In a 4D Euclidean space with metric ds = dx + dy + dz + dw,...

 

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In a 4D Euclidean space with metric ds² = dx² + dy² + dz² + dw², we define the 3D surface x² + y² +₂²+w²= R2, with a positive constant R. This surface is called a 3-sphere. We introduce the periodic coordinates (x. 0, 9) on the 3-sphere, defined through x = Y 2 = Rsin x sin cos p. Rsin x sin sin p Rsin x cos 0, such that w√√R²-x²-y²-22 = R cos x. Show that the line element of the 3-sphere geometry is given by de² = R² [dx² + sin² x (do²+ sin² 0 dp²)]. This describes the spatial geometry of a closed FLRW universe at constant co-moving time. Guidance: Obtain (dr, dy, dz, dw) in terms of (dx, de, de) on the surface of the 3-sphere, then substitute in ds² = dr² +dy² + dz² + dw² and work through the algebra. In a 4D Euclidean space with metric ds² = dx² + dy² + dz² + dw², we define the 3D surface x² + y² +₂²+w²= R2, with a positive constant R. This surface is called a 3-sphere. We introduce the periodic coordinates (x. 0, 9) on the 3-sphere, defined through x = Y 2 = Rsin x sin cos p. Rsin x sin sin p Rsin x cos 0, such that w√√R²-x²-y²-22 = R cos x. Show that the line element of the 3-sphere geometry is given by de² = R² [dx² + sin² x (do²+ sin² 0 dp²)]. This describes the spatial geometry of a closed FLRW universe at constant co-moving time. Guidance: Obtain (dr, dy, dz, dw) in terms of (dx, de, de) on the surface of the 3-sphere, then substitute in ds² = dr² +dy² + dz² + dw² and work through the algebra.

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To derive the line element of the 3sphere we start by expressing the ... View the full answer