Ricci Tensor & Trace: Exploring the Relationship Between Tensors & Traces

In summary, we discussed tensor contraction and how it relates to the trace of a matrix. We also learned that all even rank tensors can be contracted down to a nxn matrix, but odd rank tensors will be contracted to other odd rank tensors and ultimately a vector. We also discussed the Weyl tensor and its tracelessness in terms of tensor contractions.
  • #1
waht
1,501
4
Just wondering if Traces can be applied to tensors.

If the Ricci tensor is Rii then is sums over diagonal elements.

So technically, can you say the trace of the Riemann tensor is the Ricci tensor?
 
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  • #2
well what you are referring to is Tensor Contraction. A tensor contraction sums much like the trace of a matrix and gets rid of two indicies. Think of a matrix being a rank 2 tensor being contracted to a rank 0 (scalar) tensor. That is what a trace is, a specific form of a tensor contraction (if the matrix is made of tensor elements).
Here is the mathworld site:
http://mathworld.wolfram.com/TensorContraction.html
So in that case the Ricci tensor is contracted from the Riemann Tensor.
 
  • #3
Thanks alot,

For a trace, the matrix has to be n x n, right. So, can all tensors with a rank greater or equal to 2 be written as n x n matrices?

Sorry I'm asking a silly question, I didn't work with tensors in long time.
 
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  • #4
not exactly. Any tensor with an even rank can be contracted down to a nxn matrix (effectivly a rank 2 tensor), but odd rank tensors will be contracted to other odd rank tensors and ultimately a vector (n x 1 matrix).
Picture a rank 3 tensor as a n x n x n 'cubic matrix'. A contraction would basically take the trace of each level of this 'cubic matrix' and make a vector out of it.
I don't think there is a way to get rid of a single rank in the way of contraction. But all even rank tensors could be contracted (multiple times) down to a n x n matrix.
 
  • #5
Thanks, that makes a lot more sense now.
 
  • #6
One thing I should add:
The contraction can not be done with covariant or contravariant tensors , but only with mixed tensors. So, the Ricci tensor [tex]R_{ij}=R^k_{k,ij}[/tex] is the contraction of Riemann's tensor [tex]R^l_{k,ij}[/tex] (you can not contract [tex]R_{lk,ij}[/tex], one index should be rised).
Further, if you want to contract the Ricci tensor [tex]R_{ij}[/tex], you need first to rise the index 'i' and then contract [tex]g^{ki}R_{ij}[/tex] by k=j. The result will be the scalar curvature R.
 
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  • #7
gvk said:
One thing I should add:
The contraction can not be done with covariant or contravariant tensors , but only with mixed tensors. So, the Ricci tensor [tex]R_{ij}=R^k_{k,ij}[/tex] is the contraction of Riemann's tensor [tex]R^l_{k,ij}[/tex] (you can not contract [tex]R_{lk,ij}[/tex], one index should be rised).
Further, if you want to contract the Ricci tensor [tex]R_{ij}[/tex], you need first to rise the index 'i' and then contract [tex]g^{ki}R_{ij}[/tex] by k=j. The result will be the scalar curvature R.


Weyl tensor [tex]C_{abcd}[/tex] is traceless in the sense [tex]g^{ab} g^{cd} C_{abcd} = 0[/tex]? Am I right?
 
  • #8
Omega137 said:
Weyl tensor [tex]C_{abcd}[/tex] is traceless in the sense [tex]g^{ab} g^{cd} C_{abcd} = 0[/tex]? Am I right?

Sorry; I've made a mistake in the indexing...

I should have said

[tex]C_{abcd}[/tex] is traceless in the sense [tex]g^{ac} C_{abcd} = 0[/tex] or eqvivalently [tex] C^a_{bad} = 0 [/tex] for arbitrary [tex] b , d [/tex]

Sorry for the mistake...
 

What is the Ricci tensor?

The Ricci tensor is a mathematical object that describes the curvature of a space. It is a symmetric rank-2 tensor that is derived from the Riemann curvature tensor.

What is the significance of the Ricci tensor?

The Ricci tensor plays a crucial role in Einstein's theory of general relativity, which describes the gravitational interactions between objects in the universe. It is used to calculate the Einstein field equations, which relate the curvature of space to the distribution of matter and energy within it.

How is the Ricci tensor related to the trace?

The trace of the Ricci tensor is defined as the sum of its diagonal elements. In general relativity, this trace is related to the energy-momentum tensor, which describes the distribution of matter and energy in space. The trace can also be used to calculate the Ricci scalar, which is a scalar quantity that represents the overall curvature of space.

What is the difference between the Ricci tensor and the Riemann curvature tensor?

The Ricci tensor is a contraction of the Riemann curvature tensor, which is a rank-4 tensor that describes the full curvature of a space. The Ricci tensor is obtained by summing over two indices of the Riemann tensor, which reduces its complexity and makes it easier to work with in calculations.

What is the physical interpretation of the Ricci tensor?

The Ricci tensor represents the curvature of space and how it changes in the presence of matter and energy. It is used to calculate the gravitational effects of objects in space, such as the bending of light by massive objects or the motion of planets around a star. It is also used in cosmology to study the overall structure and evolution of the universe.

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