
|---|---|
| Name | Kerr–Newman |
| Type | Solution of Einstein–Maxwell equations |
| Discovered | 1965 |
| Discoverers | Roy Kerr, Ezra Newman, Alfred Schild, Brandon Carter |
| Parameters | mass, angular momentum, electric charge |
| Metric | Rotating charged black hole |
Kerr–Newman
The Kerr–Newman solution is an exact, stationary, axisymmetric solution of the Einstein field equations coupled to Maxwell's equations describing an electrically charged, rotating black hole. It generalizes the Schwarzschild metric, the Reissner–Nordström metric, and the Kerr metric by incorporating both angular momentum and electric charge, and it underpins theoretical studies in general relativity, black hole thermodynamics, and classical limits of quantum field theory in curved spacetime. The solution is central to discussions connecting the Carter constant, the Noether theorem, and properties of event horizons appearing in the Penrose process and the Hawking radiation framework.
Introduction
The Kerr–Newman spacetime was obtained in the context of efforts by Roy Kerr, Ezra Newman, Alfred Schild, and Brandon Carter to extend rotating solutions after the discovery of the Kerr metric and the charged, nonrotating Reissner–Nordström case. The solution is stationary and axisymmetric like the Kerr–Newman family, admits an asymptotically flat metric related to the Minkowski space limit, and possesses conserved quantities akin to those present in analyses by Noether, Carter, and Wald. The metric plays a role in tests of cosmic censorship conjecture, comparisons with the Majumdar–Papapetrou solution, and studies involving the Ernst equation and solution-generating techniques used by Belinski–Zakharov and Geroch.
Metric and properties
The Kerr–Newman metric is written in Boyer–Lindquist coordinates derived from transformations used in the Kerr metric and shares structure with metrics in the Newman–Penrose formalism developed by Ezra Newman and Roger Penrose. The line element depends on mass parameter M, angular momentum per unit mass a (related to Brandon Carter's analysis), and electric charge Q appearing in the electromagnetic four-potential first considered in works by Ernst and Carter. Key surfaces include the outer and inner horizons determined by roots analogous to those in the Reissner–Nordström case, an ergosphere bounded by the static limit related to processes first studied in the context of the Penrose process and discussions by Zeldovich, and a ring singularity analogous to structures in the Kerr metric studied by Israel and Siklos. The solution conserves energy and angular momentum in asymptotic regimes examined in analyses by Arnowitt–Deser–Misner and in quasi-local mass frameworks influenced by Brown and York.
Derivation and solution
Derivations employ techniques from the Newman–Penrose formalism, complex coordinate transformations similar to Newman–Janis algorithms, and solution-generating methods involving the Ernst potential used by Ernst, Harrison, and others. Original derivations built on earlier charged solutions like Reissner–Nordström and rotating vacuum solutions like Kerr metric by applying complex transformations attributed to Newman and Janis and exploiting algebraically special classifications introduced by Aleksey Petrov. The electromagnetic field is given by a vector potential consistent with James Clerk Maxwell's equations in curved space and matches asymptotic multipole expansions discussed by Kip Thorne and Geroch. This approach links the solution to inverse scattering methods developed by Belinski–Zakharov and integrability studies by Maison.
Physical interpretation and astrophysical relevance
Physically the Kerr–Newman solution models an isolated, rotating, charged compact object in asymptotically flat spacetime, and its multipole moments connect to frameworks used by Kip Thorne, R. O. Hansen, and Geroch to characterize astrophysical bodies. While astrophysical black holes are expected to be nearly neutral due to charge neutralization by surrounding plasma studied in Blandford–Znajek processes and Stuart Shapiro analyses, the Kerr–Newman metric remains a theoretical laboratory for examining magnetospheres similar to those in Pulsar and AGN models, for analyzing charged particle dynamics in contexts studied by Valeri Frolov and Brian Punsly, and for probing interactions with scalar fields and particle production as in Stephen Hawking's work. Comparisons to numerical results from simulations by groups using codes influenced by Einstein Toolkit and studies of inspiral in binary black hole mergers provide tests of strong-field predictions that relate to observations from LIGO, Virgo, and Event Horizon Telescope imaging.
Stability, perturbations, and thermodynamics
Perturbative analyses of Kerr–Newman spacetime draw on the Teukolsky equation introduced by Saul Teukolsky, quasinormal mode studies explored by Kostas Kokkotas and Emanuele Berti, and superradiant scattering first discussed by Zeldovich and applied by Press and Teukolsky. Thermodynamic properties follow the laws of black hole mechanics formulated by Bardeen, Carter, and Hawking and the statistical insights of Stephen Hawking and Bekenstein. Entropy and temperature formulas for rotating charged holes parallel those for Kerr metric and Reissner–Nordström cases and feature in holographic considerations linked to AdS/CFT correspondence work by Juan Maldacena and microstate counts inspired by Strominger–Vafa. Stability results often require analyses by numerical relativity teams including groups led by Frans Pretorius and perturbative frameworks by Subrahmanyan Chandrasekhar.
Extensions and related spacetimes
Generalizations include embedding in de Sitter or anti-de Sitter backgrounds producing Kerr–Newman–de Sitter solutions studied in contexts by G. W. Gibbons, charged dilatonic rotating solutions arising in string-inspired models by Gibbons–Maeda and Garfinkle–Horowitz–Strominger, and higher-dimensional rotating charged solutions studied by Myers–Perry and in supergravity frameworks by K. S. Stelle and Chris Hull. Connections to solution-generating techniques involve the inverse scattering approach by Belinski, dualities used in string theory by Joe Polchinski and Juan Maldacena, and geometric analyses influenced by Stephen Hawking and Geroch. The metric informs studies of naked singularity scenarios relevant to debates initiated by Roger Penrose and cosmic censorship investigations pursued by Christodoulou and Dafermos.