Definition: For an abstract object at the spot
of the manifold
, the Lie derivative
along the vector
is defined as:
.
For the tangent tensor fields defined as
we can use the following basic formulas:
,
.
These lead to following expressions for the components of the -tensor field
:
.
Note: the imprecise/incorrect but universally accepted notation is .
Examples:
,
.
A consequence of this last example is the case when $S_{\mu\nu}$ are components of the metric tensor $g_{\mu\nu}$, where one gets:
.
Note: in all above expressions for the Lie derivatives of the tensor components, it is possible to replace the ordinary partial derivatives with the corresponding covariant derivatives
.
—-
References: T1249 (1997.05.04)
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