DUALITY OF TIME:

Complex-Time Geometry and Perpetual Creation of Space

by Mohamed Haj Yousef

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3.3.4  The FLRW Metric

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In 1922, Alexander Friedmann (1888-1925) published a paper in which he describes a homogeneous, isotropic expanding or contracting Universe, based on General Relativity. Although his paper was refereed by Einstein and published in the prestigious physics journal Zeitschrift fur Physik, it remained relatively unnoticed by his contemporaries. Eventually Einstein acknowledged the correctness of Friedmann’s calculations, but also failed to appreciate the physical significance of his predictions. In 1927, Georges Lemaitre (1894-1966) arrived independently at similar results, which were soon supported by the observational evidence for the expansion of the Universe obtained by Edwin Hubble (1889-1953). Lemaitre’s results were noticed in particular by Eddington, and in his paper was translated into English in 1931.

The problem was investigated further by Howard Robertson (1903-1961) and Arthur Walker (1909-2001), who rigorously proved, in 1935, that the now called FLRW metric is the only one on a space-time that is spatially homogeneous and isotropic, but this is a geometric result that is not tied specifically to the equations of General Relativity.

The FLRW metric is an exact solution of Einstein’s field equations which describes a homogeneous, isotropic, expanding or contracting Universe, that is path connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein’s field equations are only needed to derive the scale factor of the Universe as a function of time. This model is also called the Standard Model of modern cosmology, which is also associated with the further developed Lambda-CDM model, as described in section 6.6.

The FLRW metric starts with the assumption of homogeneity and isotropy of space, and assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is:

(3.12)

whereranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates. In reduced-circumference polar coordinates the spatial metric has the form:

(3.13)

Here,,is a constant representing the curvature of space, which may be taken to have units of length-2, andis like the Schwarzschild radius, as well as the other parameters, as we have seen in section 3.3 above.

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