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| Schwarzschild-de Sitter | |
|---|---|
| Name | Schwarzschild–de Sitter metric |
| Field | General relativity |
| Discoverer | Karl Schwarzschild; Willem de Sitter |
| Year | 1916; 1917 |
Schwarzschild-de Sitter is a family of static spherically symmetric solutions of Einstein field equations describing a point mass embedded in a spacetime with positive cosmological constant, combining the works of Karl Schwarzschild and Willem de Sitter. It plays a central role in theoretical studies involving Albert Einstein's cosmological constant, quantum field theory on curved spacetime associated with Stephen Hawking radiation, and gravitational lensing analyzed since the era of Arthur Eddington and Fritz Zwicky. The solution links classical results from David Hilbert and Roy Kerr with modern investigations by researchers at institutions such as Max Planck Society and Princeton University.
The Schwarzschild–de Sitter solution extends the vacuum Schwarzschild metric introduced by Karl Schwarzschild to include a positive cosmological term first considered explicitly by Willem de Sitter in the context of early modern cosmology debates involving Alexander Friedmann, Georges Lemaître, and Édouard Goursat. It is widely discussed in reviews influenced by the work of Subrahmanyan Chandrasekhar, John Wheeler, and Roger Penrose, and serves as a pedagogical example in texts by Steven Weinberg, Misner Thorne Wheeler, and researchers at Cambridge University and Harvard University. Applications span studies by groups at Caltech, Stanford University, and University of Cambridge addressing perturbation theory and numerical relativity linked to projects like Event Horizon Telescope.
In static coordinates the line element is written similarly to the Schwarzschild form introduced by Karl Schwarzschild and exploits the spherical symmetry explored by Friedrich Bessel and Simon Newcomb in geodesy contexts; it contains parameters tied to mass associated historically with Karl Schwarzschild and the cosmological constant examined by Albert Einstein and Willem de Sitter. The metric's radial function involves roots whose structure relates to algebraic techniques used by Évariste Galois and analytic methods by Bernhard Riemann; studies by Kip Thorne, Vladimir Belinski, and Andrei Linde analyze stability and perturbations. The solution's invariants and curvature scalars are computed in works citing Isaac Newton-style potentials and modern tensor calculus developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita, and are used in tests of gravitational redshift akin to experiments by Pound Recoil, Robert Pound, and Glen Rebka.
The spacetime exhibits multiple Killing horizons analogous to features in metrics studied by Roy Kerr and Jacob Bekenstein, including a black hole event horizon and a cosmological horizon discussed in connection with the de Sitter horizon analyzed by Gibbons Hawking and Stephen Hawking. Penrose diagrams used by Roger Penrose, Brandon Carter, and Ezra Newman illustrate causal structure and are central to causal set approaches advocated by Renate Loll and Rafael Sorkin. The interplay of horizons connects to thermodynamic discussions pioneered by Jacob Bekenstein and experiments conceptually linked to Hermann Bondi's ideas on energy in general relativity; global structure is compared to asymptotically flat spacetimes studied by Arnowitt Deser Misner.
Timelike and null geodesics follow equations comparable to those analyzed for Schwarzschild and Kerr spacetimes by researchers such as S. Chandrasekhar, Brandon Carter, and Saul Teukolsky. Orbital dynamics incorporate precession effects reminiscent of classic measurements involving Mercury (planet) and historical observations by Urbain Le Verrier, and light deflection relevant to work by Arthur Eddington and Einstein; recent numerical studies are performed by groups at Max Planck Institute for Gravitational Physics and Kavli Institute for Particle Astrophysics and Cosmology. Stability of circular orbits is examined in parallels to accretion disk analyses by Katherine Blundell and relativistic jet models associated with Heinz Florian and Roger Blandford.
Surface gravity and associated temperatures for each horizon are treated using methods from black hole thermodynamics developed by Stephen Hawking, Jacob Bekenstein, and James York Jr., with entropy discussions influenced by ideas from Leonard Susskind and Gerard 't Hooft. Semi-classical analyses referencing Quantum Field Theory techniques attributed to Richard Feynman and path integral approaches developed by Gibbons Hawking provide the basis for evaluating Hawking-like radiation and backreaction studies pursued at Perimeter Institute and Institute for Advanced Study. Connections are made to cosmological temperature scales relevant to research by George Smoot and John Mather.
Various coordinate systems—such as static coordinates, Painlevé–Gullstrand-like coordinates inspired by Allvar Gullstrand and Paul Painlevé, and Kruskal-like extensions following Martin Kruskal and George Szekeres—are used to extend the manifold across horizons, mirroring techniques applied by Roger Penrose and Brandon Carter. Analytic continuations and maximal extensions are discussed in studies at University of Cambridge and Princeton University and relate to Euclidean methods by Gibbons Hawking and instanton approaches associated with Sidney Coleman. Numerical coordinate implementations appear in codes developed at NASA and European Space Agency centers for relativistic simulations.
Schwarzschild–de Sitter models inform investigations into isolated astrophysical objects within expanding cosmological backgrounds studied by Alan Guth, Andrei Linde, and researchers of inflation (cosmology), and are used in lensing calculations by teams including Kip Thorne and Massimo Meneghetti. They provide simplified arenas for quantum gravity proposals from Loop Quantum Gravity groups led by Carlo Rovelli and string-theory analyses connected to Edward Witten and Juan Maldacena. Observational implications are considered in the context of surveys by Sloan Digital Sky Survey, Dark Energy Survey, and missions such as Planck (satellite) and Euclid (spacecraft), while theoretical work continues at institutions like CERN and Perimeter Institute.