This is a copy of a conversation that I had with a student on a forum concerning the WP article Ladder_paradox. It is copied exactly apart from the correction of typos and removal of irrelevant personal chat.
- Hi I was recently reading an article on the 'pole in the barn paradox', and I don't understand it at all. How is it possible for both observers to be right?(Here's the wiki entry: http://en.wikipedia.org/wiki/Ladder_paradox )Thanks in advance
The Wikipedia article is not very clear at the moment. Are you referring to the version with two doors to the garage?
- Yeah, one at each end.
OK, let us start with the experimental setup. The ladder, in its own rest frame, is 20 m long and the garage in its own rest frame is 10 m long. We arrange for the ladder to travel through the garage at sufficient speed for it to be contracted to be just under 10 m long. We fit the garage with two doors which will close and open simultaneously. For possible future explanation, let us say that we do this by setting up a flashbulb in the centre of the garage and having an optical receiver at each end which controls the doors. In the reference frame of the garage, the ladder is contracted to 10m so that just as it is passing through the garage we briefly simultaneously close both doors whilst the ladder is inside the garage then open them again before the ladder hits them. OK so far?
- Yep ^_^
From the reference frame of the ladder, the garage is in motion and thus its length is contracted to 5 m whilst the ladder is stationary and has a length of 20m. Clearly the ladder will not fit in the garage. The problem is that we know that the doors must close whilst the ladder is passing through the garage. If they do, it would appear that they must hit the ladder. This would cause a massive explosion.The paradox is that the doors do not hit the ladder, and there is no explosion when the experiment is considered in the garage frame. Thus, at first sight it would appear that, depending on which frame of reference you consider, the experiment will have a different outcome. Do you understand the paradox?
Just to be clear, by 'Do you understand the paradox?' I mean, 'Do you see what the problem to be solved is?', not, 'Do you understand the solution'; I am coming to that.
- Yes, I understand the paradox
OK. The resolution of paradox is due to what is known as the relativity of simultaneity. I will talk about this in two stages, firstly I will say what it is and how it resolves the paradox. This is where many explanations stop because the student is expected to know about the relativity of simultaneity. Then, if you wish, I can show you how it arises the from basic postulates of special relativity.
The relativity of simultaneity is a basic result of Einstein's special theory of relativity and it says that spatially separated events that are simultaneous in one frame of reference may not be simultaneous in a frame relative motion. We set up the doors to open simultaneously in the garage frame. That means that, in general, they will not be simultaneous in a frame that is in motion with respect to the garage. In particular, the doors will not close simultaneously in the ladder frame. In fact, the back door closes first first (and then immediately opens) just as the front end of the ladder is about to hit it. Later, in the ladder frame, just as the back end of the ladder has passed through it , the front door of the garage closes. Note that these two things, the front door closing as the back end of the ladder passes it and the back door closing just as the front end of the ladder is about to hit it, both also happen in the garage frame. The difference is that in the garage frame they are simultaneous and in the ladder frame they are not.The result is that the ladder passes through the door in both reference frames without getting hit by either of the doors and no animals are hurt. Assuming the relativity of simultaneity, does this make sense?
- Yes, that makes sense thanks!
Do you want me to explain how the relativity of simultaneity follows from Einstein's two postulates?
- Yes, please
First the two postulates. Towards the end of the 1800s, scientists were doing many experiments with light, which they assumed to travel through some hypothetical aether. These experiments produced some unexpected results which Einstein was able to explain by making two postulates. The postulates were basically starting points which, if assumed, lead to a number of startling conclusions (such as the equivalence of mass and energy). Since then experiments have verified these results.
Einstein's two postulates may be stated (see Wikipedia) as:
1) The laws of physics are the same in all inertial frames of reference, and
2) The speed of light in free space has the same value c in all inertial frames of reference.
The first postulate is really quite simple and natural to understand. It basically says that no reference frame is the stationary one and that all inertial (non-accelerated) motion is relative. There is nothing new about this, it was stated by Galileo in 1632 but Einstein needed to restate it because, at the time, there was generally believed to be an aether which specified a true stationery frame. In the context of the ladder paradox, this postulate just tells us that the garage frame and the ladder frame are equally valid
The second postulate is truly weird though. It states that, in an inertial frame, the speed of light will always be measured to have the value c. This is clearly impossible if space and time are how they were once thought to be. It means that, if you run towards someone at c/2 and shine a laser at them, they will still measure the speed of light from your laser to be c. If they run towards you at a speed of c/2 (relative to their original rest frame) they will still measure the speed of light to be c! In the ladder paradox it means that regardless of what the source is or how it is moving, the speed of light will always be measured as c, in both frames. OK so far?
- ..yep
We used a flashbulb in the centre of the garage with optical sensors to operate the doors. In the garage frame, the light travels at speed c equal distances to the two door sensors so the doors close simultaneously.
In the ladder frame, the garage, flashbulb, and sensors are all moving but the light from the flashbulb still travels at speed c in our frame. The rear door sensor is moving towards the light at c/2 and the front door sensor is moving away from the light at the same speed. As the light moves the sensors move so the rear door sensor will meet the light first. In this particular case, we can just add up the speeds, so the rear door sensor is meeting the light at 3c/2 and the front door sensor at c/2. These are known as closing speeds and, as they do not represent the speed of a single object in an inertial frame, they can be more then c, in fact up to 2c. A single object, energy, or information though can never travel faster than c in an inertial frame. So by applying the second postulate we see that events that are simultaneous in the garage frame cannot be simultaneous in the ladder frame. This is a version of Einstein's famous train gedanken experiment.
- Oh, right! Thanks!
You do not sound too convinced about that last bit. Do you see why I just added two speeds? Do you see how Einstein's second postulate leads inevitably to simultaneity being different in different frames?
- I think so. Relative to the observer (with the ladder), the doors are moving in different directions from/to the light source and so the light hits them at different times; but relative to the other observer, the doors are not moving and therefore both receive the light simultaneously? Is that ok, sort of?
Yes that is it. Good luck with your studies. Martin Hogbin (talk) 09:42, 16 March 2014 (UTC)