1. Consider a diagonal metric in two dimensions ds = 91(dx)2 + 922(dx) where 911 and...

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1. Consider a diagonal metric in two dimensions ds = 91(dx)2 + 922(dx) where 911 and 922 are functions of r and 2. (a) Show that the Christoffel symbols are r = K 1 2911 T12 = I 12/2 1 0922 2922 x ag11 31 1 2911 1 2911922 8911 (0x) + 0911 dx 22 (b) Remember the Gaussian curvature for a diagonal metric can be writ- ten as ; r ; T2 = 1//1 K = 1 911 922 2922 0x 0x 2911 1 g22 x 1 ag11 2922 x 1 2922 8 922 1 911 922 + (x) 2911 81 8x1 R1212 9 + {[(). 922 1 22 + +)1 g11 From this expression and the Christoffel symbols in (a), show that the relation of the Gaussian curvature and the Riemann curvature tensor is given by x in 2 dimensions, and g = 911922 is again the determinant of the diagonal metric. 1. Consider a diagonal metric in two dimensions ds = 91(dx)2 + 922(dx) where 911 and 922 are functions of r and 2. (a) Show that the Christoffel symbols are r = K 1 2911 T12 = I 12/2 1 0922 2922 x ag11 31 1 2911 1 2911922 8911 (0x) + 0911 dx 22 (b) Remember the Gaussian curvature for a diagonal metric can be writ- ten as ; r ; T2 = 1//1 K = 1 911 922 2922 0x 0x 2911 1 g22 x 1 ag11 2922 x 1 2922 8 922 1 911 922 + (x) 2911 81 8x1 R1212 9 + {[(). 922 1 22 + +)1 g11 From this expression and the Christoffel symbols in (a), show that the relation of the Gaussian curvature and the Riemann curvature tensor is given by x in 2 dimensions, and g = 911922 is again the determinant of the diagonal metric.

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This problem is asking you to work with a twodimensional diagonal metric and to compute the Christoffel symbols and Gaussian curvature from it Lets br... View the full answer