Question on the value of speed of light, non-inertial frames and equivalence principle

Question

One of the axioms of special relativity concerns on the value of speed of light measuread by a family of inertial observers. They must measure $c$.

Now, the global inertial frame is lost if the observer $\mathcal{O}'$ is accelerated (say in a constant proper acceleration movement). Therefore, the stationary observer $\mathcal{O}$ would make the usage of a rindler coordinate system to depict the movement of $\mathcal{O}'$ in spacetime. It is interesting to note that using rindler coordinates (specifically in a pseudo-paradox called bell's spaceship paradox) to define velocity, the one realize that the speed of light isn't $c$, but it can vary as: $c<0$, $c=0$, $c>0$. This of course is due to curvilinear coordinate system, and locally every observer will agree on value $c$ (since in a local portion of the accelerated world-line, we can see a straigth line and therefore a momentary inertial frame).

This question summons the very intuitive/layman reasoning behind the EP and the light beam. I can understand the reasoning behind the curved path of light seen by a accelerated observer in a ship and in a gravitational field. But I'm a bit confused about the following:

1. the elevator represents the local nature of spacetime: it is the tangent space and threfore we are talking about special relativity.
2. if even inside this local portion of spacetime the observer sees a cuverd path of light, this means that she could measure diferent values of speed of light just like in rindler coordinates

Therefore, the question is: if even locally the path of light is curved (considering the didatic picture of the EP) why every textbook on general relativity says that locally every observer should measure $c$?

In other words, if we are inside the einstein elevator, we are talking about small regions and short times, therefore we are inertial frames. Why we would see the path of light to be curved?

Answer 1

since in a local portion of the accelerated world-line, we can see a straigth line and therefore a momentary inertial frame …

if we are inside the einstein elevator, we are talking about small regions and short times, therefore we are inertial frames

I think this is the key mistake. An accelerated worldline is not locally inertial.

The idea that spacetime is locally flat means that at each event in spacetime there exists a coordinate system such that the metric is $ds^2=-dt^2+dx^2+dy^2+dz^2$ to first order. In other words, there exist coordinates such that, if you expand the components of the metric in a Taylor series about any event, all of the first order terms vanish.

It does not mean that all coordinate systems satisfy that condition locally.

In the case of Rindler coordinates $$ds^2=-(1+ax)^2 dt^2+dx^2+dy^2+dz^2$$ The series expansion of the $dt^2$ component is $1+2ax+...$ so the first order term is $2a\ne0$. And therefore it is not locally inertial.

So in Rindler coordinates the coordinate speed of light may be different from $c$ and the path of light may not be a straight line when expressed in coordinates.

The equivalent statement that does hold in all coordinate systems is that light is a null geodesic in all coordinates and in all spacetimes.