Bending of light in a gravitational field & the principle of equivalence

I have a few conceptual issues following a standard thought experiment to argue why light bends in a gravitational field and I'm hoping I can clear them up here.

Consider an observer in a lift in free-fall in a uniform gravitational field and an observer at rest in the uniform gravitational field. Suppose that a laser is fired from a fixed height on the inside of the lift from one side to the other. According to the equivalence principle, inside the lift the laws of special relativity apply and the observer in the lift will observe the laser beam to travel from one side of the lift to the other in a straight line (at a fixed height). However, the stationary observer outside the lift will observe the lift to accelerate downwards at some constant acceleration $g$....

Now, I understand it up to this point, but doesn't one have to invoke the principle of (general) relativity, i.e. that all observers are equivalent, in order to complete the argument? To me this seems to make sense, and (personally) I would complete the argument as follows:

... Now, if the stationary observer (outside of the lift) were to observe the laser beam to propagate in a straight line then it would arrive at the other side of the lift at a different height to that observed by the observer inside the lift. This, however, would distinguish between reference frames, violating the principle of (general) relativity. In order for the principle of (general) relativity to hold we must therefore conclude that the (external) stationary observer observes the light to arrive at the other side of the lift at the same height as that recorded by the observer inside of the lift. As the lift is accelerating downwards at $g$ relative to the (external) stationary observer, this implies that the laser beam must follow a curved path. Thus light "bends" in the presence of a uniform gravitational field.

Would this argument be correct at all? (I feel I may be over thinking the issue!).

• Yes, this is pretty much the standard argument. – Javier Dec 2 '15 at 16:24
• @Javier So is it correct then that the principle of relativity is implicitly used in the argument? (several texts that I've read on the subject don't explicitly mention it, but I don't see how the argument can proceed otherwise) – Will Dec 2 '15 at 16:29
• It is very explicitly used in the argument. – DilithiumMatrix Dec 2 '15 at 17:08
• @DilithiumMatrix Ah ok, I probably haven't been looking in the right places. Are there any texts that you would suggest? Is what I wrote in my original post correct though? – Will Dec 2 '15 at 17:19
• Absolutely, that sounds very solid. What type of references are you looking for exactly? Something that talks conceptually about GR? A good textbook? – DilithiumMatrix Dec 2 '15 at 17:47