Variation of ladder/barn paradox: can observer on ladder see that barn was too small?

Question

Image Source and Extract: Wikipedia Ladder and Barn Paradox

The simplest version of the problem involves a garage, with a front and back door which are open, and a ladder which, when at rest with respect to the garage, is too long to fit inside. We now move the ladder at a high horizontal velocity through the stationary garage. Because of its high velocity, the ladder undergoes the relativistic effect of length contraction, and becomes significantly shorter. As a result, as the ladder passes through the garage, it is, for a time, completely contained inside it. We could, if we liked, simultaneously close both doors for a brief time, to demonstrate that the ladder fits.

So far, this is consistent. The apparent paradox comes when we consider the symmetry of the situation. As an observer moving with the ladder is travelling at constant velocity in the inertial reference frame of the garage, this observer also occupies an inertial frame, where, by the principle of relativity, the same laws of physics apply. From this perspective, it is the ladder which is now stationary, and the garage which is moving with high velocity. It is therefore the garage which is length contracted, and we now conclude that it is far too small to have ever fully contained the ladder as it passed through: the ladder does not fit, and we can't close both doors on either side of the ladder without hitting it. This apparent contradiction is the paradox.

I just read about the ladder/barn paradox and was wondering about the following variation:

The barn is equipped with some electronics to measure whether the ladder is completely contained, and will switch on a light (on the barn's roof) if the ladder is inside.

Now would an observer on the ladder, looking back at the barn after he's gone through, see the light turn on? I assume he should be able to see the light, which moves quicker than the ladder, if it was turned on.

From what I've read in posts here, and on https://en.wikipedia.org/wiki/Ladder_paradox, I would guess that the barn electronics will find the ladder small enough and switch on the light. So the observer on the ladder will be surprised by this, which contradicts his own experience.

Is that simply the kind of surprises the observer has to live with (when using such transport :-) or is there some fault in my reasoning?

Answer 1

Think about how these electronics would work. You'd have to have a sensor that watches each door and sends signals to a central unit with things like "the front of the ladder entered the rear door", "the back of the ladder exited the front door", etc. You'd then have to calibrate the central unit so that it turns on the light if it receives a signal that the back of the ladder has entered one door of the barn before the front of the ladder has exited the other door.

But one of the main consequences of Einstein's postulates is that observers moving relative to each other do not, in general, observe the same amount of time to elapse between spatially separated events. In fact, for certain pairs of events A & B, one observer can say that A occurred before B while another observer says that B occurred before A. This is what's going on with the ladder-in-the-barn paradox; in one frame, the front of the ladder exits the barn before the back of the ladder enters the barn, while in the other frame, these events are reversed in order.

The sensors don't really get change this fundamental difficulty. In the ladder's reference frame, the sensors & the control unit would appear to be "miscalibrated" — meaning that the light would turn on in some cases if the front of the ladder left the barn before the back of the ladder entered the barn. There would be a certain time window involved—the back would still have to enter the barn within 1 millisecond (say) of the front exiting the barn. But since the person in the ladder frame sees the sensors as "miscalibrated", they wouldn't be surprised to see the light come on even when the ladder wasn't in the barn (in their reference frame.)

Answer 2

The two points of view - one from the barn frame and one from the ladder frame - are NOT incompatible according to special relativity. This is not a big deal as there is no causality issue related to the sequence of events.

Contrary to what is alluded to in the question, the observers in the ladder frame would NOT be surprised to learn of the conclusion of the observers in the barn frame: presumably the ladder people can Lorentz-transform the coordinates of their events to the barn frame, and find that the barn people have rightly triggered their light.

Conversely, if the ladder people had a light, the barn people would not be surprised to find that the ladder light was never triggered.

The point here is that both groups of observers know the other has a different coordinate system. One would be no more surprised than if two pedestrian gave each other walking instructions to move up/down/right/left based on maps where north was up on one map, and right on the other map: of course the instructions would be different.