A couple questions about the Riemann Tensor, definition and convention

[BiGyElLoWhAt]

TL;DR Summary

What is the dx^rho in the definition representative of? Why the order of indices, (wikipedia), compare pdf convention to wikipedia convention.

According to Wikipedia, the definition of the Riemann Tensor can be taken as $R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}$. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I understand the commutator representing parallel transport, as well as the partial sigma being the basis vectors being acted on. The thing here is that I'm not exactly sure what the dx is supposed to represent. My gut tells me that we're separating the "coefficients" bit of the basis vectors from the direction, and only taking the commutator of the direction, but that feels hand wavy and not 100% (possibly 0%) correct. Can someone elaborate? I know I put "I" as the level, but any explanation between "I" and "A" would be fine with me, noting that I am looking for a more conceptual understanding of this piece.

Q2/3:
I have been self studying GR and am working through the Ricci tensor calculations for the Schwarzschild metric. I am using a PDF to check my answers and have a couple issues (PDF). 1) First off, I did the contraction myself. I used the definition given in the 'coordinate expression' section of the Wikipedia page. I am (or at least was) under the assumption that the standard convention for contracting the riemann to the ricci was with the top and middle indices, and that's what I did. In the PDF linked, when I calculated R_tt or R_00, I got exactly negative of the answer (the contraction is on page 15 of the document (page number), the R_00 is on page 17). I looked up and the author did the contraction with the rightmost bottom index. Without seeing the definition they used prior to the contraction, it's hard for me to tell if I used a different order convention or simply used the wrong index in my contraction. Since swapping the order of a commutator gives the negative result, I am very much so inclined to believe that is what is at the heart of my negative sign issue, but I don't really just want to brush it off without knowing for sure. 2) Whats up with the order given on Wikipedia for the indices. Presumably it's to keep the symmetric indices together, but that is not how I would personally order my indices, were I to attempt to derive the Riemann. Is it just a convenience thing?

I would very much so appreciate someone that has experience with this matter looking at what is in the PDF and Wikipedia and letting me know about the conventions used, as well as the other questions.

(I know since they're homogeneous diff eq's for the schwarzschild vacuum solution, the negative signs won't really matter, but in general I feel they should)

Thanks in advance.

 

[Orodruin]

Summary:: What is the dx^rho in the definition representative of? Why the order of indices, (wikipedia), compare pdf convention to wikipedia convention.

According to Wikipedia, the definition of the Riemann Tensor can be taken as Rσμνρ=dxρ[∇μ,∇ν]∂σ.

You dropped the parentheses, which were important.
$$
R^\rho_{\sigma\mu\nu} = dx^\rho ([\nabla_\mu, \nabla_\nu]\partial_\sigma)
$$
Everything in the parenthesis is to be considered the argument of the one-form $dx^\rho$, which projects out the $\rho$ component of its vector argument.

Re: signs. You need to check all of the sign conventions used. There are typically three sign choices that are relevant in GR and reading texts that use different ones can really get confusing. The sign of the Ricci tensor is one of those choices.

The ordering of the indices is pretty much by standard convention.

 

[guru_4]

https://rpubs.com/lizhi1800/Riemann_curvature_tensor_2nd_kind

Here is a full code treatment of Riemman tensor.