A couple questions about the Riemann Tensor, definition and convention

In summary, the Riemann tensor is a one-form that projects out the ##\rho## component of its vector argument. It is important to keep in mind the order of the indices when contracting the tensor, and the convention used for signs.
  • #1
BiGyElLoWhAt
Gold Member
1,622
131
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.
 
Physics news on Phys.org
  • #2
BiGyElLoWhAt said:
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.
 
  • Like
Likes BiGyElLoWhAt

Related to A couple questions about the Riemann Tensor, definition and convention

1. What is the definition of the Riemann Tensor?

The Riemann Tensor is a mathematical object that describes the curvature of a manifold in terms of its metric tensor. It is used in the study of differential geometry and general relativity.

2. What are the components of the Riemann Tensor?

The Riemann Tensor has 256 components in total, but due to its symmetries, only 20 of these are independent. These 20 components are further divided into 4 groups, each representing a different type of curvature.

3. What is the convention for writing the Riemann Tensor?

The Riemann Tensor is typically written in index notation, with four indices representing the four dimensions of spacetime. The convention for the order of these indices varies, but a common convention is to write the tensor as Rαβγδ, where α, β, γ, and δ can take on values from 0 to 3.

4. How is the Riemann Tensor related to the Ricci Tensor and Scalar Curvature?

The Ricci Tensor is a contraction of the Riemann Tensor, meaning it is formed by summing over two of the indices of the Riemann Tensor. The scalar curvature is then obtained by contracting the Ricci Tensor once more. This relationship is known as the Ricci decomposition of the Riemann Tensor.

5. What are some applications of the Riemann Tensor?

The Riemann Tensor is essential in the study of general relativity and the theory of gravity. It is used to describe the curvature of spacetime, which is directly related to the distribution of matter and energy. The Riemann Tensor also plays a crucial role in the formulation of Einstein's field equations, which describe the relationship between matter and spacetime curvature.

Similar threads

  • Differential Geometry
Replies
34
Views
2K
  • Special and General Relativity
Replies
10
Views
780
  • Differential Geometry
Replies
3
Views
3K
  • Differential Geometry
Replies
7
Views
3K
  • Special and General Relativity
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Special and General Relativity
Replies
8
Views
3K
  • Special and General Relativity
Replies
1
Views
1K
Replies
1
Views
3K
Back
Top