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Milne model
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The Milne model was a special relativistic cosmological model proposed by Edward Arthur Milne. It is a special case of the FLRW model in the limit of zero mass density, and it obeys the cosmological principle. However its predictions of Otot ~ 0 and negative spatial curvature contradict observations which show Otot ~ 1 and zero spatial curvature.
more general Friedmann, Lemaitre, Robertson, Walker Metric (Friedmann-Robertson-Walker metric) reduces to the Milne universe for a spacetime that is a pure vacuum without matter, radiation, or a cosmological constant.
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The Milne model was a special relativistic cosmological model proposed by Edward Arthur Milne. It is a special case of the FLRW model in the limit of zero mass density, and it obeys the cosmological principle. However its predictions of Otot ~ 0 and negative spatial curvature contradict observations which show Otot ~ 1 and zero spatial curvature.
Milne metric
The more general Friedmann, Lemaitre, Robertson, Walker Metric (Friedmann-Robertson-Walker metric) reduces to the Milne universe for a spacetime that is a pure vacuum without matter, radiation, or a cosmological constant. Milne cosmology therefore corresponds to a cosmological solution to the Einstein equations for :
.
The Milne metric follows when the scale factor of the FRLW metric is a constant over time,
yielding:
where:
gives the proper motion distance between points in this universe.
and appropriate constants are subsumed into relevant coordinates. The Milne metric is therefore simply a restatement of the Minkowski metric.
Whereas the FLRW metric explains all of the redshift-distance correlation of galaxies by the metric expansion of space (and only the noncorrelated data is explained by Peculiar velocity, Milne's model has no expansion of space, thus all of the redshift (except that caused by local gravitational effects) is explained by a recessional velocity associated with the hypothetical "explosion". Milne developed this model independent of general relativity but with awareness of special relativity. Since objects in Milne's universe have mass-energy, the model Milne uses violates general relativity, in particular the principle that mass-energy curves spacetime. Milne's universe is also incompatible with certain cosmological observations, in particular it makes no prediction of the cosmic microwave background radiation nor the abundance of light elements which are hallmark pieces of evidence that cosmologists agree support Big Bang cosmology over alternatives.
Milne's density function
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 universe as homogeneous and isotropic in the sense of the cosmological principle.
Unless the universe modeled has zero density, Milne's proposal does not follow the predictions of general relativity for the curvature of space caused by global matter distribution, as seen in, for example statistics associated with large-scale structure.
Differences between Milne model and other models
In order to explain the existence of matter in the universe, Milne proposed a physical explosion of matter which would not affect the universe's geometry. This is in contrast to the metric expansion of space that is the hallmark feature of many of the more famous cosmological models including the Big Bang and Steady State models. Milne's universe shares a superficial similarity to Einstein's static universe in that the metric of space is not time-dependent. Unlike Einstein's initial cosmology, Milne's proposal directly contradicts the Einstein equations for cosmological scales. Special relativity becomes a global property of Milne's universe while general relativity is confined to a local property. The reverse is true for standard cosmological models, and most scientists and mathematicians agree that the latter is self-consistent while the former is mathematically impossible.
Edward Arthur Milne predicted a kind of event horizon through the use of this model: "The particles near the boundary tend towards invisibility as seen by the central observer, and fade into a continuous background of finite intensity." The horizon arises naturally from length contraction seen in special relativity which is a consequence of the speed of light upper bound for physical objects. In Milne's universe, the velocities of objects asymptotically approach this upper bound while the distance to these objects approaches the speed of light multiplied by the time since the initial explosion of material. Beyond this distance, objects do not lie in the observable part of the Milne universe.
At the time Milne proposed his model, observations of the universe did not appear to be in a homogeneous form. This, to Milne, was a deficiency inherent in the competing cosmological models which relied on the cosmological principle that demanded a homogeneous universe. “This conventional homogeneity is only definite when the motion of the particles is first prescribed.” With present observations of the homogeneity of the universe on the largest scales seen in the cosmic microwave background and in the so-called "End of Greatness", questions about the homogeneity of the universe have been settled in the minds of most observational cosmologists.
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