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Calculate curvature by coordinate component method

by PhyPsy
Tags: index notation, riemann tensor
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PhyPsy
#1
Oct1-11, 12:39 PM
P: 39
I'm trying to follow the math in Wald's General Relativity where he starts out with the equation for covariant derivative:
[itex]\nabla[/itex]b[itex]\omega[/itex]c = [itex]\partial[/itex]b[itex]\omega[/itex]c - [itex]\Gamma[/itex]dbc[itex]\omega[/itex]d

He uses that to derive the equation for a double covariant derivative:
[itex]\nabla[/itex]a[itex]\nabla[/itex]b[itex]\omega[/itex]c = [itex]\partial[/itex]a([itex]\partial[/itex]b[itex]\omega[/itex]c - [itex]\Gamma[/itex]dbc[itex]\omega[/itex]d) - [itex]\Gamma[/itex]eab([itex]\partial[/itex]e[itex]\omega[/itex]c - [itex]\Gamma[/itex]dec[itex]\omega[/itex]d) - [itex]\Gamma[/itex]eac([itex]\partial[/itex]b[itex]\omega[/itex]e - [itex]\Gamma[/itex]dbe[itex]\omega[/itex]d)

Now, using the Riemann tensor definition Rabcd[itex]\omega[/itex]d = [itex]\nabla[/itex]a[itex]\nabla[/itex]b[itex]\omega[/itex]c - [itex]\nabla[/itex]b[itex]\nabla[/itex]a[itex]\omega[/itex]c, this equation is derived:
Rabcd[itex]\omega[/itex]d = [itex]\partial[/itex][a[itex]\partial[/itex]b][itex]\omega[/itex]c - [itex]\omega[/itex]d[itex]\partial[/itex][a[itex]\Gamma[/itex]db]c - [itex]\Gamma[/itex]dc[b[itex]\partial[/itex]a][itex]\omega[/itex]d - [itex]\Gamma[/itex]e[ab]([itex]\partial[/itex]e[itex]\omega[/itex]c - [itex]\Gamma[/itex]dec[itex]\omega[/itex]d) - [itex]\Gamma[/itex]ec[a[itex]\partial[/itex]b][itex]\omega[/itex]e - [itex]\Gamma[/itex]ec[a[itex]\Gamma[/itex]db]e[itex]\omega[/itex]d

I know that the term [itex]\partial[/itex][a[itex]\partial[/itex]b][itex]\omega[/itex]c cancels, and, due to the symmetry of the metric connection, [itex]\Gamma[/itex]e[ab]([itex]\partial[/itex]e[itex]\omega[/itex]c - [itex]\Gamma[/itex]dec[itex]\omega[/itex]d) also cancels, but the next step in the book also has a couple other terms cancelled out:
[itex]\Gamma[/itex]dc[b[itex]\partial[/itex]a][itex]\omega[/itex]d, and
[itex]\Gamma[/itex]ec[a[itex]\partial[/itex]b][itex]\omega[/itex]e

I don't see how these terms cancel. Can someone help me?
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