general relativity - Equivalence principle and acceleration vs a gravitational field

I picked this up on the net:

Einstein came to realize the principle of equivalence, and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field.

When I am being accelerated I will pick up speed over time. This will have a noticeable effect, e.g. there will be an increasing blue-shift in the light of stars which lie in the direction of acceleration. This will not happen if I am just in a gravitational field and I could well distinguish acceleration from gravity.

I suspect that being completely physically equivalent may be a bit of an overstatement and is only true as long as I don't look outside?

Or is the solution, that I cannot really know, whether those stars are being accelerated too? So when I feel the gravity and I see a constant blue shift, I might conclude that I am being accelerated, but all these stars are being accelerated the same as me?

Or is there a way to interpret gravity as acceleration, which also leads to an increasing red/blue shift?

Answer

First, it turns out that there are no uniform gravitational fields so the equivalence principle holds only locally.

But, for the sake of argument, let's assume that a uniform gravitational field can exist.

Now, consider the situation where an astronaut is in a rocket and the rocket's accelerometer reads a constant, non-zero value.

According to the equivalence principle, the following two perspectives are equivalent and thus indistinguishable by experiment:

(1) There is no gravitational field and the rocket is absolutely accelerating (due to some type of motor)

(2) The rocket is stationary (due to the same motor) in a uniform gravitational field

Note that in case 2, the stars are freely falling in the uniform gravitational field and thus, their speed relative to the stationary rocket increases with time just as in the first case.

So, it is not the case that you can distinguish the two perspectives. In both cases, the stars "in front" of the rocket will become more and more blue-shifted with time.