Milne model

The Milne universe is a special case of a more general Friedmann–Lemaître–Robertson–Walker model (FLRW). The Milne solution can be obtained from the more generic FLRW model by demanding that the energy density, pressure and cosmological constant all equal zero and the spatial curvature is negative.{{Citation needed|date=April 2017}} From these assumptions and the Friedmann equations it follows that the scale factor must depend on time coordinate linearly.{{Cite book |last=Carroll |first=Sean |title=Spacetime and geometry: an introduction to general relativity |date=2004 |publisher=Addison Wesley |isbn=978-0-8053-8732-2 |edition=1st |location=San Francisco |page=341}}{{Cite book |last=Mukhanov |first=V. F. |url=https://books.google.com/books?id=1TXO7GmwZFgC |title=Physical foundations of cosmology |date=2005 |publisher=Cambridge University Press |isbn=978-0-521-56398-7 |edition=1st |location=Cambridge, UK; New York |page=27 |oclc=ocm61440784}}

Setting the spatial curvature and speed of light to unity the metric for a Milne universe can be expressed with hyperspherical coordinates as:{{Cite book |last1=Misner |first1=Charles W. |title=Gravitation |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John Archibald |publisher=W. H. Freeman |year=1971 |isbn=978-0-7167-0334-1 |edition=1st |location=San Francisco |page=721}}

:$ds^2\; =\; dt^2-t^2(d\; \backslash chi\; ^2+\backslash sinh^2\{\backslash chi\}\; d\backslash Omega^2)\backslash$

where

:$d\backslash Omega^2\; =\; d\backslash theta^2+\backslash sin^2\backslash theta\; d\backslash phi^2\backslash$

is the metric for a two-sphere and

:$\backslash chi\; =\; \backslash sinh^\{-1\}\{r\}$

is the curvature-corrected radial component for negatively curved space that varies between 0 and $+\backslash infin$.

The empty space that the Milne model describes{{Citation needed|reason=Milne does not describe his model as empty space|date=April 2017}} can be identified with the inside of a light cone of an event in Minkowski space by a change of coordinates.

Milne developed this model independent of general relativity but with awareness of special relativity. As he initially described it, the model has no expansion of space, so all of the redshift (except that caused by peculiar velocities) is explained by a recessional velocity associated with the hypothetical "explosion". However, the mathematical equivalence of the zero energy density ($\backslash rho\; =\; 0$) version of the FLRW metric to Milne's model implies that a full general relativistic treatment using Milne's assumptions would result in a linearly increasing scale factor for all time since the deceleration parameter is uniquely zero for such a model.

Milne proposed that the universe's density changes in time because of an initial outward explosion of matter. Milne's model assumes an inhomogeneous density function which is Lorentz Invariant (around the event t=x=y=z=0). When rendered graphically Milne's density distribution shows a three-dimensional spherical Lobachevskian pattern with outer edges moving outward at the speed of light. Every inertial body perceives itself to be at the center of the explosion of matter (see observable universe), and sees the local universe as homogeneous and isotropic in the sense of the cosmological principle.

In order to be consistent with general relativity, the universe's density must be negligible in comparison to the critical density at all times for which the Milne model is taken to apply.

{{reflist}}

• [http://world.std.com/~mmcirvin/milne.html Milne Cosmology: Why I Keep Talking About It] {{webarchive |url=https://web.archive.org/web/20060912154238/http://world.std.com/~mmcirvin/milne.html |date=12 September 2006 }} - a detailed non-technical introduction to the Milne model
• {{Cite book |last=True Wegener |first=Mogens |url=https://www.researchgate.net/publication/352325104_NON-STANDARD_RELATIVITY_A_Philosopher's_Handbook_of_Heresies_in_Physics_New_Concise_Edition_revised_20210609 |title=Non-Standard Relativity: a Philosopher's Handbook of Heresies in Physics |date=2021 |publisher=Books on Demand |isbn=978-8743031420}} A thorough historical and theoretical study of the British Tradition in Cosmology, and one long celebration of Milne.

{{Relativity}}

Category:Obsolete theories in physics

Category:Exact solutions in general relativity

Category:Minkowski spacetime

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