Formula for light deflection in elevator

[wmikewells]

I was just wondering if there is a simple formula for determining how much a horizontal light beam deflects in an accelerating elevator. The horizontal light beam would start at one side of the elevator and hit the other side of the elevator. In a non-accelerating elevator, the light beam would hit the opposite side at height H which would be equal to the starting height of the light beam. In an accelerating elevator, the light beam would hit the opposite side at height H minus some distance D.

I guess it would only depend on the following factors:

1. width of the elevator
2. rate of acceleration

However, I have not been able to find such a formula on-line.

 

[Dale]

How long does it take for the light to cross the elevator? How far has the elevator moved during that time?

 

[wmikewells]

Thanks for the reply. I would think that the speed of the elevator would not matter (only the rate of acceleration) and that the time for the light to travel across the elevator would be dependent on the width of the elevator alone. But since I am not an expert on general relativity, I could be wrong. I am only concerned about the reference frame of the elevator (simple case and not the additional complexity from some other reference frame) , and the elevator is traveling vertically up (not horizontally). I am just concerned about the deflection downward, which I think would be visible to a person in the elevator.

If they matter, let's say that the elevator is 2 meters across, so the time to cross the elevator is 2 meters / 3 * 10^8 meters/second = 6.7 * 10^-7 seconds. And that the elevator starts at 0.0 meters / second. When the light beam is fired, the elevator starts to accelerate.

Please let me know if my assumptions that the formula to derive the deflection downward on the opposite wall is only dependent on the width of the elevator and the rate of acceleration.

 

[wmikewells]

Please let me know if my assumptions that the formula to derive the deflection downward on the opposite wall is only dependent on the width of the elevator and the rate of acceleration.

... are wrong.

 

[bcrowell]

In units where c=1, the only possible relevant variables are a (the acceleration) and w (the width of the elevator). Since the effect has to be odd in a and even in w, the result has to be of the form $(...)aw^2+(...)$, where the first ... represents a unitless constant, and the second ... represents higher-order terms. In Newtonian mechanics, the first ... would equal $1/(2c^2)$. The c² has to be there because of units, but the 1/2 is suspect, because, e.g., there are different theories of gravity that make different predictions about the deflection of starlight by the sun -- only GR gets the unitless factor right.

How long does it take for the light to cross the elevator? How far has the elevator moved during that time?

The time is determinable from the variables explicitly listed in the OP. The distance the elevator has moved is zero, because we're talking about the elevator's frame.

 

[wmikewells]

Thanks for the overview. Not so simple after all. I was out on an afternoon bike ride, and it occurred to me that there might be a simple way for me to picture it. It does not reduce the complexity of the formula, but it makes it easier for me to picture how the light beam would deviate.

Let's say there is another elevator accelerating in tandem right next to the original elevator. If the tandem elevator stopped accelerating at the moment the light beam was fired, the tandem elevator would preserve the spot the light beam would hit on the opposite wall. In other words, the distance between the elevators would be the distance the light beam would deviate in the still accelerating elevator. In a simple example, let's say the accelerating elevator was able to gain a centimeter edge over the coasting elevator in the time it took the light to travel across accelerating elevator. So the light beam would appear to deviate down by a centimeter in the accelerating elevator.

Is it as simple as that? Of course, for large rates of acceleration, calculating the distance between the elevators would have to take into account relativistic effects. I am guessing that is where the extra complexity comes into play.