Diagonal connection problem

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In summary, the conversation is about using a formula to prove that the off-diagonal components of the Christoffel symbol of the second kind are equal to zero when the metric is diagonal. The formula being used is \Gamma^a_{bc} = 1/2 g^{ad}(\partial_bg_{dc} + \partial_cg_{bd} - \partial_dg_{bc}), and it is discussed how to manipulate it and use the fact that the metric is diagonal in order to prove this. The conversation also briefly touches on the use of the natural log in the Hobson book and suggests trying to differentiate it.
  • #1
ehrenfest
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Homework Statement


I am trying to show that the connection [tex] \Gamma^a_{bc} [/tex] is equal to 0 when the metric g_ab is diagonal. Will the formula
[tex] \Gamma^a_{bc} = 1/2 g^{ad}(\partial_bg_{dc} + \partial_cg_{bd} - \partial_dg_{bc}) [/tex] be of use? How can I manipulate that equation and use the fact that the metric is diagonal?

Homework Equations


The Attempt at a Solution

 
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  • #2
Why do you think the connection is zero if the metric is diagonal? It's not even true. Look at spherical coordinates.
 
  • #3
It comes right out of my book!
 

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  • #4
No it doesn't; your book doesn't ask you to prove all of the components are zero: just some of them.
 
  • #5
OK. So it only wants me to prove that the off-diagonal components are zero (that statement in parentheses was pretty important). Anyway, I still have the same two questions as in the first post.
 
  • #6
If [itex] \Gamma [/itex] is meant to be the Riemann - Christoffel (or the metric) connection, then, yes, that's the formula you should use.
 
  • #7
Yes. It is the Christoffel symbol of the second kind. So am allowed to do this:

[tex] \Gamma^a_{bc} = 1/2 (\partial_bg^{ad}g_{dc} + \partial_cg^{ad}g_{bd} - \partial_dg^{ad}g_{bc}) [/tex]
?

Then the first term becomes the Kronecker delta, I think.

If not, how should I use the fact that the metric is diagonal?
 
  • #8
ehrenfest said:
Yes. It is the Christoffel symbol of the second kind. So am allowed to do this:

[tex] \Gamma^a_{bc} = 1/2 (\partial_bg^{ad}g_{dc} + \partial_cg^{ad}g_{bd} - \partial_dg^{ad}g_{bc}) [/tex]
?
Why do you think, for all d, that gad is a constant with respect to the b, c, and d-th variables?



If not, how should I use the fact that the metric is diagonal?
Any way you can imagine. You could substitute zero for the off-diagonal terms. You could decompose the metric into a linear combination of simpler tensors. Et cetera.
 
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  • #9
I see. You just replace d with a.
 
  • #10
Now, can someone explain to me where in the world this natural log comes from at the bottom of attached page of the Hobson book?
 

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  • #11
Did you try differentiating it?
 

1. What is a diagonal connection problem?

The diagonal connection problem is a mathematical problem that involves connecting all the points on a grid with diagonal lines without any lines crossing or overlapping.

2. What is the significance of the diagonal connection problem?

The diagonal connection problem has important applications in computer science, particularly in the field of circuit design and optimization. It also has implications in graph theory and geometry.

3. What is the difficulty level of solving the diagonal connection problem?

The difficulty level of solving the diagonal connection problem depends on the size of the grid. As the grid size increases, the complexity of the problem also increases, making it more challenging to find a solution.

4. What are some known algorithms for solving the diagonal connection problem?

Some known algorithms for solving the diagonal connection problem include the depth-first search algorithm, the breadth-first search algorithm, and the A* algorithm. These algorithms use different strategies to find a solution to the problem.

5. Are there real-life applications of the diagonal connection problem?

Yes, there are real-life applications of the diagonal connection problem, such as in the design of microchips, circuit boards, and other electronic components. It is also used in the planning of transportation routes, such as in city planning and logistics.

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