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2. The Robertson Walker Metric

The only metric compatible with Hubble's findings and the Cosmological Principle is the Robertson Walker Metric (RWM) with the corresponding line element,

Equation 4 (4)

For a brief explanation consider the following:

There are a few subtleties that must be discussed. First, the r that appears in the line element (4) is not the radius of the universe. The r is a dimensionless, comoving coordinate that ranges from zero to one for k = 1. The measurable, physical distance is given by the RWM above. Choosing a frame common to two distinct points, one obtains,

Equation 4b

for their separation. Where dtheta and dphi are zero, because one has freedom to arrange the axis and ds2 represents their separation in spacetime. Thus, their spatial separation is found by considering spacelike hypersurfaces, that is dt2 = 0. Thus, their separation is

Equation 4c

Evidently for a k = 0 flat universe, the distance is simply,

Equation 4c

Thus, a(t) has units of length and depends on the geometry of the spacetime.

The next issue is that of curvature. The curvature of the universe is determined by the amount of energy and matter that is present. The space is one of constant curvature determined by the value of k. Because any arbitrary scaling of the line element (4) will not affect the sign of k, we have the following convention (11)

11 When the metric is invariant under multiplication by a scale factor, the metric is said to be conformally invariant. Back.

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