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TheMilne model was aspecial-relativisticcosmologicalmodel of the universe proposed byEdward Arthur Milne in 1935.^[1] It is mathematically equivalent to a special case of theFLRW model in the limit of zeroenergy density and it obeys thecosmological principle^{[citation needed]}. The Milne model is also similar toRindler space in that both are simple re-parameterizations of flatMinkowski space.

Since it features both zero energy density and maximally negativespatial curvature, the Milne model is inconsistent withcosmological observations^{[citation needed]}. Cosmologists actually observe the universe'sdensity parameter to be consistent withunity and its curvature to be consistent withflatness.^[2]

The Milne universe is a special case of a more generalFriedmann–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]} From these assumptions and the Friedmann equations it follows that the scale factor must depend on time coordinate linearly.^[3][4]

Setting the spatial curvature and speed of light to unity the metric for a Milne universe can be expressed with hyperspherical coordinates as:^[4][5]

${\displaystyle d{s}^2=d{t}^2-t^2(d{\chi }^2+{\sinh }^2\chi d{\mathrm{\Omega }}^2)}$

where

${\displaystyle d{\mathrm{\Omega }}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2}$

is the metric for a two-sphere and

${\displaystyle \chi ={\sinh }^{-1}r}$

is thecurvature-corrected radial component for negatively curved space that varies between 0 and${\displaystyle +\mathrm{\infty }}$.

The empty space that the Milne model describes^{[citation needed]} can be identified with the inside of a light cone of an event in Minkowski space by a change of coordinates.^[4]

Milne developed this model independent ofgeneral relativity but with awareness ofspecial relativity. As he initially described it, the model has no expansion of space, so all of the redshift (except that caused bypeculiar velocities) is explained by arecessional velocity associated with the hypothetical "explosion". However, the mathematical equivalence of the zeroenergy density (${\displaystyle \rho =0}$) version of theFLRW metric to Milne's model implies that a full general relativistic treatment using Milne's assumptions would result in a linearly increasingscale factor for all time since thedeceleration 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 sphericalLobachevskian 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 (seeobservable universe), and sees the local universe as homogeneous and isotropic in the sense of thecosmological principle.

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

• Milne Cosmology: Why I Keep Talking About ItArchived 12 September 2006 at theWayback Machine - a detailed non-technical introduction to the Milne model
• True Wegener, Mogens (2021).Non-Standard Relativity: a Philosopher's Handbook of Heresies in Physics. Books on Demand.ISBN 978-8743031420. A thorough historical and theoretical study of the British Tradition in Cosmology, and one long celebration of Milne.

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