Mathematics of General Relativity - Tensorial Derivatives - The Lie Derivative

The Lie Derivative

Another important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection. Whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the Lie derivative uses a congruence from a vector field to achieve the same purpose. The idea of Lie dragging a function along a congruence leads to a definition of the Lie derivative, where the dragged function is compared with the value of the original function at a given point. The Lie derivative can be defined for type (r, s) tensor fields and in this respect can be viewed as a map that sends a type (r, s) to a type (r, s) tensor.

The Lie derivative is usually denoted by, where is the vector field along whose congruence the Lie derivative is taken.

The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. The Lie derivative of a scalar is just the directional derivative:

Higher rank objects pick up additional terms when the Lie derivative is taken. For example, the Lie derivative of a type (0, 2) tensor is

More generally,

\begin{align} &\mathcal L_X T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} = X^c(\nabla_cT^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) - \\ & \quad (\nabla_cX ^{a_1}) T ^{c \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\nabla_cX^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} + \\ & \quad (\nabla_{b_1}X^c) T ^{a_1 \ldots a_r}{}_{c \ldots b_s} + \ldots + (\nabla_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c}
\end{align}

In fact in the above expression, one can replace the covariant derivative with any torsion free connection or locally, with the coordinate dependent derivative, showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.

One of the main uses of the Lie derivative in general relativity is in the study of spacetime symmetries where tensors or other geometrical objects are preserved. In particular, Killing symmetry (symmetry of the metric tensor under Lie dragging) occurs very often in the study of spacetimes. Using the formula above, we can write down the condition that must be satisfied for a vector field to generate a Killing symmetry:

\begin{align} \mathcal L_X g_{ab} &= 0 \\ \Leftrightarrow \nabla_a X_b + \nabla_b X_a &= 0 \\ \Leftrightarrow X^c g_{ab,c} + X^c_{,a} g_{bc} + X^c_{,b} g_{ac} &= 0
\end{align}

Read more about this topic:  Mathematics Of General Relativity, Tensorial Derivatives

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