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hw 04 solution - 1 Catalin Constantinide PHYS 600 HW4...

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Catalin Constantinide PHYS 600 - HW4 SOLUTIONS PROBLEM 1 H a L replace d x a = ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m dx m and d x b = ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n dx n inside ds 2 = h ab d x a d x b to obtain ds 2 = h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n dx m dx n so we can identify g mn H x L = h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n The inverse metric is g mn H x L = h ab x m ÅÅÅÅÅÅÅÅÅÅÅÅÅ ∑x a x n ÅÅÅÅÅÅÅÅÅÅÅÅ ∑x b H b L we' ll use the definition of the affine connection G mn l = x l ÅÅÅÅÅÅÅÅÅÅÅÅÅ ∑ x a 2 x a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ x m x n and the result from H a L ; then g mn ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ x l = ÅÅÅÅÅÅÅÅÅÅÅÅÅ x l h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n = h ab 2 x a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ x l x m ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n + h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m 2 x b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ x l x n g ln ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ x m = ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x l ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n = h ab 2 x a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ x m x l ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x n + h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x l 2 x b ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ x m x n - g ml ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ x n = - ÅÅÅÅÅÅÅÅÅÅÅÅ x n h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x l = -h ab 2 x a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ x n x m ∑ x b ÅÅÅÅÅÅÅÅÅÅÅÅÅ x l - h ab ∑ x a ÅÅÅÅÅÅÅÅÅÅÅÅÅ x m 2 x
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