The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1


**1.1. Lemma.**


If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ is any symmetric two tensor, then
$$
\delta\left[\operatorname{DEin}\left({ }^{(4)} g\right) \cdot ^{4} h\right]=0
$$
where $\delta=\delta\_{(4) g}$ is the divergence with respect to ${ }^{(4)} g$.


They gave the following proof:


The contracted Bianchi identities assert that $\delta \operatorname{Ein}\left({ }^{(4)} g\right)=0$. Differentiation gives the identity
$$
\*\*\left[\text { D } \delta\left({ }^{(4)} g\right) \cdot ^{4}h\right] \cdot \operatorname{Ein}\left({ }^{(4)} g\right)+\delta\left[D \operatorname{Ein}\left({ }^{(4)} g\right) \cdot^{4} h\right]=0\*\*
$$
where $\delta\left({ }^{(4)} g\right)=\delta\_{(4)}$ indicates the functional dependence of $\delta$ on ${ }^{(4)} g$, and $\left[\mathrm{D} \delta\left({ }^{(4)} g\right) \cdot ^{4} \mathrm{h}\right]$ is the linearized divergence operator acting on Ein $\left({ }^{(4)} g\right)$. The lemma follows since $\operatorname{Ein}\left({ }^{(4)} g\right)=0$.


Sadly, I cannot see how they found the identity in the second sentence of the proof. Could you give me some hints/suggestions on how to derive the identity in the second sentence of Lemma 1.1?


Any help/suggestions would be highly appreciated. Thanks so much.


1*Fischer, Arthur E.; Marsden, Jerrold E.; Moncrief, Vincent*, [**The structure of the space of solutions of Einstein’s equations. I: One Killing field**](http://www.numdam.org/item?id=AIHPA_1980__33_2_147_0), Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 33, 147-194 (1980); [eudml](https://eudml.org/doc/76090). [ZBL0454.53044](https://zbmath.org/?q=an:0454.53044), [MR605194](https://mathscinet.ams.org/mathscinet-getitem?mr=605194).



Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper
I presume the formula you are asking about is the long one highlighed by $\*$ $\*$ in your question, while the standard "contracted Bianchi identity" $\delta \operatorname{Ein}\left({ }^{(4)} g\right)=0$ poses no mystery to you. The Longer equation is simply obtained by using the Leibniz rule while applying the functional derivative $\mathrm{D}$ that last identity. I would consider that a pretty clear explanation, but possibly that is not the point you are confused about.


Perhaps you are wondering why the Leinbniz rule applies to $\delta$, which is an operator. A quick way to see why is just to write the corresponding expression in coordinates. There is no need to be very explicit, the following schematic form suffices:
$$
 \delta \mathrm{Ein} = \partial \cdot \mathrm{Ein} + \Gamma \cdot \mathrm{Ein} .
$$
That is, the covariant divergence operator $\delta(-)$ consists of a coordinate derivative part $\partial\cdot(-)$ and of the part containing multiplication by Christoffel symbols $\Gamma\cdot(-)$. Unsurprisingly, the coordinate derivative part has no dependence on the metric, hence formally $\mathrm{D}\partial = 0$. So $\mathrm{D}\delta$ comes entirely from the Christoffel part, which quite obviously obeys the Leibniz rule.


If that's still not where your confusion lies, please clarify your question further.


