 In this video, we explore another apparent paradox presented by relativity theory, the so-called pole and barn paradox, also known as the ladder paradox, or the length contraction paradox. Suppose we have a red pole, and we have a blue barn. The pole's length is 2, and the barn's length is 1.6. The pole is longer than the barn. Let's refer to the left side of the barn as the front door, and the right side is the back door. The right side of the pole will call its head, and the left side its tail. Since the pole is longer than the barn, if we throw the pole through the barn at some velocity v, some part of it will always be visible. At any given time, the barn can hide the pole's head, or its tail, but not both. Obviously, a barn cannot fully contain a pole that is longer than the barn. At least that's how things work in a non-relativistic world. But the coordinate transformations we developed in video two of this series tell us that if you and I are moving relative to each other, then two events you perceive to take place at the same time, I may perceive to take place at different times. Moreover, the coordinate transformations contain a dilation factor, beta. To be specific, we'll assume a velocity of three-fifths the speed of light, which gives a dilation factor of five-fourths. If we naively consider only the time coordinates, this implies that if I measure a time interval delta tau, you measure a time interval delta t that is larger than what I measure by a factor of five-fourths. But from my point of view, my time interval is larger than yours by the same factor. So we seem to have the absurd prediction that two time intervals are each larger than the other. We saw the resolution of this in the Twin Paradox video. Likewise, if we naively consider only the space coordinates, it appears that if I measure a distance delta xi, you measure a distance delta x, which is larger by a factor of five-fourths. At the same time, I see my measured distance as being larger than yours. Again, we have an apparent absurdity, two lengths, each of which is longer than the other. Relativity tells us that the absurdities are an illusion that arises from naively thinking about space and time separately. As an illustration of what can happen when space and time get mixed up, consider this hover toy gliding across my floor and underneath a wooden beam. The toy is wider than the beam. So, even though any particular point on the toy will at some time be hidden by the beam, the entire toy cannot hide behind the beam at a given time. With a little video processing, however, we can make it appear that for a moment the toy is fully hidden by the beam. We see the front end disappear followed by the back end, and only after that does the front end reappear. To create this illusion, we first delete the right half of the video. Then we take a time-delayed version of the video and delete its left half. By stitching these two halves together, we create a video in which events on the left side actually happened after the apparently simultaneous events on the right side. This makes it appear that the hover toy completely disappears behind the beam. According to relativity theory, an analogous effect comes into play when a rapidly moving object is observed, making the object appear to contract in the direction of motion. Back to our pole and barn scenario. We consider the barn to be at rest and use Latin letters for its coordinates. Specifically, the front door is at x equals 2.4 and the back door at x equals 4. The pole is moving at three-fifths the speed of light, and we use Greek letters for its coordinates. Specifically, the tail is at xi equals zero and the head at xi equals two. Flooding the relation between the coordinate systems using blue for the barn and red for the pole, we get the sort of space-time diagram we've seen in previous videos. The numerical labels apply to the blue coordinates. Recall that the green line with slope of unity represents the speed of light. The back door is located at x equals four and the front door at x equals 2.4. The tail is at xi equals zero and the head at xi equals two. In the barn frame, at time t equals four, the pole head will coincide with the back door and the pole tail with the front door. The pole will appear to fit inside the barn. From the pole's perspective, these events happen at different times. Tau equals two and 3.2 respectively. Let's now use a space-time diagram to visualize the pole's journey through the barn from the pole's perspective. The thick blue lines represent the closed back door and closed front door. At time t equals four in the blue frame, the back door opens and the front door closes. At a given time in pole coordinates, the pole occupies the thick red line. As it moves through time, its head eventually arrives at the back door just as it opens. The pole continues through the barn and eventually its tail passes through the front door just as it closes. Now let's abstractly animate what this would look like from the pole's perspective. The pole's head is on the right and its tail is on the left. We put five clock lights on the pole that flash in unison every second of pole time. The head and tail clock readings are also shown numerically. The barn is represented by a rod with the back door at right and the front door at left. An additional ball is used to indicate when a door is closed. No ball indicates an open door. The pole's head enters the front door, passes through the barn to the back door just as it opens. Then the tail reaches the front door just as that closes. Notice that the lights flash in unison. The pole makes it through a shorter barn because in its reference frame, the back door opens before the front door closes. Now let's look at things in the barn's reference frame. At a given barn time t, the pole occupies the vertical thick red line. It appears contracted enough that the whole thing can fit inside the barn at time t equals four when the back door opens and the front door closes. The barn sees the contracted pole approaching. Notice that the pole's lights don't flash in unison. The entire pole enters the barn. The back door opens and the front door closes. A snapshot of the pole completely inside the barn illustrates the mixing of time and space we discussed previously. The head clock reads pole time 2.0 while the tail clock reads pole time 3.2. As we move from left to right across the frame we move back in pole time. This is very much analogous to the video trick we use to make the hover toy hide behind the wooden beam. So the way space and time get mixed up when objects are in relative motion allows a pole to appear to fully fit within a barn even though the pole is longer than the barn. Now the obvious question seems to be, is the pole really completely inside the barn? After all in one reference frame it is and yet in another it isn't. So which is it? Really? Well, in the barn's reference frame the pole is really inside the barn. If you set up synchronized cameras along the length of the barn and had them take pictures at time t equals four you would see the entire pole inside the barn. You would also see the clocks on the pole showing different times. At the same time in the pole's reference frame the pole is really never fully inside the barn. What quote really happens depends on the reference frame. Welcome to the world of relativity. Okay, okay, enough with the hand waving mumbo jumbo. What if you never open the back door? In the pole's frame its head will crash into the back door causing the pole to come to a stop while its tail is still outside the front door. But in the barn's frame the pole would crash and come to rest with its tail already inside the front door. So we'd be standing in a field looking at a barn and a pole at rest with its tail both inside and outside the barn. Impossible. So relativity presents a paradox. Well, it is impossible. And it's not a paradox. The subtle issue is that an object can't just stop all at once. The collision of the pole's head with the barn's back door would send out a shock wave. It's the force of that shock wave that would stop the other parts of the pole. And nothing, including that shock wave, can travel faster than light. So let's plot a light path from the impact back towards the front door. Recall that light always travels at an angle of plus or minus 45 degrees on our space time diagrams. This light path intersects the path of the pole's tail well inside the barn. And this point of intersection is the same regardless of the reference frame. So in fact, both reference frames would agree that the pole's tail was inside the barn before it came to rest, or vaporized, or whatever. Visualized in the pole's frame the shock wave would be released before the tail was inside the barn, but would reach the tail only after it was well inside. In the barn's frame the shock wave would be released just as the tail entered the front door, and would reach the tail when it was well inside. In fact, both reference frames would agree on the tail's position in the barn and clock reading when the shock wave reached it.