Gauge transformation in cosmological perturbation

[Figaro]

Based on this lecture notes http://www.helsinki.fi/~hkurkisu/CosPer.pdf

For a given coordinate system in the background spacetime, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. As indicated in figure 2, the coordinate system ${\hat x^α}$ associates point $\bar P$ in the background with $\hat P$, whereas ${\tilde x^α}$ associates the same background point $\bar P$ with another point $\tilde P$.

1) My understanding here is that both coordinate ${\hat x^α}$ and ${\tilde x^α}$ give the same point in the background with different coordinate value, but what does equation (4.2) ( $\tilde x^α (\tilde P) = \hat x^α (\hat P) = \bar x^α (\bar P)$) mean? By this, does that mean there is an abuse of notation such that $\bar x^α (\bar P)$ is "THE" point?

The coordinate transformation relates the coordinates of the same point in the perturbed spacetime

$\tilde x^α (\tilde P) = \hat x^α (\tilde P) + ξ^α$
$\tilde x^α (\hat P) = \hat x^α (\hat P) + ξ^α$

2) What does he mean by this? Based on my search there is one statement saying,
A gauge transformation is different from a general coordinate transformation $Q (x)$ → $\tilde Q (x)$. In a gauge transformation, $\tilde Q$ and $Q$ are both calculated at the same coordinate value corresponding to two different space-time points, while in a general coordinate transformation both Q and x are transformed so that we are dealing with the values of a quantity at the same space-time point observed in the two systems.

Based on my understanding ,in a gauge transformation we use $\tilde x^α$ and $\hat x^α$ to evaluate the same point in the perturbative spacetime, it yields different spacetime points in the "background spacetime"? For a general coordinate transformation if we transform $Q → \tilde Q$ but evaluate at the same point $x$, the "TRUE" location of the point is not changed, just the coordinate system/observer's perspective just like in the passive rotations. But if we also transform $x → \tilde x$ then we not only change the perspective but also the "TRUE" position of the point, but in both observers's coordinate frame they see that the point has the same position with respect to their coordinate frame, is this correct? It seems to contradict the statement above that "we are dealing with the values of a quantity at the same space-time point observed in the two systems"

 

[cristo]

I'm not really sure what he means by that equation. The author is just trying to say that there is not a unique way in which to define the mapping between a point on the background spacetime and points on the perturbed spacetime. Therefore your second equations relate the coordinate systems on the perturbed spacetime to one another.

There are two ways of thinking about perturbation theory; namely the active and passive approaches. The author of this review uses the passive approach (where points are evaluated at the same "physical" point; i.e. point in the background spacetime), while the approach you are talking about in the last paragraph is the active approach (where points are evaluated at the same coordinate point). Either are valid ways of thinking about things.

 

[MyCosmicSchool]

In the cosmological perturbations, there are two scenarios.
1) Transformation at a given point, P. In this case, the physical location of the point in 4-spacetime is the same but their coordinates are different in different gauges.
2) Transformation at a given set of coordinates. In this case, the coordinates are fixed but the three points are physically separated in 4-spacetime.
This lecture series clarifies your doubt:

Watch the first 10 lectures (~ 2 hours) and the question will be answered. Let me know if anyone needs any further help in cosmological perturbations. It is not easy for beginners. Enjoy the ride!

 

[haushofer]

Welcome in the pure hell called diffeomorphism invariance, general covariance, passive and active transformations.

 

[haushofer]

But in trying to contribute: is it an idea to rephrase this whole gauge business first for perturbed solutions to the maxwell equations? Maybe that gives some insight.