The Robertson-Walker geometry (to be discussed in Chapter 9) has a spacetime interval ds = dt-a(t)...

 

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The Robertson-Walker geometry (to be discussed in Chapter 9) has a spacetime interval ds² = dt²-a(t)² [dx² + sin² x (do²+ sin²0do²)] where the coordinates are xº = t, x¹ = x, x² = 0, x³ = 0, and a(t) is a function of time. (a) Calculate all the Christoffel symbols Im with purely spatial indices; that is, with indices from 1 to 3. (b) A covariant vector with components b = (0, 1, 0, 0) is initially located at X = Xo in the equatorial plane 0 = π/2. We parallel-transport this vector instantaneously (dt = 0) once around a closed circle in the equatorial plane (x = Xo, 0 = π/2, p = 0 to 27). Calculate the final vector bu that results from this parallel transport. Đ The Robertson-Walker geometry (to be discussed in Chapter 9) has a spacetime interval ds² = dt²-a(t)² [dx² + sin² x (do²+ sin²0do²)] where the coordinates are xº = t, x¹ = x, x² = 0, x³ = 0, and a(t) is a function of time. (a) Calculate all the Christoffel symbols Im with purely spatial indices; that is, with indices from 1 to 3. (b) A covariant vector with components b = (0, 1, 0, 0) is initially located at X = Xo in the equatorial plane 0 = π/2. We parallel-transport this vector instantaneously (dt = 0) once around a closed circle in the equatorial plane (x = Xo, 0 = π/2, p = 0 to 27). Calculate the final vector bu that results from this parallel transport. Đ

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A To calculate the Christoffel symbols Gamma k lm with purely spatial indices we can start by taking the derivative of the metric tensor gkl with resp... View the full answer