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Metric Tensors and Christoffel Symbols

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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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 ...

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.

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