Remarks:
In the following explanation 4-dimensional space-times $M$ equipped with a metric of signature (3,1) are considered.
There are several Wikipedia pages treating frames (sometimes called tetrads or Vielbeins) in GR. See for example, here, here and here
There is a very good introductory chapter on the subject in chapter 5 of these notes by: R. Aldrovandi and J. G. Pereira.
A frame in GR means a set of four vector fields $\mathbf{e}_a: M \rightarrow TM$, $a=0, 1,2,3$ satisfying the constraint equation:
$\mathbf{g} = \eta^{ab} \mathbf{e}_a \mathbf{e}_b$,
where $\mathbf{g}$ is the inverse metric tensor and $\eta^{ab}$ is the flat Lorentzian metric.
These vector fields can be thought of as the mapping of the coordinate vectors of some given Mikowski space through the local coordinate system to the tangent space. In physical terms, we associate each such a frame with a local observer.
Now, basically, we can work with the components of the frame vector fields instead of the metric, but one observes that the frame fields have 16 components, while the metric has (due to its symmetry) only 10 components.
This redundancy is due to the fact that the frame fields are not unique and a new set of frame fields $\mathbf{e}^{\prime}_a$ satisfying
$\mathbf{e}^{\prime}_a = M_a^b(x) \mathbf{e}_b$
satisfies the same constraint, where $M_a^b(x)$ is a Lorentz transformation matrix (i.e. satisfying $M_a^b(x) M_c^d(x) \eta_{bd} = \eta_{ab}$)
Please observe that we can choose a non-constant Lorentz transformation depending on the location on the manifold, for this reason, these transformations are called local Lorentz transformations.
Now the dimension count checks: 16 frame components = 10 Metric components + 6 Lorentz transformations at every point.
This formalism may seem merely a change of variables, but this is not the whole story.
First, the local Lorentz transformations can be viewed as sections of a principal $SO(3,1)$ bundle over $M$ (This bundle is called $SO(M, \mathbf{g})$.
Thus this formulation is a formulation of GR as a gauge theory.Now, since we can allow the local Lorentz transformations to depend on the coordinates, this formalism allows to define accelerating frames, simply by taking thelocal Lorentz transformations to depend on time.
Secondly, in the standard formulation of GR allows to can define classical fields as sections of bundles whose local transformations are functions of the coordinate transformation (diffeomorphisms) of the base manifold.
These bundles are called natural bundles, for example the coordinate transformation of the tangent bundle is the Jacobian matrix of the coordinate transformations of the base manifold. (Similarly, the inverse Jacobian matrix for the cotangent bundle).
Thus the standard formulation of GR allows the definition of vector fields, tensor fields etc. but not spinor fields, which are very important in physics.
Spinor bundles are not natural, but there is no natural way to define a general coordinate transformation of a spinor field given a diffeomorphism of the base manifold.
However, if the base manifold $M$ has a spin structure, then the frame formalism allows to define spinor fields as follows: Since $M$ is spin, $SO(M, \mathbf{g})$ can be lifted to a spin bundle $Spin(M, \mathbf{g})$ , then a spinor bundle is the associated bundle corresponding to a fundamental spinor representation, and spinor fields are sections of the spinor bundle.
This construction can be performed in local coordinates follows:
First, we can form the dual frame $\mathbf{e}^a: M \rightarrow T^{*}M$ by requiring:
$\langle \mathbf{e}^a, \mathbf{e}_b \rangle = \delta^a_b$
The dual frame can be used to define the frame components of any vector field $\mathbf{V}$:
$V^a = \langle \mathbf{e}^a, \mathbf{V} \rangle$
Conversly, one can form the "curved" components of a vectors using the original frame. For example, consider the Dirac matrices $\{\gamma^a\}$ generating the Clifford algebra $Cl(3,1)$.
Then their curved components are given by:
$\gamma^{\mu} = \gamma^a e_a^{\mu}$
More generally, one uses the metric $\mathbf{g}$ to lower "curved indices", the inverse metric to raise "curved indices".
and similarly, the Lorentz metric $\mathbf{\eta}$ for the flat indices. One uses the frame vectors and their dual to replace curved indices with flat indices and vice-versa.
Next ,the spin connection
is defined as:
$\omega_{\mu}^{ab} = e^a_{\nu}(\partial_{\mu} e^{\nu b}+ e^{\sigma b}\Gamma^{\nu}_{\sigma \nu})$
where, $\Gamma^{\nu}_{\sigma \nu}$ is the Levi-Civita connection.
It is not difficult to verify (by looking at the local Lorentz transformations) that $\omega_{\mu}^{ab}\sigma_{ab}$ is a connection on $Spin(M, \mathbf{g})$, where $\sigma_{ab}$ are the generators of the fundamental spinor representation.
- Using the above data, the fully covariant Dirac equation on $M$ takes the form:
$-i \gamma^{\mu} D_{\mu} \psi + m \psi = 0$,
where $D_{\mu}$ is the covariant derivative associated with the spin connection
$ D_{\mu} = \partial_{\mu}-i\omega_{\mu}^{ab}\sigma_{ab}$
Thus the fully covariant Dirac equation looks just like the Dirac equation coupled to a gauge field given by the spin connection.
Classical fields where this construction is possible are sections of bundles called "gauge natural bundles".
It is important to mention that the solution of the fully covariant Dirac equation depends on the frame fields, but observable quantities such as the number of bound states for example, depend only on the metric.
Update:
Since local observers are identified with points on the fibers of the frame bundle, then all frames are inertial because they can be obtained from the action of a Lorentz transformation on a single frame (i.e. point on the fiber).
The parameters of the lorentz transformation are the velocity vector and the orientation of the frame. It is explicit in the equations that we can allow variable Lorentz transformations, i.e., Lorentz transformations dependent
on the local coordinates of the base manifold, in particular on the time coordinate.
Now I'll divide my answer into two parts:
Particles: Suppose the four velocity vector of a particle moving on a geodesic is given by $V^{\mu} = \frac{dx^{\mu}}{d\tau}$, ($\tau$ is any parameter along the path) then the frame coordinates of this vector are: $V^a = e^a_{\mu} V^{\mu}$ and the components of the velociy measured by an observer moving with a velocity defined by the Lorenntz matrix $M(x)$ are
$V^{\prime b} = M^b_a(x) V^a$. Again, a variable $M$ indicates an accelerating frame.
Fields: The equations of motion will be covariant with respect to these transformations, because for sections of natural bundles, the frame vectors
do not appear in the equations of motion, while in the case of gauge natural sections such as spinors these (variable Lorentz) transformations will appear as gauge transformations and the equations of motion are constructed to be gauge invariant.
Thus, the equations of motion are not affected by local Lorentz transformations, or in other words, Physics looks the asme to all observers even if they are accelerating.
Given a $n+1$ dimensional manifold $M$ one by definition has charts or coordinates that are homeomorphic to $\mathbb{R}^{n+1}$. This is independent of any Lorentzian or Riemaniann metric on $\cal{M}$.
Now if the manifold $M$ admits a Lorentzian metric $g$ then the coordinates use to define the manifold also define the components of the metric.
A common way to define a spacetime even if it is not know the whole manifold $M$ is to work locally. Even if the topology of $M$ is not $\mathbb{R}^{4}$, locally it is. Then using the local (flat) coordinates one define locally a metric and imposing some symmetries and physical conditions one can arrive to relevant metrics such as the Schwarzschild metric.
However, another way to find solutions to Einstein's equations is to see them as an initial value problem. That is given on a hypersurface $\Sigma$ the first and second fundamental form one can determine a Lorentzian manifold $M$ using Einstein's equations.
Regarding your comment of how to make the sense of coordinates. In this case we have chosen an initial n-dimensional $\Sigma$ where we know the coordinates by definition. Then, also one define a lapse function $N$ and shift vector $\beta$ on $\Sigma$. These choices define locally a chart with topology $M={\mathbb{R}} \times {\Sigma}$. The choice of the lapse and shift sometimes are related with physical observers, but others are used for mathematical convenience such as the harmonic slicing. See section 9 This define the $n+1$- dimensional coordinates on $M$. In fact this is the topology of all globally hyperbolic spacetimes.
However, this is only a local characterization. If one wants the total manifold then one is interested in maximal extensions. For example, the chart you used has a well know coordinate singularity at $r=2M$ and therefore one needs to change the chart and coordinates to cover the full spacetime. Notice that the existence of the singularity is responsible for the change in the global topology.
The concept of maximal extension is related to geodesic completeness or the well-posedness of Einstein equations (no curvature singularities). If this two criteria are equivalent is part of the Strong Cosmic Censorship.
The global case which correspond to the question of the maximal extension of the space-time as as a well-posed problem of Einstein's equations is an active area of research. The Strong Cosmic Censorship deals with the uniqueness of global solutions.
Best Answer
Let me first mention that a maximal atlas contains all possible coordinates you can choose on the manifold. None of them need to cover the entire manifold and often none of them actually do (for example, there is no coordinate system that covers the entire Earth at once: you'll always miss at least one point).
Let us now consider the Schwarzschild spacetime. We equip it with a coordinate system that holds for $r > 2 M$, $t \in \mathbb{R}$. We notice the metric still seems reasonable for $r < 2M$, but at $r = 2 M$ some components either vanish or diverge and we can't really use it there. Since we are typically on situations with $r > 2M$, let us start considering only the $r > 2M$ region.
When doing computations on this region, we notice that there are geodesics that go up to $r = 2M$. This happens in infinite static time $t$, but in finite proper time. This prompts us to investigate that region. In order to do so, we can make a change of coordinates to, for example, Kruskal–Szekeres coordinates. We are doing this transformation just in the $r > 2M$ region, where everything works fine.
Now here's the interesting bit: the apparent singularity at $r = 2M$ is no longer appearing on our new set of coordinates. If we compute the geodesic motion, we'll see a body passing through the region which would be previously identified as $r = 2M$ and keeps going. If, out of curiosity, we decide to make a similar coordinate transformation for $r < 2M$, we find out the body has entered that region which previously we did not know if was physical.
Now we notice that although we initially started with only $r > 2M$, we now realize that there was still spacetime beyond that region because geodesics come and go from places we weren't previously describing. Hence, in our new charts, we include these new regions. I like to parallel differential geometry with cartography and imagine this process as an explorer who had a map of some jungle, for example. One day, they go further than the region shown in the map. Of course, nothing changed in the jungle or in the manifold, but now they might need to use different maps to understand this different place.
Being a bit more direct:
You don't necessarily need to think of the coordinate transformation as being singular. You can just make it for $r > 2M$, $\theta \in (0,\pi)$, where it is not singular, and then notice that geodesics will still go through the region you thought was singular, so you just update your new coordinates to cover that region. F
For example, pick spherical coordinates. $\theta = 0$ and $\theta = \pi$ are singular. Yet, you can perform the coordinate transformation to Cartesian with $\theta \in (0,\pi)$ and afterwards realize that you have geodesics going through $x^2 + y^2 = 0$, so you simply admit $x = 0$ and $y = 0$ as a possible value for your coordinates. In the case for Schwarzschild, we did that, and also noticed this ended up allowing us to cover the region with $r < 2M$ as well in a single attempt.
As I mentioned in the beginning, a maximal atlas consists of all possible coordinate systems that allow for smooth transitions among themselves. We typically choose to assume maximal atlases out of convenience. When you change coordinates, you don't change the atlas nor the manifold. In a more cartographic language, the Earth won't change if you one day decide you want to use a Gall–Peters projection instead of a Mercator projection. The original choice of coordinates was made out of pure convenience and is not an intrinsic property of the manifold.
There is a minor caveat, however.
Maximal Extensions
I initially asked for us to consider spacetime only for $r > 2M$, as that was the available region for our coordinates which we considered physical. After performing a coordinate transformation, we realized there were other regions our geodesics could access and we chose to consider them as part of spacetime as well. In a more technical jargon, we found an extension of the spacetime. In the particular case of the Schwarzschild solution, we found the unique maximal analytic extension of the Schwarzschild spacetime. In other words, the unique analytic manifold that coincides with the Schwarzschild metric in the region $r > 2M$ and cannot be further enlarged.
The thing is: initially, if we so desired, we could consider the original spacetime to be only the region for $r > 2M$ (i.e., we could consider spacetime to be only the right wedge of the Schwarzschild spacetime's Penrose diagram). In this situation, once we changed coordinates and allowed them to run over to values different than $r > 2M$, we ended up in a larger manifold (larger in the sense it contains the original spacetime as a submanifold).
Typically, we assume all spacetimes to be maximally extended, i.e., we assume that it is impossible to find another spacetime that is larger than the one we're considering (see Hawking & Ellis 1973, Sec. 3.1, for some more detail on the technical bits). From a physical point of view, this means demanding from the start that if a geodesic can reach some point, then that point should belong to spacetime. It is the view that we screwed up by taking the Schwarzschild coordinates too seriously: they are just an arbitrary choice of coordinates and never had the obligation of covering up the whole manifold. It is the view that we should recognize the poles are a part of Earth, even if their longitude is ill-defined. The view that the jungle exists regardless of whether our maps cover it.