Deriving a Metric: Show that the metric for a 3-sphere (i.e. a three- dimensional sphere embedded...

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Deriving a Metric: Show that the metric for a 3-sphere (i.e. a three- dimensional "sphere" embedded in 4-dimensional space) of radius Reis de²= dr² + R² sin² (r/Re) (d0² + sin² 0 do²). Hint: The 3-sphere can be written as the surface w²+x² + y² + z² = R² in 4-dimensional Cartesian space. This surface can be parameterized with a generalization of spherical coordinates: W = Z = X = Y = Recos X Resin x cos X Resin x sin Resin x sin cos o sin o Begin with the distance relation in four dimensions, change coordinates to the above, and then change variables once more, setting r = Rex. Show your work, or you will not receive full credit for the problem. Deriving a Metric: Show that the metric for a 3-sphere (i.e. a three- dimensional "sphere" embedded in 4-dimensional space) of radius Reis de²= dr² + R² sin² (r/Re) (d0² + sin² 0 do²). Hint: The 3-sphere can be written as the surface w²+x² + y² + z² = R² in 4-dimensional Cartesian space. This surface can be parameterized with a generalization of spherical coordinates: W = Z = X = Y = Recos X Resin x cos X Resin x sin Resin x sin cos o sin o Begin with the distance relation in four dimensions, change coordinates to the above, and then change variables once more, setting r = Rex. Show your work, or you will not receive full credit for the problem.

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ANSWER To derive the metric for a 3sphere of radius Re we will follow the steps outlined in the hint ... View the full answer