Xjyuk

My questions concern the role of Lorentz transformations in Special Relativity and General Relativity, as described in the following fragment of the series of GR lectures: https://www.youtube.com/watch?v=iFAxSEoj6Go&list=PLFeEvEPtX_0S6vxxiiNPrJbLu9aK1UVC_&index=14&t=0s#t=102m50s 
(lecture 13 of International Winter School on Gravity and Light 2015 by Frederic Schuller, can be found also here: https://gravity-and-light.herokuapp.com/lectures)

In short, it says that:

1. The role of Lorentz transformations is exactly the same in SR and GR.



2. Namely, Lorentz transformations relate the frames of any two observers at the same point $p in M$ and as such are the change of the basis of the tangent space at $p$, $T_p M$.




3. Therefore, it is conceptually wrong to think of them as acting on the points of the spacetime manifold $M$ as transforming $x^mu to x'^mu = Lambda^mu_nu x^nu$.

Here are my questions:

1. Is there any physics textbook that follows consistently this way of thinking? People usually use $x^mu to x'^mu$ as the formula for the Lorentz transformations without mentioning that this is in any way improper.




2. How to think about the Lorentz invariance of laws, e.g. of the Maxwell equations? It was historically an important observation that the Maxwell equations are not Galilei invariant but Lorentz invariant, which led to the construction of SR. But checking the invariance of the equations amounts to checking how the equations behave when we change $x^mu to x'^mu$ $-$ at least this was always presented to me in this way.




3. The transformation $x^mu to x'^mu$ also seems to be used in the derivation of Noether's theorems.




4. If Lorentz transformations take place in the tangent space and translations take place in the spacetime, then how does it make sense to talk about the Poincaré group that encompasses them all?

The local Lorentz transformation are acting on the tangent space $T_p(M)$ to the curved GR manifold at each point $p$. The idea of a "tangent space" is a formal way of ascribing a flat space to the neighbourhood of $p$ in which we are so close to $p$ that we don't notice the curvature. This is in the same way that when we draw of map of a town, we do not notice that the surface of the Earth is a sphere and not an infinite plane. The $x^mu$ are coordinates in this neighbourhood. Each point $p$ has it's own neighbourhood with their origin at $p$. Although the neighbourhoods of nearby $p$'s will overlap, when we get sufficiently far away we can no longer maintain the convenient fiction that we are in a flat space.

The equivalence principle says that each point $pin M$ has a sufficiently small neighbourhood in which we don't notice the curvature and so, for example, the flat space Maxwell equations can be used. These flat-space equations are Lorentz invariant, so each point has its own attached group of Lorentz tranformations that act on the local coordinates $x^mu$ just as they do in SR.