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The General Solution of Bianchi Type III Vacuum Cosmology
General Relativity and Quantum Cosmology · Submitted 17 Jul 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The second order Ordinary Differential Equation which describes the unknown part of the solution space of some vacuum Bianchi Cosmologies is completely integrated for Type III, thus obtaining the general solution to Einstein's Field Equations for this case, with the aid of the sixth Painlev\'{e} transcendent $P_{VI}$.
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The abstract, sentence by sentence
The second order Ordinary Differential Equation which describes the unknown part of the solution space of some vacuum Bianchi Cosmologies is completely integrated for Type III, thus obtaining the general solution to Einstein's Field Equations for this case, with the aid of the sixth Painlev\'{e} transcendent $P_{VI}$. For particular representations of $P_{VI}$ we obtain the known Kinnersley two-parameter space-time and a solution of Euclidean signature. The imposition of the space-time generalization of a "hidden" symmetry of the generic Type III spatial slice, enables us to retrieve the two-parameter subfamily without considering the Painlev\'{e} transcendent.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The second order Ordinary Differential Equation which describes the unknown part of the solution space of some vacuum Bianchi Cosmologies is completely integrated for Type III, thus obtaining the general solution to Einstein's Field Equations for this case, with the aid of the sixth Painlev\'{e} transcendent $P_{VI}$. | result 0.55 | 0.92 | 0.91 | asserted 1.00 |
| s1 | For particular representations of $P_{VI}$ we obtain the known Kinnersley two-parameter space-time and a solution of Euclidean signature. | result 0.99 | 0.86 | 0.08 | asserted 1.00 |
| s2 | The imposition of the space-time generalization of a "hidden" symmetry of the generic Type III spatial slice, enables us to retrieve the two-parameter subfamily without considering the Painlev\'{e} transcendent. | result 0.41 | 0.84 | 0.10 | asserted 1.00 |
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The paper
Original abstract
The second order Ordinary Differential Equation which describes the unknown part of the solution space of some vacuum Bianchi Cosmologies is completely integrated for Type III, thus obtaining the general solution to Einstein's Field Equations for this case, with the aid of the sixth Painlev\'{e} transcendent $P_{VI}$. For particular representations of $P_{VI}$ we obtain the known Kinnersley two-parameter space-time and a solution of Euclidean signature. The imposition of the space-time generalization of a "hidden" symmetry of the generic Type III spatial slice, enables us to retrieve the two-parameter subfamily without considering the Painlev\'{e} transcendent.
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How the literature uses it
6 arXiv papers cite this in 12 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 1 paper
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It is worth noticing that all types apart from VIII and IX have an isometry group with a 2-dimensional Abelian subgroup, and when this group acts orthogonally-transitively the Einstein vacuum equations in any signature are integrable and can be solved by the twistor methods of Mason and Woodhouse and collaborators, [ref] , following [ref] and [ref] [In particular all the examples in [ref] and [ref] do have orthogonally transitive $G_2$ actions, and this is in a sense the reason why the integrations there lead to Painlevé equations.]. 5pt In this paper we set out to find real non-diagonal cohomogeneity-one Einstein metrics.
Non-diagonal four-dimensional cohomogeneity-one Einstein metrics in various signatures · Introduction · 0.81
Cites it as background 3 papers
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These geometries have been examined analytically, exploring their rich behavior, for different matter content of the universe and for different cosmological scenarios (e.g., [ref] ).
Qualitative analysis of Kantowski-Sachs metric in a generic class of $f(R)$ models · Introduction · 0.83
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The general classical solution of most Bianchi types in vacuum has been given in [ref] .
Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities · Classical treatment · 0.83
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Furthermore, amongst the Bianchi subclass the simplest but still very interesting geometries are the Bianchi I [ref] and Bianchi III [ref] .
Dynamical analysis of anisotropic scalar-field cosmologies for a wide range of potentials · Introduction · 0.91
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