Up or Down? A Question about Relativity The standard text discussion of the equivalence principle and the bending of light rays goes like this. We, as observers watch a glass elevator ascending. A light beam is fired from left to right. The acceleration of the elevator causes the light beam to curve downward. Since the elevator passengers can't discern the difference between an acceleration and a (uniform) gravitational field, ergo, a gravitational field bends a light ray.
Fine, but consider the standard text derivation of time dilation---which posits that the light beam would be carried along with the elevator. Here, of course, the argument assumes inertial frames so the elevator is rising at a constant rate.
Now consider this. Suppose an elevator passenger affixes a small laser to the left hand elevator wall at precisely its midpoint and focuses it on a point midway up the right hand wall. Where do we "stationary" observers outside the elevator see the ray hit the right hand wall: at its midpoint, above it, or below it? If we buy the equivalence principle argument it will clearly be below (even if the elevator speed is constant) because of the upward movement of the elevator. On the other hand, the midpoint is a physical thing which can be identified by, say, a paint blob---so one could reason that the light beam should hit the right hand wall at its midpoint.
I don't think simultaneity issues enter the picture here. The only issue is where does the light beam strike the right hand wall?
 A: An observer outside the elevator would not see the laser beam as being parallel to the floor of the elevator.
Imagine you are playing ball with a friend. You stand to the west of him and he runs from south to north as you throw him the ball. You throw the ball as he is directly east of you; in order for him to catch the ball, you must throw it northeast, since he is moving. However, in his frame of reference, the ball is directly west of him the entire time, because his reference for "west" moves with him. So he says the ball flew west-to-east and you say it flew southwest-to-northeast. Those lines are not parallel.
Likewise, since the elevator in the textbook example is moving upward, you outside the elevator must "lead the target" to shine a laser through the window on one side and hit a dot parallel to it on the far wall, because light has a finite speed and the dot moves as the light is crossing the space. You will see the laser beam pointing "up and across," while your friend in the elevator sees it "straight across" from the window to the dot.
The only real difference between the ball and laser scenarios is that a ball has a variable speed and light speed is the same in all reference frames; so in the ball scenario, you and your friend would disagree on the speed of the ball but not the time of flight, whereas in the elevator he would disagree with the timing but not the speed, and that is where time dilation arises.
A: The ray will hit the right hand wall at its midpoint, and both the passenger and the outside observer will agree on this.  
The equivalence principle states that the laws of Physics in a free fall and in an inertial moving elevator are the same, meaning that a passenger cannot expect an experiment to distinguish between the two (but deduce its motion relatively outside objects).
The following is not true:

The acceleration of the elevator causes the light beam to curve downward.

This statement refers to how external observers in an inertial frame would see the light: if the elevator moves with constant velocity, they would see the ray hit the opposite wall following a straigth line; if the elevator has constant acceleration, the ray draws a parabola, but always hitting the midpoint on the right wall.
This is exactly the same you expect if the experiment is done in a train using a ball: it is just a matter of kinematics, of the way the light and the elevator motions are combined.  
Another classical (and more violent!) example is about the monkey and the hunter: the hunter wants to shot the monkey that is hanging to a tree branch, but he knows that at exactly the same time the bullet exits from his gun, the monkey will drop from the branch. Where should the hunter aim at?
The bullet and the monkey are the laser ray and the passenger in the elevator, the hunter the external observer. The hunter has to aim at the monkey, as the two will be moving in the same frame. The bullet will bend as seen by the hunter, because it has both horizontal and vertical velocities.
EDIT: 
thank to the discussion with bright magus below, I add to this answer this link because it treats the problem exposed in the question in a nice way. In the same site, this page treats the equivalence principle and the concept of light bending. Each page reports also references to dig out more details on the matter.
A: When he  fires he will see the white spot where it was the light time away so it will have moved into the right position when it is hit at instant later.
By the way I dont understand fedino's remark about light speed having a vertical component. Are we talking about the photons momentum vector, the Poyinting vector or what? 
A: This an attempt to give a more detailed explanation since the question really is quite fundamental and has mostly been explained by referring to the impossibility of a co-moving observer detecting any effects of the non accelerated linear motion whatever the speed might be. Its the same as saying you must just trust Einstein without explaining the mechanism of how photons ejected along the x-axis of the frame, which is steadily moving up in the y-direction, seem to stay on the x-axis according to the co-moving observer. 
As I see it each ejected photon aquires a velocity component v in the y direction (just like a bullet would) and thus stays on the x-axis. Since its total speed is c, its velocity in the x-direction becomes c/gamma (Pythagoras) and it will thus take longer to hit the wall at x = L, the travel time being gamma*L/c. However the co-moving observers clock is time dilated so he will observe a travel time L/c just as if v was zero.
Hope this makes sense.
