hw6 - that satisFes the condition R 2 = x 2 + y 2 + z 2 + w...

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Physics 3002 Problem Set 6, due 3/25/09 Lam Hui 1. Ryden problem 6.4. 2. Ryden problem 3.3. 3. In class, we have derived the (spatial) metric for the two-dimensional closed universe (known as a 2-sphere), by embedding it in three-dimensional space (basically thinking of the 2-sphere as the surface of a ball). Here, you are to derive the (spatial) metric for the three-dimensional closed universe, by embed- ding it in four-dimensional space. Let us start with a four-dimensional space with a metric of the form ds 2 = dx 2 + dy 2 + dz 2 + dw 2 . Consider a 3-sphere
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Unformatted text preview: that satisFes the condition R 2 = x 2 + y 2 + z 2 + w 2 . Show that the metric on the 3-sphere is given by ds 2 = dr 2 + R 2 sin 2 ( r/R )[ d 2 + sin 2 d 2 ]. The rela-tion between x, y, z, w is x = R sin ( r/R ) sin cos , y = R sin ( r/R ) sin sin , z = R sin ( r/R ) cos , w = R cos ( r/R ). What are the natural ranges for r , and so that the 3-sphere is covered exactly once? (You might Fnd it helpful to review the natural ranges of coordinates on the 2-sphere.)...
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