Eikonal Blog

2010.01.05

Lie Derivative

Filed under: mathematics — Tags: — sandokan65 @ 16:14

Definition: For an abstract object {\cal T}(x) at the spot x of the manifold M, the Lie derivative {\cal L}_{\xi} along the vector \xi is defined as:

{\cal L}_{\xi}{\cal T}(x):\equiv \lim_{\epsilon\rightarrow 0} \frac1{\epsilon} \left[{\cal T}(x+\epsilon\xi)-{\cal T}(x)\right].

For the tangent tensor fields defined as
{\cal T}(x) = T_{\mu_1\cdots\mu_p}{}^{\nu_1\cdots\nu_q}(x)  dx^{\mu_1} \otimes \cdots \otimes dx^{\mu_p} \otimes \partial_{\nu_1} \otimes \cdots \otimes \partial_{\nu_q}
we can use the following basic formulas:
{\cal L}_{\xi} dx^{\mu} = (\partial_{\alpha}\xi^\mu) dx^\alpha,
{\cal L}_{\xi} \partial_{\nu} =  - (\partial_{\nu}\xi^{\beta}) \partial_{\beta}.

These lead to following expressions for the components of the (p,q)-tensor field {\cal T}:

({\cal L}_{\xi} T)_{\mu_1\cdots\mu_p}{}^{\nu_1\cdots\nu_q}(x) = \xi^\alpha\partial_\alpha T_{\mu_1\cdots\mu_p}{}^{\nu_1\cdots\nu_q} + \sum_{i=1}^{p} T_{\mu_1\cdots\hat{\mu_i}\alpha\mu_p}{}^{\nu_1\cdots\nu_q}(x) (\partial_{\mu_i}\xi^\alpha) - \sum_{j=1}^{q} T_{\mu_1\cdots\mu_p}{}^{\nu_1\cdots\hat{\nu_j}\beta\nu_q}(x) (\partial_{\beta} \xi^{\nu_j}).

Note: the imprecise/incorrect but universally accepted notation is {\cal L}_{\xi} T_{\mu_1\cdots\mu_p}{}^{\nu_1\cdots\nu_q}(x).

Examples:

  • {\cal L}_{\xi} \varphi = \xi^\alpha\partial_\alpha \varphi,
  • {\cal L}_{\xi} V_{\mu} = \xi^\alpha\partial_\alpha V_{\mu} + V_{\alpha} (\partial_\mu \xi^\alpha),
  • {\cal L}_{\xi} A^{\nu} = \xi^\alpha\partial_\alpha A^{\nu} - A^{\alpha} (\partial_\alpha \xi^\mu),
  • {\cal L}_{\xi} S_{\mu\nu} = \xi^\alpha\partial_\alpha S_{\mu\nu} +S_{\alpha\nu} \partial_\mu\xi^\alpha + S_{\mu\alpha} (\partial_\nu \xi^\alpha) = \partial_\mu (S_{\alpha\nu} \xi^\alpha) + \partial_\nu (S_{\alpha\mu} \xi^\alpha) - (\partial_\mu S_{\alpha\nu} + \partial_\nu S_{\alpha\mu}  - \partial_\alpha S_{\mu\nu} ).

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:

{\cal L}_{\xi} g_{\mu\nu} = \partial_\mu \xi_\nu  + \partial_\nu \xi_\mu  - 2 \Gamma_{\mu\alpha\nu}\xi^{\alpha} = \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu.

Note: in all above expressions for the Lie derivatives of the tensor components, it is possible to replace the ordinary partial derivatives \partial_\mu with the corresponding covariant derivatives \nabla_\mu.

—-
References: T1249 (1997.05.04)

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