The Lie derivative (named after Norwegian mathematician Sophus Lie) is a differentiable (not Riemannian) version of differentiating with respect to a vector field [1]. It evaluates the change in a tensor quantity as it “flows” along a given vector field.
The Lie derivative generalizes a function’s directional derivative to higher rank tensors. Unlike the covariant derivative, which requires a connection on all spacetime, the Lie derivative only requires a curve [2].
Lie derivatives can be defined for functions, vector fields, differential forms and tensor fields. These aren’t much different from each other. For example, forms and tensor fields are really just extensions of vector fields. Lie derivatives are especially useful on a curved space and have many applications in physics and cosmology as well as in the study of symmetries of differential equations.
Notation
£XT [3].
Where:
- X = a vector field,
- T = a general tensor field.
Examples of Lie Derivative
The basic idea of a Lie derivative is a directional derivative on a differentiable manifold that depends on a vector field, but not on any particular choice of metric or connection [3].
The Lie derivative evaluates the change of a tensor field, along the flow induced by another vector field. As a simple example, imagine a fast-flowing river with two neighboring fluid elements (perhaps salt water and fresh water) connected by a vector measuring their separation. The Lie derivative tells us what would happen to the vector as the salt and fresh water followed the water’s flow lines.
Lie derivatives happen naturally in fluid flow problems and can simplify calculations. In a simple Newtonian physics context, a function can be dragged along by a fluid flow, or Lie derived by the flow-generating vector field. The Lie derivative of the function here is the directional derivative of the function along the vector field, where the rate of change of the function is measured by a comoving observer [6].
References
[1] Lecture 2 – Basic Concepts II – Lie Groups (2009). Retrieved March 16, 2021 from: https://www2.math.upenn.edu/~brweber/Lectures/USTC2011/Lecture%202%20-%20Basic%20Concepts%20II.pdf
[2] Symmetry (2013). Retrieved March 16, 2021 from: http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/GRKillin
g.pdf
[3] Lie Derivatives. Retrieved March 16, 2021 from: http://www.phy.olemiss.edu/~luca/Topics/l/lie_derivative.html
[4] Grøn, Ø., & Hervik, S. (2007). Einstein’s General Theory of Relativity: With Modern Applications in Cosmology. Springer Science & Business Media.
[5] Lie derivatives. Retrieved March 16, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.447.2246&rep=rep1&type=pdf