# Modified Lie bracket [closed]

In a paper by Barnich, they use a different definition of the Lie bracket for vector fields at null infinity. Can somebody please give me the intuition behind using this Lie bracket in Equation 4.12:

$$\left[\xi_1,\,\xi_2\right]_M=\left[\xi_1,\,\xi_2\right]-\delta^g_{\xi_1}\xi_2+\delta^g_{\xi_2}\xi_1$$ where $\delta^g_{\xi_1}\xi_2$ denotes the variation in $\xi_2$ under the variation of the metric induced by $\xi_1$, $\delta^g_{\xi_1}g_{\mu\nu}=\mathcal{L}_{\xi_1}g_{\mu\nu}$.

Can somebody please write the variation terms explicitly, as to how to compute it.

The paper I am referring to is arxiv.org/abs/1001.1541

The reason to modify the lie bracket is the field dependence of $$\zeta^\mu$$.
For the full Einstein theory, the remaining parameters $$\zeta^\mu$$ persevering both gauge fixing and boundary conditions generally depend on spactime metric, i.e. $$\zeta^\mu=\zeta^\mu[g].$$ Notice that the commutator of two symmetry transformations is $$[\delta_{\zeta_1},\delta_{\zeta_2}]g_{\mu\nu}=(\delta_{\zeta_1}\delta_{\zeta_2}-\delta_{\zeta_2}\delta_{\zeta_1})g_{\mu\nu}.$$ The single variation of metric is given by original lie bracket $$\delta_{\zeta_2}g_{\mu\nu}=\mathcal{L}_{\zeta_2}g_{\mu\nu}.$$ Owning to the field-dependence of $$\zeta_2$$, the second time symmetry transformation is imposed as \begin{align} \delta_{\zeta_1}\delta_{\zeta_2}g_{\mu\nu}&=\mathcal{L}_{\delta_{\zeta_1}\zeta_2}g_{\mu\nu}+\mathcal{L}_{\zeta_2}\delta_{\zeta_1}g_{\mu\nu}\\ &=\left(\mathcal{L}_{\delta_{\zeta_1}\zeta_2}+\mathcal{L}_{\zeta_2}\mathcal{L}_{\zeta_1}\right)g_{\mu\nu}\, . \end{align} Finally, we have $$[\delta_{\zeta_1},\delta_{\zeta_1}]=\mathcal{L}_{[\zeta_2,\zeta_1]+\delta_{\zeta_1}\zeta_2-\delta_{\zeta_2}\zeta_1}\, ,$$ or say, the modified lie bracket corresponding to field variation is $$[\zeta_1,\zeta_2]_{M}=[\zeta_1,\zeta_2]-\delta_{\zeta_1}\zeta_2+\delta_{\zeta_2}\zeta_1\, ,$$ where $$\delta_{\zeta_1}\zeta_2[g]=\zeta_2[\mathcal{L}_{\zeta_1}g]\, .$$
• Thank you Lain! Will you provide us with an explicit example for a metric like one of black hole solutions? If $\xi_{2}[g]$ is an asymptotic killing vector preserving asymptotic fall-off conditions of a given metric, then one should first calculate $\mathcal{L}_{\xi_{1}}g$ after that checks how $\xi_{2}$ preserves fall-off conditions for $\mathcal{L}_{\xi_{1}}g$? Commented Aug 6 at 19:51