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I am completely unable to follow the following sequence of working between equations 2 and 3. AFAIK the final answer is correct, but the intermediate steps seem to be a "casserole of nonsense". I would appreciate feedback from anyone who can follow this, or who can correct it if it is wrong . . .

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Now suppose we embed a Euclidean three-dimensional space (x,y,z) in a four-dimensional space (w,x,y,z) whose metric is

http://mathpages.com/rr/s7-01/7-01_files/image004.gif [Broken]

where k is a fixed constant equal to either +1 or -1. If k = +1 the four-dimensional space is Euclidean, whereas if k = -1 it is pseudo-Euclidean (like the Minkowski metric). In either case the four-dimensional space is "flat", i.e., has zero Riemannian curvature. Now suppose we consider a three-dimensional subspace comprising a sphere (or pseudo-sphere), i.e., the locus of points satisfying the condition

http://mathpages.com/rr/s7-01/7-01_files/image005.gif [Broken]

From this we have w2 = (1 - r2)/k = k - kr2, and therefore

http://mathpages.com/rr/s7-01/7-01_files/image006.gif [Broken]

Substituting this into the four-dimensional line element above gives the metric for the three-dimensional sphere (or pseudo-sphere)

http://mathpages.com/rr/s7-01/7-01_files/image007.gif [Broken]

Taking this as the spatial part of our overall spacetime metric (2) that satisfies the Cosmological Principle, we arrive at

http://mathpages.com/rr/s7-01/7-01_files/image008.gif [Broken]

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# Mistake in cosmology chapter in "Reflections on Relativity"?

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