Contracting Tensors: Multiply by g^αρgασ?

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In summary, the conversation is about contracting Riemann curvature tensor to Ricci tensor and whether or not to multiply by g^{\alpha\rho}g_{\alpha\sigma}. The person asking the question is confused about the ordering of indices and is seeking clarification. However, the other person advises against starting a new thread and suggests continuing the discussion in the existing one.
  • #1
pleasehelpmeno
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When contracting [itex]R^{\sigma}_{ \mbox{ }\mu\nu\rho}[/itex] to [itex]R_{\mu\nu}[/itex]

Should one multiply by [itex]g^{\alpha\rho}g_{\alpha\sigma}[/itex]? I often get confused with ordering of indices and such like
 
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  • #3
there was no real answer, ignore the ricci tensor i just mean in general then, I don't really understand the order in which the indices should go, if it even matters at all.
 
  • #4
If you're not satisfied with the answers, you should say that in the same thread, not start a new one.
 
  • #5


I understand your confusion with the ordering of indices when contracting tensors. To answer your question, yes, you should multiply by g^{\alpha\rho}g_{\alpha\sigma} when contracting R^{\sigma}_{ \mbox{ }\mu\nu\rho} to R_{\mu\nu}. This is known as the metric tensor and it is used to raise and lower indices in tensor equations. In this case, the g^{\alpha\rho} term will raise the lower index \rho in R^{\sigma}_{ \mbox{ }\mu\nu\rho}, while the g_{\alpha\sigma} term will lower the upper index \sigma in R_{\mu\nu}. This will result in the indices being in the correct order for the contraction to be performed. It is important to pay attention to the ordering of indices in tensor equations, as it can greatly affect the final result. I suggest practicing with simpler tensor equations to become more familiar with the ordering of indices and the use of the metric tensor.
 

1. What is the purpose of contracting tensors?

The purpose of contracting tensors, specifically by multiplying by g^αρgασ, is to simplify and reduce the complexity of mathematical equations involving tensors. This process helps in solving problems related to relativity, quantum mechanics, and other fields of physics.

2. How is the contraction of tensors done?

The contraction of tensors is done by multiplying indices that are repeated in the same term. In the case of multiplying by g^αρgασ, the repeated indices are α and α, and the result is the contraction of the two tensors into a scalar value.

3. What is the significance of g^αρgασ in tensor contraction?

g^αρgασ is a metric tensor, which is used to define the geometry of space and time in the theory of relativity. In tensor contraction, it acts as a simplifying factor that reduces the complexity of equations involving tensors.

4. Can tensors be contracted in any order?

No, tensors cannot be contracted in any order. The order of contraction is important as it affects the final result. In the case of multiplying by g^αρgασ, the order of contraction must be αρ first, followed by ασ, to get the correct result.

5. Are there any other methods of tensor contraction?

Yes, there are other methods of tensor contraction, such as Einstein summation notation and index notation. These methods involve using indices and summation symbols to represent the contraction of tensors, rather than explicitly writing out the repeated indices.

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