Confused about derivatives of the metric

  • Thread starter quasar_4
  • Start date
  • #1
290
0

Main Question or Discussion Point

Hi,

I am incredibly confused about second derivatives of the metric. I know that in general, the covariant derivative of a vector is given by

[tex] \nabla_a v^b = \partial_a v^b + \Gamma^b_{ac}v^c [/tex]

and I think I understand how to generalize to higher rank tensors (just decompose into an outer product of vectors and covectors and use Leibniz, right?).

But, if we have the metric tensor, and we want to take the 2nd derivative

[tex] \partial_a \partial_b g_{cd} = g_{cd, ab} [/tex]

do I just differentiate the components of g twice or are there Christoffel symbols hiding in there too?? They are just partial derivative symbols and not covariant derivatives, so I don't see how/why we'd do anything but differentiate components of the metric.

But I've been trying to compute a curvature tensor for hours now and I cannot for the life of me get the answer that Maple gets, so I must not understand how to compute 2nd derivatives of the metric... :grumpy:
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
Indeed, to calculate
[tex]
\partial_a \partial_b g_{cd} = g_{cd, ab}
[/tex]
you simply take the second derivatives of the components of g. The Christoffel symbols only come in when you want to calculate the covariant derivative
[tex]
\nabla_a \nabla_b g_{cd} = g_{cd; ab}.
[/tex]
Usually, in GR, we choose the Christoffel symbols such that
[tex]
\nabla_a g_{bc} = g_{bc;a} = 0,
[/tex]
the so-called metric compatible connection.
 

Related Threads on Confused about derivatives of the metric

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
987
  • Last Post
Replies
3
Views
3K
Replies
13
Views
3K
  • Last Post
Replies
5
Views
736
Replies
2
Views
751
  • Last Post
Replies
1
Views
1K
Replies
4
Views
660
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
23
Views
9K
Top