# Gauge invariance in GR perturbation theory

## Main Question or Discussion Point

I have been following [this video lecture] on how to find gauge invariance when studying the perturbation of the metric.

Something is unclear when we try to find fake vs. real perturbation of the metric.

We use an arbitrary small vector field to have the effect of a chart transition map or coordinate transformation.

When the change in the metric is given by Lie derivative of the metric along the vector field it is said that the change in the metric components is due to the coordinate change and therefore fake.

But it is unclear to me why we apply "the change" of the $$\delta g_{ab}$$ along the arbitrary small vector field $$\xi$$ and conclude that

$$\Delta_{\xi}\delta g_{ab}=\mathcal{L}_{\xi} g_{ab}$$

**Q1: Shouldn't it be that
$$\delta g_{ab}=\Delta_{\xi} g_{ab}=\mathcal{L}_{\xi} g_{ab}$$**

**Q2: Also, we assume that when
$$\Delta_{\xi}\delta g_{ab}=0$$ then the $$\delta g_{ab}$$ is not fake and therefore $$g_{ab}+\delta g_{ab}$$ an objective new metric, but how?**

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