The spatial part of the Robertson-Walker metric is ds = a (1) | dr. 0 d6]...

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The spatial part of the Robertson-Walker metric is ds² = a² (1) | dr.² ²0 d6²] +r² do2+r² sin² 0 a (a) For positive k, what is the allowed range of the r coordinate? Define a new coordinate by the transformation r = (1/√) sin (√), and rewrite the metric using it. Such coordinates are called hyperspherical coordinates. Ex- plain why they show that the space has the geometry of a three-dimensional sphere. (b) Find an analogous transformation for negative k. Use it to find the ratio of circumference to radius for a circle at coordinate distance = 10/✓k. Compare with the equivalent ratio for a flat geometry. The spatial part of the Robertson-Walker metric is ds² = a² (1) | dr.² ²0 d6²] +r² do2+r² sin² 0 a (a) For positive k, what is the allowed range of the r coordinate? Define a new coordinate by the transformation r = (1/√) sin (√), and rewrite the metric using it. Such coordinates are called hyperspherical coordinates. Ex- plain why they show that the space has the geometry of a three-dimensional sphere. (b) Find an analogous transformation for negative k. Use it to find the ratio of circumference to radius for a circle at coordinate distance = 10/✓k. Compare with the equivalent ratio for a flat geometry.