Question from Weinberg's cosmology text

[blueeyedblond]

I am trying to learn gauge issues in cosmology from Weinberg's "Cosmology". However, I am completely stumped at how he gets 5.3.4 from 5.3.3. I was doing great until that point and now I'm going crazy. Any help would be highly appreciated!

 

[atyy]

He doesn't.

He wants to show that 5.3.5 (i) reduces to 5.3.4 in the absence of gravitation (ii) further that 5.3.5 is a scalar. (i) is obvious, (ii) he follows the reasoning 5.2.12 and 5.2.13.

 

[blueeyedblond]

He doesn't.

He wants to show that 5.3.5 (i) reduces to 5.3.4 in the absence of gravitation (ii) further that 5.3.5 is a scalar. (i) is obvious, (ii) he follows the reasoning 5.2.12 and 5.2.13.

atyy,

Thanks so much for taking the time to reply!

However, I was referring to the recent book "Cosmology" by Weinberg and not the older "Gravitation and Cosmology" which is what you seem to be referring to to. (For a moment I thought you're a crazy person!)

I think I've figured out the answer to my question, but if you have any insights on that you're very welcome to let me know.

Sorry for the confusion!

 

[atyy]

Ah ha ha ha! Sorry about that! (But I am a crazy person )

 

[arkajad]

I don't know if this will help you or not, but (5.3.4) is nothing but the minus Lie derivative of the unperturbed metric with respect to the vector field $$\epsilon^\lambda(x)$$. So it describes the rate of change of the metric dragged along the trajectories of $$\epsilon$$ with respect to the original metric.

 

[blueeyedblond]

I don't know if this will help you or not, but (5.3.4) is nothing but the minus Lie derivative of the unperturbed metric with respect to the vector field $$\epsilon^\lambda(x)$$. So it describes the rate of change of the metric dragged along the trajectories of $$\epsilon$$ with respect to the original metric.

That is absolutely correct. And the key to deriving it is to use the two unnumbered equations below 10.9.6 in Weinberg's "Gravitation and Cosmology" book.