Milne Metric
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. From these assumptions and the Friedmann equations it follows that the scale factor must depend on time coordinate linearly.
Setting the spatial curvature and speed of light to unity the metric for a Milne universe can be expressed with hyperspherical coordinates as:
where
is the metric for a two-sphere and
is the curvature-corrected radial component for negatively curved space that varies between 0 and .
The empty space that the Milne model describes 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 version of the FLRW metric to Milne's model implies that a full general relativistic treatment using Milne's assumptions would result in an increasing scale factor and associated metric expansion of space with the unique feature of a linearly increasing scale factor for all time since the deceleration parameter is uniquely zero for such a model.
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