R computation from 1 independent Riemann tensor component

This means that when contracting with the metric, the position of the indices does not matter as long as they are in the middle.
  • #1
1,251
11
We have ##R^{1}_{212}## as the single independent Riemann tensor component, and I'm after ##R##. From symmetry properties and contracting we can attain the other non-zero components.

The solution then states that ##R_{11}=R^{1}_{111} + R^{2}_{121}=R^{2}_{121}## .

I thought it would have been
##R_{11}=R^{1}_{111} + R^{2}_{121}+R^{2}_{112}+R^{2}_{211}##

All I can think of then is that the upper indicie can only contract with the middle indice? I've never heard this before though, is that what's going on here?

Similarlly for ##R_{22}=R^{2}_{222} + R^{1}_{212}=R^{2}_{121}## is the solution.

The components ##R^{2}_{211}, R^{2}_{112}, R^{1}_{221}, R^{1}_{122}, ##are the unused components with two 2s, two 1s - if the upper indice only contracts with the middle, these would not come into the formula for any ricci vector...

If someone could let me know if I'm on the right or wrong track here,
cheers !
 
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  • #3
TSny said:

Ok. so it looks like only the middle indice contracts. But doesn't this differ then from when we contract vectors with the metric, here the position of the indices isn't important is it? When we contract with the metric all that matters is upper and lower? Thanks.
 
  • #4
binbagsss said:
Ok. so it looks like only the middle indice contracts. But doesn't this differ then from when we contract vectors with the metric, here the position of the indices isn't important is it? When we contract with the metric all that matters is upper and lower? Thanks.

The Ricci tensor is defined as ##R_{ab}\equiv R^c_{~acb}## the contraction is in the middle index.
 
  • #5


I would respond by saying that both you and the solution are correct, but you are considering different things. The solution is correct in terms of the specific independent Riemann tensor component that was given, which is ##R^{1}_{212}##. This component only involves the first and second indices, so when contracting with the first index to get ##R_{11}##, only the components with two 1s and one 2 (such as ##R^{1}_{111}## and ##R^{2}_{121}##) will contribute. The other components with two 2s and one 1 (such as ##R^{2}_{211}## and ##R^{2}_{112}##) will not contribute to the calculation of ##R_{11}##.

However, you are also correct in that the components with two 2s and one 1 are not completely unused. They will contribute to other independent Riemann tensor components, such as ##R^{2}_{222}## and ##R^{1}_{221}##, which can then be used to calculate other Ricci tensor components like ##R_{22}##. So while they may not directly contribute to the calculation of ##R_{11}##, they are still important in the overall calculation of the Riemann and Ricci tensors.

In summary, both you and the solution are correct, but you are considering different aspects of the problem. It is important to understand the specific components and their relationships when working with tensors.
 

1. What is R computation from 1 independent Riemann tensor component?

R computation from 1 independent Riemann tensor component is a mathematical process used to calculate the curvature of a space. It involves using the Riemann tensor, which is a mathematical object that describes the curvature of a space, to calculate the curvature at a single point in that space. This is done by plugging in the values of the tensor's components at that point into a specific equation.

2. Why is R computation from 1 independent Riemann tensor component important?

R computation from 1 independent Riemann tensor component is important because it allows us to understand the curvature of a space at a specific point. This information is crucial in fields such as general relativity, where the curvature of space-time affects the motion of objects and the behavior of gravity. It also helps us to understand the overall geometry of a space and its properties.

3. What are the steps involved in R computation from 1 independent Riemann tensor component?

The first step in R computation from 1 independent Riemann tensor component is to identify the space and the point at which we want to calculate the curvature. Then, we need to determine the values of the Riemann tensor components at that point. Next, we plug these values into the equation for R computation, which involves multiplying the components by specific matrices and summing them together. Finally, we solve the equation to get the value of the curvature at that point.

4. Are there any applications of R computation from 1 independent Riemann tensor component?

Yes, there are many applications of R computation from 1 independent Riemann tensor component. As mentioned before, it is crucial in understanding the behavior of gravity and the geometry of space in general relativity. It is also used in other fields such as cosmology, astrophysics, and differential geometry. Additionally, it has applications in computer graphics and computer vision, where it is used to calculate the curvature of surfaces.

5. Is R computation from 1 independent Riemann tensor component difficult to understand?

R computation from 1 independent Riemann tensor component can be challenging to understand at first because it involves complex mathematical concepts and equations. However, with proper study and practice, it can be comprehended by anyone with a strong foundation in mathematics and physics. It is a fundamental concept in many fields of science and is worth investing time and effort to understand it thoroughly.

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