Math 2.0

Relativity and gravitational theory Relativistic cosmology Example

Mixmaster universe

Builds on Bianchi classificationKasner metricGravitational singularityContinued fraction

Example

Let ω1,ω2,ω3\omega^1, \omega^2, \omega^3 be left-invariant one-forms on the group SU(2)≅S3SU(2) \cong S^3. Assume dω1=ω2∧ω3d\omega^1 = \omega^2 \wedge \omega^3 and its cyclic permutations. Then their dual vector fields span the Lie algebra of Bianchi type IX. The mixmaster universe is the family of vacuum metrics, with Λ=0\Lambda = 0, of the form ds2=−dt2+a(t)2(ω1)2+b(t)2(ω2)2+c(t)2(ω3)2ds^2 = -dt^2 + a(t)^2 (\omega^1)^2 + b(t)^2 (\omega^2)^2 + c(t)^2 (\omega^3)^2 on I×S3I \times S^3. Here a,b,ca, b, c are positive functions of tt, and the speed of light is 11. The Einstein equations reduce to a system of ordinary differential equations for aa, bb and cc.

Check that each slice t=constt = \text{const} is homogeneous. The left-invariant forms are invariant under the left action of SU(2)SU(2) on itself. This action is transitive, so it gives isometries of the metric that move any point of a slice to any other. When a=b=ca = b = c the slices are round three-spheres, as in a closed Friedmann–Lemaître–Robertson–Walker model; in general they are anisotropic.

Remarks

Toward the initial singularity, a generic solution passes through an infinite sequence of epochs. In each epoch it is close to a Kasner metric with exponents p1=−u1+u+u2,p2=1+u1+u+u2,p3=u(1+u)1+u+u2,p_1 = \frac{-u}{1 + u + u^2}, \qquad p_2 = \frac{1 + u}{1 + u + u^2}, \qquad p_3 = \frac{u(1 + u)}{1 + u + u^2}, for a parameter u≥1u \ge 1; one checks p1+p2+p3=1p_1 + p_2 + p_3 = 1 and p12+p22+p32=1p_1^2 + p_2^2 + p_3^2 = 1. In the approximation of Belinski, Khalatnikov and Lifshitz, one epoch follows another by u↦u−1u \mapsto u - 1 while u≥2u \ge 2, and a new era begins by u↦1/(u−1)u \mapsto 1/(u - 1) when 1<u<21 < u < 2. So the lengths of the successive eras are the partial quotients of the continued fraction expansion of the initial uu, which makes the dynamics chaotic. The name was given by Charles Misner.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

Show the path as a graph
Associative algebra, concept, no evidence levelAssociative algebraconceptCenter of a Lie algebra, concept, no evidence levelCenter of a Lie algeb…conceptCentre of a group, concept, no evidence levelCentre of a groupconceptCommutator of a ring, concept, no evidence levelCommutator of a ringconceptContinued fraction, concept, no evidence levelContinued fractionconceptEndomorphism, concept, no evidence levelEndomorphismconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGravitational singularity, concept, no evidence levelGravitational singula…conceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie algebra, concept, no evidence levelLie algebraconceptLie group, concept, no evidence levelLie groupconceptLie ideal, concept, no evidence levelLie idealconceptMatrix, definition, no evidence levelMatrixdefinitionMatrix, concept, no evidence levelMatrixconceptNilpotent group, concept, no evidence levelNilpotent groupconceptRicci curvature, concept, no evidence levelRicci curvatureconceptRing, concept, no evidence levelRingconceptSemidirect product of Lie algebras, concept, no evidence levelSemidirect product of…conceptSkew-symmetric matrix, concept, no evidence levelSkew-symmetric matrixconceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptVector space, definition, no evidence levelVector spacedefinitionHeisenberg group, example, no evidence levelHeisenberg groupexampleJacobi identity, statement, no evidence levelJacobi identitystatementKasner metric, example, no evidence levelKasner metricexampleSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionLie algebra of an associative algebra, example, no evidence levelLie algebra of an ass…exampleMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectGeneral linear Lie algebra, example, no evidence levelGeneral linear Lie al…exampleHeisenberg algebra, example, no evidence levelHeisenberg algebraexampleDeterminant, definition, no evidence levelDeterminantdefinitionOrthogonal Lie algebra, example, no evidence levelOrthogonal Lie algebraexampleSpecial linear group, object, no evidence levelSpecial linear groupobjectSpecial linear Lie algebra, example, no evidence levelSpecial linear Lie al…exampleBianchi classification, statement, no evidence levelBianchi classificationstatementMixmaster universe, example, no evidence levelMixmaster universeexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Associative algebra concept
  2. Center of a Lie algebra concept
  3. Centre of a group concept
  4. Commutator of a ring concept
  5. Continued fraction concept
  6. Endomorphism concept
  7. General linear group object
  8. Gravitational singularity concept
  9. Identity matrix object
  10. Lie algebra concept
  11. Lie group concept
  12. Lie ideal concept
  13. Matrix definition
  14. Matrix concept
  15. Nilpotent group concept
  16. Ricci curvature concept
  17. Ring concept
  18. Semidirect product of Lie algebras concept
  19. Skew-symmetric matrix concept
  20. Trace of a matrix concept
  21. Vector space definition
  22. Heisenberg group example
  23. Jacobi identity statement
  24. Kasner metric example
  25. Square matrix definition
  26. Transpose definition
  27. Linear map definition
  28. Lie algebra of an associative algebra example
  29. Multilinear map definition
  30. Orthogonal matrix definition
  31. Alternating multilinear map definition
  32. Orthogonal group object
  33. General linear Lie algebra example
  34. Heisenberg algebra example
  35. Determinant definition
  36. Orthogonal Lie algebra example
  37. Special linear group object
  38. Special linear Lie algebra example
  39. Bianchi classification statement
  40. Mixmaster universe example

Further reading

  1. Web https://en.wikipedia.org/wiki/Mixmaster_universe
About this page
TagCRD6KQCTD6
URIm2://CRD6KQCTD6
Kindexample
WritingW0 stub
MSC83F05, 83C75, 83C15
Instance ofExact solutions in general relativity
UsesBianchi classification, Kasner metric, Gravitational singularity, Continued fraction
Same as
Changed3 November 2026

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