Lorentz transformation in GR

Question

I try to do basics computations of SR with the heavier formalism of GR to see if I understand it well.

Change of coordinates is spacetime: changes of coordinates in space time are change of coordinate maps in the $(\mathbb{R}^4,\eta)$ Lorentzian manifold. For the cartesian coordinates we have one global map and it's the identity. If we want to go to other coordinate we take another atlas of $(\mathbb{R}^4,\eta)$ then we perform the changes of coordinate as seen in differential geometry courses.

Change of coordinates in tangent spaces: tangent spaces have a natural basis given by the coordinate on the manifold. To change coordinates in tangent spaces, it's the same thing as for general vector spaces: we do a linear combination of the basis vectors, then deduced how the components change, etc.

Problem: when we are talking about Lorentz boosts is the $x$ direction in RR, we usually write \begin{equation} \begin{bmatrix} \gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix} \begin{bmatrix} t\\x\\y\\z \end{bmatrix}= \begin{bmatrix} t'\\x'\\y'\\z' \end{bmatrix} \end{equation} with usual notations. We usually say that we "change of coordinate" from $(t,x,y,z)$ to $(t',x',y',z')$.

1. Since it's a linear transformation between four-vectors, is it a change of coordinates in a tangent space?

2. Aren't the $(x,y,z,t)$ suppose to be the coordinates in spacetime? I always saw $x^\mu$ as being the $\mu$-th component of a coordinate map $x:U\subset\mathcal{M}\to\mathbb{R}^4$, $\mathcal{M}$ a manifold.

3. More generally, what is really the role of Lorentz transformation in curve spacetime? What does "reference frame" really means is this context?

I would love to read about that but I didn't see anything in the classical GR references.

Answer 1

Lorentz transformation is a change of basis of tangent space. Tangent space is defined to mean a vector space associated with a point in spacetime. One should not confuse this with coordinates. Coordinates do not define vectors in general curved spacetimes. Thus we can apply Lorentz transformation to vectors, like momentum, but not to coordinates.

A reference frame consists of the reference matter, the apparatus, and the procedures, required to determine a spacetime coordinate system.

A coordinate system is a mapping from physical events to coordinates with the form $(x^0, x^1, x^2, x^3)$

Usually $(x^0, x^1, x^2, x^3) = (t, x, y, z)$where $t$ is the time of the event and $(x, y, z)$ describes the position of the event, or we may be using polar coordinates $(r,\theta, \phi)$, and we may also use more general coordinates.