Covariant and Lie Derivatives

Notation is the metric, and are the Christoffel symbols. is the covariant derivative, and is the partial derivative with respect to . is a scalar, is a contravariant vector, and is a covariant vector. is a scalar density of weight 1, and is a scalar density of weight w. (Note that is a density of weight 1, where is the determinant of the metric. It follows that is a scalar.)

Covariant Derivatives

Covariant derivatives are defined by

(1) The extension to tensors of different types is straightforward. Note the useful result for the divergence of a vector field: Lie Derivatives

Lie derivatives with respect to a vector field are defined by

(2) The extension to tensors of different type is straightforward. Lie derivatives do not rely on the presence of a metric for their definitions. When a metric is present, they can be written in terms of covariant derivatives by simply replacing with in the formulas above. Note the useful result for the Lie derivative of the metric tensor: A Killing vector field satisfies .