The ladder paradox (or paradox barns ) is a mind experiment in special relativity. It involves a ladder, parallel to the ground, traveling horizontally and therefore experiencing long contraction of Lorentz. As a result, the ladder fits inside the garage which is usually too small to hold it. On the other hand, from the point of view of an observer moving with a ladder, it is a moving garage, so it is a garage that will be contracted to a smaller size, so it can not hold the ladder. This apparent paradox comes from the false assumption of absolute synchrony. The ladder fits the garage only if both ends are simultaneously inside the garage. The paradox is solved when it is assumed that in relativity, the relative relative to each observer, makes the answer whether the ladder fits in the garage is also relative to each observer.
Video Ladder paradox
Paradox
The simplest version of the problem involves a garage, with open front and rear doors, and a ladder that, when resting in connection with the garage, is too long to go inside. We are now moving the ladder with high horizontal speed through the stationary garage. Due to its high speed, the ladder experiences a relativistic effect of long contraction, and becomes significantly shorter. As a result, when the ladder passes through the garage, for a while, it is entirely contained within it. We can, if we like, simultaneously close both doors for a short time, to show that the ladder fits.
So far, this is consistent. The apparent paradox occurs when we consider the symmetry of the situation. As the observer moves with the staircase traveling at a constant velocity within the frame of reference of garage inertia, this observer also occupies an inertial framework, where, by the principle of relativity, the same laws of physics apply. From this perspective, it is a stairway that is now stationary, and a garage that moves at high speed. Therefore the garage is contracted in length, and we now conclude that it is too small to fully have a ladder as it passes: the ladder does not fit, and we can not close both doors on either side of the ladder. without hitting him. This real contradiction is a paradox.
Maps Ladder paradox
Resolution
The solution to the apparent paradox lies in the relativity of simultaneity: what an observer observes (eg by garage) are two simultaneous events that in reality can not be simultaneously performed by another observer (eg by a ladder). When we say that the "fit" ladder in the garage, we mean exactly is that, at a certain time, the position of the staircase and the front position of the ladder are inside the garage; in other words, the front and back of the ladder are in the garage simultaneously. Since simultaneity is relative, then, two observers disagree about whether the ladder is suitable or not. For the observer in the garage, the rear end of the ladder is in the garage at the same time with the front end of the stairs, so the ladder fits; but for the observer with the ladder, these two events are not simultaneous, and the ladder does not match.
The obvious way to look at this is to consider the doors, which, in the garage frame, cover for short periods that the stairs are fully inside. We are now seeing these events in a ladder frame. The first event is the front of the stairs approaching the garage exit. The door closed, and then opened again to let the front of the stairs through it. At other times, the back of the stairs goes through the entrance, which closes and then opens. We see that, since relativeity is relative, the two doors need not be closed at the same time, and the ladder does not need to enter the garage.
This situation can be illustrated further by the Minkowski diagram below. The diagram is in the garage break framework. The vertical blue-light band shows the garage in space-time, and the bright red band shows the ladder in space-time. The x and t axes are the garage space and time axis, respectively, and x? and T? is the staircase and the time-axis, respectively.
In the garage frame, the ladder at a given time is represented by a set of horizontal points, parallel to the x axis, on the red tape. One such example is the thick blue line segment, located inside the blue ribbon representing the garage, and which represents the ladder at that moment entirely inside the garage. In the ladder frame, however, the simultaneous event set lies on the line parallel to the x axis; ladder at a given time because it is represented by a cross-section of such a line with a red ribbon. One example is the thick red line segment. We see that such a line segment is never entirely within the blue ribbon; That is, the ladder is never fully located in the garage.
Closing the stairs in the garage
In a more complicated version of the paradox, we can physically trap the ladder once fully inside the garage. This can be done, for example, by not opening the exit door again after we close it. In the garage frame, we assume the exit can not move, so when the ladder hit it, we say that it stopped instantly. By this time, the entrance has also been closed, so the stairs are trapped inside the garage. Since its relative velocity is now zero, it is not long contracted, and is now longer than the garage; it must bend, yell, or explode.
Again, the puzzle comes from considering the situation from the ladder frame. In the above analysis, in its own frame, the ladder is always longer than the garage. So how do we close the door and trap it inside?
It should be noted here the general feature of relativity: we have concluded, taking into account the framework of the garage, that we indeed trap the ladder in the garage. Therefore, this must be true in any frame - it can not happen that the ladder is locked in one frame but not on the other. From the ladder frame, we know that there must be an explanation of how the stairs are trapped; we just have to find the explanation.
The explanation is that, although all parts of the ladder simultaneously slowed to zero within the frame of the garage, because of the relative harmony, the corresponding decelerations in the ladder frame are not simultaneous. Instead, each part of the ladder slows down sequentially, from front to back, until finally the back of the ladder decreases its speed, which is already inside the garage.
Due to the long contraction and widening of time both controlled by the Lorentz transformation, the ladder paradox can be seen as the physical correlation of the twin paradox, in which one instance of a pair of twins leaves the earth, moves at a speed for a period, and returns to the earth a little younger than the twin earth. As in the case of a ladder trapped inside a barn, if there is no preferable frame of reference - each moves only relative to the other - how could it be a traveling twin rather than a younger stationer (only because it is a ladder rather than a more shed short)? In both cases it is the distinguished distinguishing phenomena: the twins, not the earth (or the ladder, not the barn) that undergoes power decelerations in the temporal (or physical, in the case of ladder-barn) inertial frames.
Ladder paradox and style transmission
What if the back door (exit stair door) is permanently closed and does not open? Assume the door is so solid that the ladder will not penetrate it when it collides, so it must stop. Then, as in the scenarios described above, within the frame of reference of the garage, there are times when the ladder is actually inside the garage (that is, the back of the ladder is inside the front door), before colliding with the back door and stopping. However, from the frame of reference of the stairs, the stairs are too large to fit in the garage, so that by the time it collides with the back door and stops, the back of the stairs still has not reached the front door. This seems to be a paradox. The question is, does the back pass through the front door or not?
Difficulties arise largely from the assumption that the stairs are rigid (ie, maintaining the same shape). The stairs look stiff in everyday life. But really rigidly requires it to transfer power at infinite speed (that is, when you press one end the other end must immediately react, otherwise the ladder will change shape). This is contrary to special relativity, which states that information can run no faster than the speed of light (which is too fast for us to notice in real life, but significant in the ladder scenario). So the object can not be completely rigid under special relativity.
In this case, when the front of the ladder collides with the rear door, the back of the ladder does not yet know it, so keep moving forward (and the "compress" ladder). In both the garage frame and the ladder inertia frame, the rear end continues to move at the time of the collision, until at least the point at which the rear of the stairs enters the cone of light from the collision (ie, the point at which power moves backward at the speed of light from the point of collision will reach it). At this point the ladder is actually shorter than the original contract length, so the rear end is inside the garage. Calculations in both reference frames will show this to be the case.
What happens after the force reaches the back of the ladder (the "green" zone in the diagram) is not specified. Depending on physics, the ladder may break; or, if elastic enough, it can bend and redevelop to its original length. At fairly high speeds, any realistic material will explode violently into plasma.
Man falls into grate variation
This paradox was originally proposed and solved by Wolfgang Rindler and involved fast-paced humans, represented by sticks, falling into the bars. It is assumed that the whole rod above the bars in the reference reference frame before the downward acceleration begins simultaneously and is equally applied to each point in the rod.
From the grate perspective, the rod undergoes a long contraction and fits into the grate. However, from the standpoint of the stem, it is a long-contracted scar , which seems to be too long to fall.
In fact, the downward acceleration of the rod, which is simultaneously in the frame of reference of the test, is not simultaneous in the frame of reference of the rod. Within the frame of reference of the rod, the lower front portion of the first rod is accelerated downward (not shown in the drawing), and as time passes, more stems are subjected to acceleration downward, until the rod end is accelerated downward. This results in bending of stems within the frame of reference of the rod. It should be emphasized that, since this bending occurs in the bar framework, it is the actual physical distortion of the rod that will cause pressure to occur on the stem.
Bar and paradox rings
A very similar but simpler problem than the paradox of stems and scars, involving only the inertial framework, is the paradox of "bar and ring" (Ferraro 2007). The stem and grate paragraphs are complicated: this involves a non-inertial reference frame because at one point a human walks horizontally, and a moment later he falls down; and it involves human physical deformation (or segmented stem), because the rod is bent in one frame of reference and straight on the other. These aspects of the problem introduce complications involving rod stiffness that tend to obscure the true nature of "paradox". The "bar and ring" paradoxes are free of this complication: a bar, slightly larger than the diameter of the ring, moves up and down the right with its long horizontal axis, while the silent ring and ring plane are also horizontal. If the movement of the bar is such that the center of the bar coincides with the center of the ring at some point of time, then the bar will be Lorentz-contracted because of the forward component of its movement, and it will pass through the ring. Paradox occurs when the problem is considered within the framework of the break from the bar. The ring now moves down and to the left, and Lorentz will contract along its horizontal length, while the bar will not be contracted at all. How does the bar get through the ring?
The paradoxical resolution again lies in the relativity of simultaneity (Ferraro 2007). The length of a physical object is defined as the distance between two simultaneous events occurring at each end of the body, and because similarity is relative, so is the length. The length of this variability is only Lorentz contraction. Similarly, the physical angle is defined as the angle formed by three concurrent events , , and this angle will also be a relative quantity. In the above paradox, although the rod and the ring plane are parallel within the framework of the ring break, they are not parallel within the framework of the stem break. The rods are not channeled through the ring being contracted by Lorentz because the plane of the ring is rotated relative to the stem with sufficient quantity to let the stem pass through.
Source of the article : Wikipedia
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