When we talk about a linearly accelerating reference frame in the context of Galilean spacetime, it is absolutely clear what is meant. However, when we talk about the same in special relativity, I am honestly not even sure what it would even mean.

In setting up inertial reference frames, we consider a grid of rods and clocks, apply Einstein synchronization, and set up a notion of simultaneity. What would be the analog for accelerating reference frames? How would we define linearly accelerating reference frames a priori?

A common choice is to invoke Rindler coordinates, but there are two problems:

1. Rindler coordinates are not global, so they don't cover the entire Minkowski diagram of spacetime.
2. When we make the transformation from inertial reference coordinates $(t, x)$ to Rindler coordinates $(T, X)$, straight line worldlines are not sent to hyperbolas. The opposite transformation sends worldlines $(T, X_{0})$ to hyperbolas, but (i) the resulting hyperbolas represent objects having different proper accelerations depending on the value of $X_{0}$, and (ii) worldlines of the form $(T, X_{0}+vT)$ are not sent to hyperbolas.

In this view, it's not clear why we would be justified in calling Rindler coordinates as having anything to do with linearly accelerating reference frames.

This post asks for references about accelerating reference frames, but I'm asking more specific questions and I don't know if the references address my worries.

When we talk about a uniformly accelerating reference frame in the context of Galilean spacetime, it is absolutely clear what is meant (in a previous edit I said "linearly accelerating" but this was non-standard terminology). However, when we talk about the same in special relativity, I am honestly not even sure what it would even mean.

In setting up inertial reference frames, we consider a grid of rods and clocks, apply Einstein synchronization, and set up a notion of simultaneity. What would be the analog for accelerating reference frames? How would we define uniformly accelerating reference frames a priori?

A common choice is to invoke Rindler coordinates, but there are two problems:

1. Rindler coordinates are not global, so they don't cover the entire Minkowski diagram of spacetime.
2. When we make the transformation from inertial reference coordinates $(t, x)$ to Rindler coordinates $(T, X)$, straight line worldlines are not sent to hyperbolas. The opposite transformation sends worldlines $(T, X_{0})$ to hyperbolas, but (i) the resulting hyperbolas represent objects having different proper accelerations depending on the value of $X_{0}$, and (ii) worldlines of the form $(T, X_{0}+vT)$ are not sent to hyperbolas.

In this view, it's not clear why we would be justified in calling Rindler coordinates as having anything to do with uniformly accelerating reference frames.

This post asks for references about accelerating reference frames, but I'm asking more specific questions and I don't know if the references address my worries.