In special relativity, to derive formulas for proper time and proper distance, one uses the concept of instantaneous rest frames.
In order to calculate proper time of a massive object moving in the flat spacetime where events are described by coordinates $(T,X,Y,Z)$, one can find an inertial frame where an object remains at rest. Such frame is called an instantaneous rest frame. Spacetime in this frame can be described by metric $ds^2=-c^2d\tau^2+dx^2+dy^2+dz^2$ where $\tau$ is object's proper time and $x,y,z$ are coordinates in rest frame. Next, we equate this metric with the flat spacetime metric:
$$ds^2=-c^2d\tau^2+dx^2+dy^2+dz^2=-c^2dT^2+dX^2+dY^2+dZ^2$$
An object in its instantaneous rest frame is at rest, so $dx=dy=dz=0$. We get this formula for proper time
$$\Delta\tau=\frac1c\int\sqrt{c^2dT^2-dX^2-dY^2-dZ^2}$$
Similarly, given two spacelike-separated events $a$ and $b$, one can find an instantaneous rest frame which an inertial frame where $a$ and $b$ are simultaneous. We equate the metric of the instantaneous rest frame with metric with the flat spacetime metric:
$$ds^2=-c^2d\tau^2+dx^2+dy^2+dz^2=-c^2dT^2+dX^2+dY^2+dZ^2$$
In the instantaneous rest frame, $d\tau=0$. The formula for proper distance between $a$ and $b$ is
$$L=\int_a^b\sqrt{dx^2+dy^2+dz^2}=\int_a^b\sqrt{-c^2dT^2+dX^2+dY^2+dZ^2}$$
Now I want to understand how to generalize the above derivations of proper time and proper distance to a curved spacetime described by metric $ds^2=g_{\mu\nu}dx^\mu dx^\nu$. Why in general relativity proper time $\tau$ is defined as $d\tau=\sqrt{-ds^2}/c$ and proper distance $\sigma$ is defined as $d\sigma=ds$?
I read an answer by @Qmechanic in a related question and I am struggling to understand what is going on in point 3 in their answer. In particular, how the temporal coordinate reparametrization works.
I think in general relativity instantaneous rest frames are now called proper frames. The linked page says that proper frames can either be inertial or not inertial. I am not sure how proper frames can be used to calculate proper time and proper distance.
After reading a bit about Riemann normal coordinates suggested by @naturallyInconsistent, I think that in a curved spacetime, the metric for a spacetime experienced by an observer in a timelike geodesic is approximately flat with some negligible higher order terms. The metric for a spacetime experienced by an observer who is undergoing hyperbolic motion with constant proper acceleration is approximately Rindler with some negligible higher order terms. The metric for a spacetime experienced by an observer who is in uniform circular motion approximately looks like the one here (6e) with some negligible higher order terms. I am not sure if this sounds correct though.
For example, for an observer in some fixed Schwarzschild coordinate $(R,\Theta,\Phi)$ in Schwarzschild spacetime described by metric
$$ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2+\left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$$
their spacetime can be described by a Rindler metric
$$ds^2=-\left(\frac{ax}{c^2}\right)^2c^2d\tau^2+dx^2+dy^2+dz^2$$
for some constant $a>0$. Here $\tau$ is proper time of the observer.
To calculate proper time of the observer, one sets $x=c^2/a$ (this is the default position for a Rindler observer) and $dx=dy=dz=0$ and the above Rindler metric becomes $ds^2=-c^2d\tau^2$ and it can be equated to Schwarzschild metric. To calculate proper distance, one sets $d\tau=0$ to get $d\sigma^2\equiv ds^2=dx^2+dy^2+dz^2$ and equate $d\sigma^2$ to Schwarzschild metric.
