Calculate curvature by coordinate component method

In summary, The conversation is about a mathematical derivation using equations for covariant derivatives and Riemann tensor definition. The terms \partial[a\partialb]\omegac and \Gammae[ab](\partiale\omegac - \Gammadec\omegad) are cancelled due to the symmetry of the metric connection. The terms \Gammadc[b\partiala]\omegad and \Gammaec[a\partialb]\omegae are also cancelled, which is achieved by substituting ##e## with ##d## and using the condition ##{}_b\partial_a+{}_a\partial_b = 0##.
  • #1
PhyPsy
39
0
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 canceled 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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  • #2
We can substitute ##e## by ##d## without changing anything. Now what is left is the condition ##{}_b\partial_a+{}_a\partial_b = 0##.
 

1. What is the coordinate component method for calculating curvature?

The coordinate component method is a mathematical technique used to determine the curvature of a curve or surface in a given coordinate system. It involves calculating the second derivative of the curve or surface with respect to each coordinate variable.

2. How is the curvature calculated using the coordinate component method?

The curvature is calculated by taking the second derivative of the curve or surface with respect to each coordinate variable, and then using these values to calculate the magnitude of the curvature using a specific formula. This formula varies depending on the dimensionality of the curve or surface.

3. What information is needed to use the coordinate component method for calculating curvature?

To use the coordinate component method, you will need the equation of the curve or surface in terms of the coordinate variables, as well as the values of the first and second derivatives of the curve or surface with respect to each coordinate variable.

4. Are there any limitations to using the coordinate component method for calculating curvature?

Yes, there are some limitations to using this method. It is generally only applicable to smooth curves and surfaces, and may not be accurate for curves or surfaces with sharp corners or discontinuities. Additionally, the results may be affected by the choice of coordinate system.

5. What are some real-world applications of the coordinate component method for calculating curvature?

The coordinate component method is commonly used in fields such as physics, engineering, and mathematics to analyze the behavior of curves and surfaces. It can be applied to various real-world problems, such as calculating the curvature of a road or determining the shape of a curved surface in engineering designs.

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