No. of field equations and components or Riemann tensor?

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Discussion Overview

The discussion revolves around the relationship between the number of components of the Riemann tensor and the number of field equations in the context of curvature in space, particularly in relation to general relativity. Participants explore how these quantities compare in different dimensions, especially in the context of vacuum solutions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the implications of comparing the number of components of the Riemann tensor (N) to the number of field equations (M) in terms of curvature.
  • Another participant suggests that certain properties related to curvature may only manifest in spacetimes with more than three spatial dimensions.
  • A participant notes that in 1, 2, and 3 dimensions, the number of field equations is less than or equal to the number of components of the Riemann tensor.
  • There is a clarification regarding the indices of the Riemann tensor, with some participants indicating that they refer to the components denoted by alpha, beta, gamma, and delta.
  • One participant argues that the Riemann tensor has at least as many independent components as the Ricci tensor, and that the number of field equations corresponds to the components of the Ricci and Einstein tensors, which is less than or equal to the components of the Riemann tensor.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the relationships between the components of the Riemann tensor and the field equations, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants reference specific dimensional contexts and the structure of tensors, but there are unresolved assumptions regarding the implications of these relationships and the definitions of the terms used.

damnedcat
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no. of field equations and components or Riemann tensor??

Someone was trying to explain to me about curvature in space. From what I got from what they were saying doesn't make sense to me. I'm not sure I understand what the number of components, N, of R[tex]\alpha,\beta,\gamma,\delta[/tex] when compared to
the number of field equations, M, implies about curvature (when i say compare i mean: N>M, N<M, N=M). I thought looking at just the Riemann tensor tells u about curvature. mYBE i Just didn't get what he was saying. Any help?
 
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Was your friend referring to something like this? http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html See section 8.1.2, first para.

-Ben
 
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Something like that. He was saying that certain property can only occur for space times of more than 3 spatial dimensions. Since for instance the einstein eqn for vacuum in 1,2 and 3 dimensions have the number of field equations being less than or equal to the number of components of the Riemann tensor. I think he was talking about curvature of the space.
 


Sorry, but can you clarify by what you mean by alpha/beta/gamma/delta ?
 


nicksauce said:
Sorry, but can you clarify by what you mean by alpha/beta/gamma/delta ?

I think he means the indicies of the Riemann tensor?
 


damnedcat said:
Something like that. He was saying that certain property can only occur for space times of more than 3 spatial dimensions. Since for instance the einstein eqn for vacuum in 1,2 and 3 dimensions have the number of field equations being less than or equal to the number of components of the Riemann tensor. I think he was talking about curvature of the space.

I think this must be somewhat garbled. In any number of dimensions, the Riemann tensor, which has 4 indices, has at least as many independent components as the Ricci tensor, which has two indices and is constructed from it. The Einstein tensor has the same number of components as the Ricci tensor. The field equations have the Einstein tensor on one side. So the number of field equations will always be equal to the number of components in the Ricci and Einstein tensors, and less than or equal to the number of components in the Riemann tensor.
 

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