Riemann tensor: indipendent components

In summary, the conversation discusses the number of independent components of the Riemann curvature tensor in different types and symmetries. It is mentioned that raising or lowering indices does not change the number of independent components, but the choice of base may affect the number. The concept of independence in relation to tensors is also briefly touched upon.
  • #1
AdeBlackRune
9
0
Hi, thanks for the attention and excuse for my bad english.
I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor.
We have two kind of riemann tensor:

type (3,1) Rikml

type (4,0) Rrkml

There are also some symmetry rule:

first skew-symmetry
Rrkml = -Rkrml
secondo skew-symmetry
Rrkml = -Rrklm
(Rikml = -Riklm)​
block symmetry
Rrkml = Rmlrk
Bianchi's first identity
Rrkml+Rrlkm+Rrmlk=0​
(Rikml+Rilkm+Rimlk=0)​

Now, the (4,0) R-type has
n2(n2-1)/12
indipendent components. What about the (3,1) R-type? The indipendent component are also n2(n2-1)/12 or they are n2(n2-1)/3?

(In the case n=4, (4,1) R-type has 20 indipendent component; (3,1) R-type are 20 or 80?)
 
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  • #2
Why would acting with the metric on the Riemann tensor change the number of independent components?
 
  • #3
haushofer said:
Why would acting with the metric on the Riemann tensor change the number of independent components?

I give up. Why would it not, though it does seems reasonable that raising or lowering indices would have no effect?


Come to think of it, we could take our N independent variables, and transform to Riemann normal coordinates, where the metric is Minkowskian, without loss of generality. With the metric now having constant entries, indices are raised and lowered without changing the number of independent variables.
 
Last edited:
  • #4
I suppose that raising or lowering indices leaves unchanged the number of indipendent components, but i was not able to find a confirm of that. I thought that, given the metric tensor, the Rikml are (linear?) functions of the Rrkml component. So, can 20 indipendent quantities generate more than 20 indipendent quantities?
 
  • #5
Of course rising and lowering indicies does not change the number of independent
components of any tensor. However the number of independent components
should depend of the choice of base. For example in coordinate (holonomic)
base the number of independent components of metric tensor is 10 but
in ortonormal base it's of course 0. The number of independent components
of Riemann tensor in 4 dimension in coordinate base is 20 but what about
ortonormal base. Does the extra condidtion imposed on base give in this example any
extra condition which have to be fullfiled by Riemann tensor?
 
  • #6
We cast around this word 'indendent' so casually at times, it's sometimes hard to tell at times what is independent of what. In the case of a vector, we like to cast it as an object that is independent of coordinate system or metric. The same applies to any tensor object.

In the case of the components of a tensor, I believe we are looking for the linear independence of tensor element, and perhaps eliminate additional degrees of freedom for other reasons.
 

1. What is the Riemann tensor?

The Riemann tensor, also known as the Riemann curvature tensor, is a mathematical object used to describe the curvature of a manifold in multi-dimensional space. It was developed by the mathematician Bernhard Riemann and is an important tool in the study of differential geometry and general relativity.

2. What are the independent components of the Riemann tensor?

The Riemann tensor has 20 independent components, which means that they cannot be expressed in terms of each other. These components represent the different ways in which the curvature of a manifold can be measured, such as the change in angle of a vector or the change in length of a geodesic curve.

3. How is the Riemann tensor calculated?

The Riemann tensor is calculated using the metric tensor, which defines the distance between points in a manifold. It involves taking derivatives of the metric tensor and using them to form a complex mathematical expression. There are also certain symmetries and identities that can be used to simplify the calculation.

4. What is the significance of the Riemann tensor in physics?

The Riemann tensor is a crucial component in general relativity, as it allows for the mathematical description of the curvature of space-time in Einstein's theory of gravity. It helps to explain the effects of gravity on the motion of objects and the behavior of light in the presence of massive objects.

5. Are there any applications of the Riemann tensor outside of physics?

Yes, the Riemann tensor has applications in other areas of mathematics and engineering, such as in the study of elasticity and fluid mechanics. It is also used in computer graphics and computer vision to create realistic 3D images and models. Additionally, it has applications in data analysis and machine learning for representing and analyzing complex data sets.

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