Is the Lie derivative a tensor itself?

[spinless]

Hi everyone!

A few days ago in General Relativity class, the professor introduced the concept of Lie derivative and at the end he mentioned that the Lie derivative was a tensor itself. I've been looking everywhere, but I only find how it acts on vectors, tensors, etc. Does anyone know of any book/article/website that mentions this idea?

Thank you very much in advance!!

 

[fresh_42]

Here are some definitions:
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/#Lie-Derivatives

If $X,Y$ are vector fields, then $\mathcal{L}_X(Y)=[X,Y]$ which is again a vector field. $\mathcal{L}$ is linear in both arguments, so you may consider it a tensor. How it acts can be seen in the article I linked to.

Every derivative is a directional derivative. It measures variation in that direction. There are three main parameters: function, direction, and location. All of them can be the variable that results in different quantities: a number, a function, or even a field of directions. We abstract from function and location and consider directions alone in the case of vector fields.

 

[PeterDonis]

Moderator's note: Thread moved to the Differential Geometry forum.

 

[martinbn]

Hi everyone!

A few days ago in General Relativity class, the professor introduced the concept of Lie derivative and at the end he mentioned that the Lie derivative was a tensor itself. I've been looking everywhere, but I only find how it acts on vectors, tensors, etc. Does anyone know of any book/article/website that mentions this idea?

Thank you very much in advance!!

The Lie derivative of a tensor is a tensor. Just like the derivative of a functions is a function.

 

[haushofer]

Hi everyone!

A few days ago in General Relativity class, the professor introduced the concept of Lie derivative and at the end he mentioned that the Lie derivative was a tensor itself. I've been looking everywhere, but I only find how it acts on vectors, tensors, etc. Does anyone know of any book/article/website that mentions this idea?

Thank you very much in advance!!

You can find the general expression for the Lie derivative of a tensor with respect to some vector field $\xi$. Just apply the definition, and perform a general coordinate transformation to see whether you get inhomogenous terms. E.g., for a vector field $V$ you obtain in component form

$L_{\xi} V^{\mu} = \xi^{\lambda} \partial_{\lambda}V^{\mu} -V^{\lambda}\partial_{\lambda}\xi^{\mu}$

If you're dealing without torsion, you can actually show that this equals

$L_{\xi} V^{\mu} = \xi^{\lambda} \nabla_{\lambda}V^{\mu} -V^{\lambda}\nabla_{\lambda}\xi^{\mu}$

Similar arguments hold for Lie derivatives of general tensor fields.