Metric Tensors and Christoffel Symbols
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Problem 1. Derive the formula given below for the Christoffel symbols ?_ij^k of a Levi-Civita connection in terms of partial derivatives of the associated metric tensor g_ij.
?_ij^k = (1/2) g^kl {?_i g_lj ? ?_l g_ij + ?_j g_il }.
Problem 2. Compute the Christoffel symbols of the Levi-Civita connection associated to each of the following metrics.
(a) The metric of the unit sphere S^2 centered at the origin obtained from spherical polar coordinates (?, ?),
ds^2 = d?^2 + sin^2 ? d?^2 .
(b) The Lorentzian metric defining SR geometry,
ds^2 = ? {dt + a(r) d?}^2 + dr^2 + r^2 d?^2 + dz^2 ,
where a(r) = m r^2 + a_0 , and m and a_0 are constants.
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Solution Summary
A brief explanation of how to compute the covariant derivative of covariant 1 and 2-tensors is included at the beginning. Two problems and their solutions follow. The first derives a formula for the Christoffel symbols of a Levi-Civita connection in terms of the associated metric tensor. The second computes the Christoffel symbols of two specific metric tensors by using the formula derived in the first problem.
Solution Preview
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A brief explanation of how to compute the covariant derivative of covariant 1 and 2-tensors is included at the beginning.
The solution to Problem 1 is a ...
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