Gauge invariance in GR perturbation theory

[victorvmotti]

I have been following [this video lecture][1] 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?**

[1]:

 

[Ambitwistor]

Thank you for bringing up this interesting and important question about gauge invariance in perturbation theory.

To answer your first question, it is important to understand the difference between the Lie derivative and the covariant derivative. The Lie derivative represents the change in a tensor field along a vector field, while the covariant derivative accounts for the change in a tensor field due to a change in the underlying coordinate system. In the case of gauge transformations, we are interested in the change in the metric components due to a change in coordinates, which is why we use the covariant derivative. Therefore, it is correct to write:

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

To answer your second question, when $\Delta_{\xi}\delta g_{ab}=0$, it means that the change in the metric components due to the coordinate transformation is zero. This implies that the perturbation is not affected by the choice of coordinates, which means it is a true physical perturbation and not a fake one. In other words, the perturbation is gauge invariant. Therefore, $g_{ab}+\delta g_{ab}$ is an objective new metric that is independent of the choice of coordinates.

I hope this helps clarify your doubts. If you have any further questions, please do not hesitate to ask. Keep up the good work in studying gauge invariance in perturbation theory!