Linearized Einstein Field Equations

  • Thread starter PLuz
  • Start date
  • #1
64
0

Main Question or Discussion Point

Hi everyone,

Say that one can separate the metric of a space time in a background metric and a small perturbation such that [itex]g_{\alpha \beta}=g'_{\alpha \beta}+h_{\alpha \beta}[/itex], where [itex]g'_{\alpha \beta}[/itex] is the background metric and [itex]h_{\alpha \beta}[/itex] the perturbation.

Computing the christoffel symbols one would get, to first order in the perturbation: [tex]\Gamma^\alpha_{\beta \gamma}=\Gamma'^\alpha_{\beta \gamma}+\frac{1}{2}(h^{\alpha}_{\beta,\gamma}+h^{ \alpha }_{\gamma,\beta}-h_{\beta \gamma}\hspace{.2mm}^{,\alpha}),[/tex]

right?
Then why, in this reference, in the text right after Eq.19.23, [itex]C^\alpha_{\beta \gamma}=\frac{1}{2}(h^{\alpha}_{\beta;\gamma}+h^{ \alpha }_{\gamma;\beta}-h_{\beta \gamma}\hspace{.2mm}^{;\alpha})[/itex], is written with covariant derivatives?


Thank you
 
Last edited:

Answers and Replies

  • #2
haushofer
Science Advisor
Insights Author
2,297
674
A difference between two connections is a tensor, which can be checked by explicitly writing down the transformation of this difference. Hence you'll need covariant derivatives, not partial derivatives. Of course, these covariant derivatives should follow from your definition of the connection and your C.

So that is something which you should do first. Second, you should be very careful with lowering and raising indices underneath partial derivatives.
 
  • #3
64
0
Yes, you're absolutely right, at both things. I didn't care about the partial derivative when I raised my indexes and indeed I was being naive in the definition of C.

Thank you very much you were a life (brain) saver!
 

Related Threads on Linearized Einstein Field Equations

Replies
1
Views
3K
Replies
5
Views
654
  • Last Post
Replies
1
Views
568
  • Last Post
3
Replies
57
Views
10K
  • Last Post
Replies
9
Views
4K
  • Last Post
Replies
1
Views
471
  • Last Post
Replies
11
Views
1K
  • Last Post
Replies
1
Views
774
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
992
Top