Lie derivative with respect to anything else

In summary, the conversation is about the possibility of a different version of the lie derivative where it is taken with respect to a tensor field rather than a vector field. The person asking the question is wondering if this type of derivative exists and if it would be a reasonable concept. They also inquire if another way to describe the lie derivative could be "measuring the change in a tensor field with respect to the change in a vector field". The other person in the conversation mentions the possibility of defining the lie derivative with respect to a totally contravariant tensor as the tensor product of component-wise lie derivatives, which would result in another tensor.
  • #1
jfy4
649
3
Hi,

I have been looking around, and I can't seem to find a slightly different version of the lie derivative where the lie derivative is taken with respect to a tensor field, rather than a vector field. That is, a quantity which measures the change in a vector field, along the "flow" of a tensor field.

I am not asking about the lie derivative of a tensor, which is the change in a tensor field through the flow of a vector field.

does such a derivative exist? is this a reasonable question?

Also, would a different, but correct way to describe the Lie derivative be "it measures the change in a tensor field with respect to the change in a vector field" ?

Thanks,
 
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  • #2
I can't think of a definition of the lie derivative with respect to a covector off my head.
However, we may talk about the lie derivative with respect to a totally contravariant tensor.We could define it as the tensor product of component-wise lie derivatives. Such a quantity could be another tensor.
 

What is the Lie derivative with respect to anything else?

The Lie derivative with respect to anything else is a mathematical concept used in differential geometry and differential topology to measure how a vector field changes along the flow of another vector field. It is a generalization of the directional derivative in calculus.

What are the applications of the Lie derivative with respect to anything else?

The Lie derivative with respect to anything else has various applications in physics, particularly in the field of general relativity. It is used to study the behavior of matter and energy in curved space-time, as well as to calculate the gravitational effects of rotating bodies.

How is the Lie derivative with respect to anything else calculated?

The Lie derivative with respect to anything else is calculated using the Lie bracket, which is a mathematical operation that measures the difference between two vector fields. The Lie bracket is defined as the commutator of the two vector fields. The Lie derivative is then obtained by taking the limit as the difference between the two vector fields approaches zero.

What is the relationship between the Lie derivative with respect to anything else and Lie derivatives with respect to specific vector fields?

The Lie derivative with respect to anything else is a generalization of the Lie derivatives with respect to specific vector fields. It can be seen as a way of measuring how a vector field changes along the flow of any other vector field, rather than just a specific one. This allows for a more flexible and comprehensive understanding of vector field behavior.

Can the Lie derivative with respect to anything else be extended to other mathematical structures?

Yes, the concept of the Lie derivative with respect to anything else can be extended to other mathematical structures, such as Lie groups and Lie algebras. This allows for a more general and abstract understanding of the behavior of these structures and their associated vector fields.

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