Covariant derivative and connection of a covector field

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Discussion Overview

The discussion revolves around the derivation of the covariant derivative of a covector field (1-form) and the associated connection symbols. Participants explore the product rule for covariant derivatives, particularly in the context of tensors, and the implications of these rules on the derivation process.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses concern that the covariant derivative does not have a product rule between covector and vector fields, suggesting a need for a more rigorous derivation.
  • Several participants question the understanding of the product rule for tensors, with some asserting it is defined in certain contexts, while others are unsure of its application to covectors and vectors.
  • A participant mentions using lecture notes from their professor and Sean Carroll's notes, indicating a reliance on these resources for understanding the topic.
  • There is a discussion about the nature of the product rule as it pertains to tensor products versus the interaction between covectors and vectors.
  • One participant proposes that taking the covariant derivative of a tensor product of a one-form and a vector can be approached using the Leibniz product rule, but emphasizes the need for the covariant derivative to commute with the tensor contraction operator.
  • Another participant points out that the requirement for commutation is stated in Carroll's notes, suggesting it is an important aspect of the derivation that may not have been adequately addressed in prior discussions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the application of the product rule for covariant derivatives involving covectors and vectors. There are competing views on the definitions and implications of the product rule in this context.

Contextual Notes

Some participants note that the definitions and requirements for the covariant derivative may not be fully articulated in the resources they are using, leading to confusion about the application of the product rule and the commutation with tensor contraction.

Vyrkk
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TL;DR
Deriving the expression for the covariant derivative of a covector field in components
I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors.
What people usually do is
  1. take the covariant derivative of the covector acting on a vector, the result being a scalar
  2. Invoke a product rule to develop in "(covariant derivative of vector)(covector) + (covariant derivative of covector)(vector) = Covariant derivative of scalar"
  3. substract the covariant derivative of the vector which is already known, leaving the expression for the covariant derivative for the covector on the other side.
E.g:
- https://www.physicsforums.com/threa...-derivative-for-covectors-lower-index.689141/
- https://math.stackexchange.com/questions/1069916/covariant-derivative-for-a-covector-field
- https://math.stackexchange.com/questions/1499513/covariant-derivative-of-a-covariant-vector
Well, I find something disturbing: the covariant derivative doesn't have such a product rule between a covector field and a vector field, or even between two vector fields. The only product rule it has is the Leibniz rule (which is one of the defining property of the axiomatic definition of the covariant derivative) between a vector field and a scalar field (a function on the manifold), but definitely no such thing between a vector and a covector.
What is an explanation for this, and a more rigorous derivation?
 
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What makes you think the product rule when acting on tensors is not defined? What book are you using?
 
Orodruin said:
What makes you think the product rule when acting on tensors is not defined? What book are you using?

It's not that I think it's not defined, it's that I haven't seen it defined anywhere.

As far as I know, the covariant derivative has a Leibniz rule for a scalar field times a vector field, and for the tensor product of any two tensors.

I am using my professor's Lecture notes and complement them with Sean Caroll notes.
 
It is in the very first definition of the connection on page 55 of Carroll’s lecture notes ...
 
Orodruin said:
It is in the very first definition of the connection on page 55 of Carroll’s lecture notes ...
Well, the product rule given p. 55 as part of the definition is a product rule for tensor product, not for a one form acting on a vector field or for two vector fields
 
Vyrkk said:
Well, the product rule given p. 55 as part of the definition is a product rule for tensor product, not for a one form acting on a vector field or for two vector fields
A one form acting on a vector is nothing but a contraction of a tensor product.
 
Orodruin said:
A one form acting on a vector is nothing but a contraction of a tensor product.
Mmh finally we're getting somewhere.
So if I take the covariant derivative of a tensor product of a one-form and a vector, I can develop using the Leibniz product rule for the tensor product. Taking the contraction on each side, I get my result.

But under one condition: the contraction has to go under the covariant derivative of the tensor product, otherwise I wouldn't know what to do with this term. In other words, the covariant derivative has to commute with the tensor contraction operator. Is this right?

In that case, the commutation would have to be a requirement and an important step in the derivation, which I haven't seen stated/done in any of the aforementionned derivations
 
Vyrkk said:
In that case, the commutation would have to be a requirement and an important step in the derivation, which I haven't seen stated/done in any of the aforementionned derivations
This is stated two pages later in Carroll's lecture notes as requirement 3, right before requirement 4, which is that it should reduce to the partial derivative on scalar fields.
 
Orodruin said:
This is stated two pages later in Carroll's lecture notes as requirement 3, right before requirement 4, which is that it should reduce to the partial derivative on scalar fields.
And now I understand why this requirement is needed, despite Caroll not stating its use anywhere in the derivation. Thank you!
 

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