de Sitter–Schwarzschild solution
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| de Sitter–Schwarzschild solution | |
|---|---|
| Name | de Sitter–Schwarzschild solution |
| Field | General relativity |
| Introduced by | Willem de Sitter, Karl Schwarzschild |
| Year | 20th century |
de Sitter–Schwarzschild solution is a static, spherically symmetric vacuum solution of Einstein field equations with a positive cosmological constant, combining features of the de Sitter space and the Schwarzschild metric. It models a nonrotating, uncharged mass embedded in an expanding background and interpolates between the asymptotically flat Schwarzschild metric and the maximally symmetric de Sitter space. The solution is central in studies linking black hole physics, cosmology, and semiclassical effects in spacetimes with cosmological constant.
Introduction
The de Sitter–Schwarzschild solution arose in the context of early 20th‑century developments by Karl Schwarzschild and Willem de Sitter addressing exact solutions of the Einstein field equations, later formalized by researchers working on cosmology and relativistic gravitation. It embeds the Schwarzschild mass parameter into the de Sitter background characterized by a positive Λ, and has been explored by authors studying the interplay between Mach's principle debates, the Friedmann equations, and the structure of cosmological horizons. The spacetime is frequently invoked in analyses by scholars of Stephen Hawking and Roger Penrose on singularity theorems and thermodynamics.
Metric and derivation
The line element is given in static coordinates by a lapse function that includes the Schwarzschild term and a Λ term; derivations follow from solving the Einstein field equations with T_{μν}=0 and Λ>0 under the assumptions of spherical symmetry and stationarity used by Karl Schwarzschild and generalized in later expositions by Arthur Eddington and Georges Lemaître. Coordinate charts linking static coordinates to global de Sitter space charts have been used in treatments by Maximilian Planck and Albert Einstein-era authors to analyze geodesic structure and conserved quantities associated with Killing vectors like those discussed in works by Emmy Noether and Élie Cartan. The metric admits a mass parameter M and cosmological radius R_Λ∝Λ^{-1/2}, showing limiting cases studied in classical papers by Subrahmanyan Chandrasekhar and later reviews by John Wheeler.
Horizons and causal structure
This solution exhibits multiple Killing horizons: an inner black hole horizon and an outer cosmological horizon, whose locations depend on M and Λ; their properties are treated with techniques from the causal analysis of Roger Penrose and conformal compactification methods used by Roger Penrose and Klaus Heun in related contexts. The causal diagram shares features with the Penrose diagrams used for the Reissner–Nordström metric and Kerr metric but lacks the angular momentum structure of Kerr spacetimes; analysis of global structure draws on methods from studies by Hermann Weyl and Stephen Hawking. Extremal limits where horizons coincide correspond to parameter values analogous to extremal cases examined by Paul Ehrenfest and are important in thermodynamic interpretations following work by Jacob Bekenstein.
Physical properties and limits
In the small‑Λ limit the geometry reduces locally to the Schwarzschild solution studied by Karl Schwarzschild and in the small‑M limit it approaches de Sitter space as characterized by Willem de Sitter; these limits connect to cosmological models of the Friedmann–Lemaître–Robertson–Walker metric used by Georges Lemaître and Alexander Friedmann. Thermodynamic quantities such as surface gravities and associated Hawking temperatures generalize results obtained by Stephen Hawking and James Hartle; the entropy assignments relate to the area law emphasized in the work of Jacob Bekenstein and the holographic ideas later advanced by Gerard 't Hooft and Leonard Susskind. The solution also allows study of mass definitions in asymptotically nonflat settings, invoking concepts from ADM mass analyses by Richard Arnowitt, Stanley Deser, and Charles Misner, and quasi‑local mass constructs explored by Kip Thorne collaborators.
Stability and perturbations
Linear and nonlinear perturbation analyses of the de Sitter–Schwarzschild background build on techniques developed for black holes by Tullio Regge and Wheeler and on quasinormal mode computations refined by researchers such as Saul Teukolsky and Vitor Cardoso. Studies examine scalar, electromagnetic, and gravitational perturbations, linking to work on mode stability in Reissner–Nordström and Kerr–Newman contexts by Leor Barack and Hiroyuki Nakano. Semiclassical backreaction and evaporation scenarios draw on methods from Stephen Hawking and Bill Unruh, while numerical relativity treatments employ formalisms developed by Miguel Alcubierre and Gunnar Andersson in time evolution and constraint damping.
Applications and significance
The de Sitter–Schwarzschild solution underpins models of primordial black holes in inflationary cosmology studied by Alan Guth and Andrei Linde, and informs investigations into black hole thermodynamics in de Sitter backgrounds pursued by Ted Jacobson and Raphael Bousso. It serves as a testing ground for semiclassical gravity, quantum field theory in curved spacetime as developed by Nicolás Birrell and Paul Davies, and for conjectures relating to the holographic principle by Juan Maldacena. The geometry also appears in discussions of cosmic censorship advanced by Roger Penrose and in observational considerations linked to the Lambda cold dark matter model employed by the Planck collaboration and WMAP researchers.
Related solutions and extensions
Extensions include charged de Sitter–Reissner–Nordström spacetimes analyzed in works by Gustav Reissner and Hendrik Lorentz-inspired treatments, rotating generalizations related to the Kerr–de Sitter metric introduced following Roy Kerr's methods, and higher‑dimensional analogues studied in the context of Kaluza–Klein theory and string theory by authors such as Juan Maldacena and Andrew Strominger. Other related constructions involve Nariai limits explored by Hideki Nariai and regular black hole models investigated by Bardeen and researchers in loop quantum gravity like Carlo Rovelli and Abhay Ashtekar.