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4. Assume the universe to be represented by the two- dimensional surface of a sphere of radius R. A possible R coordinate system on the surface is (r,0), where ' is the length of the geodesic from the north pole and 0 is the angle measured relative to some arbitrary "prime meridian". The metric on the surface is given by ds2 = dr² + R² sin2(r R) do². (a) Show that the circumference of a circle of radius l' drawn on the sphere has length C = 2xRsin(r R). Then prove that for r < R this gives the expected Euclidean result. (b) Suppose you are located at the North Pole and look at an object of width w (transverse to your line of sight) at a distance l' from you. Determine the angular width A0 that the object subtends from your perspective. A9- How does this angle compare to the case of a flat surface? (Hint: Expand the result in a power series in r.) (c) Assume that "galaxies" are uniformly distributed on the surface of the sphere with a number density of n galaxies per unit area. Show that the number of galaxies inside a circle of radius l' is N = 2xnR² -cos(r/R) (Hint: Find the infinitesimal element of area on the surface, and then integrate.) How does this compare with the Euclidean case? (You may again expand in a power series and/or plot Nvs. l' for both the Euclidean and spherical cases.) 5. Now, assume that our 3-dimensional space is a hypersphere of radius R described by the metric ds² = dr² + R² sin2()R)(d02 + sin² edo? (a) What is the circumference of a circle of radius r? What is the circumference of the largest circle that can exist in this space? (b) Based on your results for part (c) of the previous problem, what would you expect for the number of galaxies inside a sphere of radius ", as compared to the Euclidean case?

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