A Decoupling of SVT Metric Perturbations

[Zag]

Hello everyone,

I have been studying perturbation theory in the context of FRW cosmologies, and so far have had a really hard time understanding why the SVT (Scalar, Vector, and Tensor) perturbations associated with the metric tensor "decouple" at first order in perturbation theory.

All references avoid explaining this crucial step and simply jump to the final results by mentioning something along the lines: "Because the perturbations decouple, we can write these equations of motion for the scalars, and these equations over here for the vectors, etc."

However, it is not clear to me at all why these 3-scalars, 3-vectors, and 3-tensors which encapsulate the perturbations should evolve independently. In fact, Einstein equations mix them all into the same equation of motion, namely the field equations of general relativity. What is the argument to separate these perturbations into different equations? Where can I find a rigorous mathematical treatment which is not outdated?

Thanks a lot! Any reference and/or comment is appreciated.Zag

 

[kimbyd]

I think "at first order" is the clue here. I don't know all of the details, but they're likely using a linearized solution, where the linear parts are independent of one another. I'm pretty sure this is a trivial statement, honestly: first-order expansion in perturbation can have no couplings, because an "$x_1 x_2$" term would be a second-order term.

Taking this approximation is useful as long as the coupling terms are small compared to the non-coupled terms. In General Relativity, this is typically true as long as your density is varying smoothly. Such linearized solutions are useful for large-scale cosmology, such as the cosmic microwave background, but tend to break down in galaxy clusters. Working with such solutions requires recognizing when they break down. So you'd want to have a good treatment of precisely where the terms that aren't first order come in before attempting to apply TeVeS to a real system.

 

[Zag]

Hello @kimbyd . Thank you for your reply.

You are right. I ended up figuring things out and, indeed, if only first order terms in the perturbations are kept, the very Einstein equations decompose into scalar, vector, and tensor contributions. Since each of these contributions must be unique by the Helmholtz-Hodge decomposition theorem, each of these components end up giving rise to its own equation of motion.Zag