Math 2.0

Relativity and gravitational theory Relativistic cosmology Example

Milne model

Builds on Friedmann–Lemaître–Robertson–Walker metricFriedmann equationsMinkowski spaceLight coneHyperboloid model

Example

The Milne model is the Friedmann–Lemaître–Robertson–Walker metric with k=−1k = -1 and scale factor a(t)=ta(t) = t. It has no matter and no cosmological constant: ds2=−dt2+t2(dχ2+sinh⁡2χ dΩ2),t>0,ds^2 = -dt^2 + t^2 \left( d\chi^2 + \sinh^2\chi \, d\Omega^2 \right), \qquad t > 0, in units with c=1c = 1.

Check with the Friedmann equations: a˙/a=1/t\dot a / a = 1/t, so the first equation reads 1/t2=−k/a2=1/t21/t^2 = -k/a^2 = 1/t^2, and the second reads a¨/a=0\ddot a / a = 0, both with ρ=p=Λ=0\rho = p = \Lambda = 0.

Remarks

The Milne model is flat: it is the interior of the future light cone of a point OO in Minkowski space. On that region, with Minkowski coordinates (T,X)(T, X) centred at OO, put t=T2−∣X∣2t = \sqrt{T^2 - \abs{X}^2}, the proper time from OO. The slices t=constt = \text{const} are the hyperboloids T2−∣X∣2=t2T^2 - \abs{X}^2 = t^2, and each carries the metric t2t^2 times the metric of hyperbolic space, as in the hyperboloid model. The comoving observers are the inertial observers that pass through OO, so the "expansion" is ordinary motion of test particles away from a common event. Its Hubble parameter is H=1/tH = 1/t and its deceleration parameter is q=0q = 0.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

Show the path as a graph
Bilinear form, concept, no evidence levelBilinear formconceptCayley–Klein metric, concept, no evidence levelCayley–Klein metricconceptCosmological constant, concept, no evidence levelCosmological constantconceptCosmological principle, concept, no evidence levelCosmological principleconceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptHyperbolic cosine, concept, no evidence levelHyperbolic cosineconceptHyperbolic geometry, concept, no evidence levelHyperbolic geometryconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLight cone, concept, no evidence levelLight coneconceptMatrix, definition, no evidence levelMatrixdefinitionMinkowski space, concept, no evidence levelMinkowski spaceconceptPerfect fluid, concept, no evidence levelPerfect fluidconceptPoincaré disk model, object, no evidence levelPoincaré disk modelobjectQuadratic form, concept, no evidence levelQuadratic formconceptScale factor, concept, no evidence levelScale factorconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptSpace form, concept, no evidence levelSpace formconceptSpacetime, concept, no evidence levelSpacetimeconceptBeltrami–Klein model, example, no evidence levelBeltrami–Klein modelexampleTranspose, definition, no evidence levelTransposedefinitionFriedmann–Lemaître–Robertson–Walker metric, example, no evidence levelFriedmann–Lemaître–Ro…exampleOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionFriedmann equations, statement, no evidence levelFriedmann equationsstatementOrthogonal group, object, no evidence levelOrthogonal groupobjectIndefinite orthogonal group, example, no evidence levelIndefinite orthogonal…exampleHyperboloid model, example, no evidence levelHyperboloid modelexampleMilne model, example, no evidence levelMilne modelexample
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. Bilinear form concept
  2. Cayley–Klein metric concept
  3. Cosmological constant concept
  4. Cosmological principle concept
  5. Einstein field equations concept
  6. Hyperbolic cosine concept
  7. Hyperbolic geometry concept
  8. Identity matrix object
  9. Light cone concept
  10. Matrix definition
  11. Minkowski space concept
  12. Perfect fluid concept
  13. Poincaré disk model object
  14. Quadratic form concept
  15. Scale factor concept
  16. Signature of a quadratic form concept
  17. Space form concept
  18. Spacetime concept
  19. Beltrami–Klein model example
  20. Transpose definition
  21. Friedmann–Lemaître–Robertson–Walker metric example
  22. Orthogonal matrix definition
  23. Friedmann equations statement
  24. Orthogonal group object
  25. Indefinite orthogonal group example
  26. Hyperboloid model example
  27. Milne model example

Further reading

  1. Web https://en.wikipedia.org/wiki/Milne_model
About this page
Tag543HJFE64X
URIm2://543HJFE64X
Kindexample
WritingW0 stub
MSC83F05, 83A05
Instance ofExact solutions in general relativity
UsesFriedmann–Lemaître–Robertson–Walker metric, Friedmann equations, Minkowski space, Light cone, Hyperboloid model
Same as
Changed3 November 2026

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