The ladder paradox (or barn-pole paradox) is a thought experiment in special relativity. It involves a ladder, parallel to the ground, travelling horizontally at relativistic speed (near the speed of light) and therefore undergoing a Lorentz length contraction. The ladder is imagined passing through the open front and rear doors of a garage or barn which is shorter than its rest length, so if the ladder was not moving it would not be able to fit inside. To a stationary observer, due to the contraction, the moving ladder is able to fit entirely inside the building as it passes through. On the other hand, from the point of view of an observer moving with the ladder, the ladder will not be contracted, and it is the building which will be Lorentz contracted to an even smaller length. Therefore, the ladder will not be able to fit inside the building as it passes through. This poses an apparent discrepancy between the realities of both observers.
This apparent paradox results from the mistaken assumption of absolute simultaneity. The ladder is said to fit into the garage if both of its ends can be made to be simultaneously inside the garage. The paradox is resolved when it is considered that in relativity, simultaneity is relative to each observer, making the answer to whether the ladder fits inside the garage also relative to each of them.
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 cannot close both doors on either side of the ladder without hitting it. This apparent contradiction is the paradox.
Figure 1: An overview of the garage and the ladder at rest
Figure 2: In the garage frame, the ladder undergoes length contraction and will therefore fit into the garage.
Figure 3: In the ladder frame, the garage undergoes length contraction and is too small to contain the ladder.
Resolution
Figure 4: Scenario in the garage frame: a length contracted ladder passing through the garageFigure 5: Scenario in the ladder frame: a length contracted garage passing over the ladder. Only one door is closed at any time
The solution to the apparent paradox lies in the relativity of simultaneity: what one observer (e.g. with the garage) considers to be two simultaneous events may not in fact be simultaneous to another observer (e.g. with the ladder). When we say the ladder "fits" inside the garage, what we mean precisely is that, at some specific time, the position of the back of the ladder and the position of the front of the ladder were both inside the garage; in other words, the front and back of the ladder were inside the garage simultaneously. As simultaneity is relative, then, two observers disagree on whether the ladder fits. To the observer with the garage, the back end of the ladder was in the garage at the same time that the front end of the ladder was, and so the ladder fit; but to the observer with the ladder, these two events were not simultaneous, and the ladder did not fit.
A clear way of seeing this is to consider the doors, which, in the frame of the garage, close for the brief period that the ladder is fully inside. We now look at these events in the frame of the ladder. The first event is the front of the ladder approaching the exit door of the garage. The door closes, and then opens again to let the front of the ladder pass through. At a later time, the back of the ladder passes through the entrance door, which closes and then opens. We see that, as simultaneity is relative, the two doors did not need to be shut at the same time, and the ladder did not need to fit inside the garage.
The situation can be further illustrated by the Minkowski diagram below. The diagram is in the rest frame of the garage. The vertical light-blue band shows the garage in spacetime, and the light-red band shows the ladder in spacetime. The x and t axes are the garage space and time axes, respectively, and x′ and t′ are the ladder space and time axes, respectively.
In the frame of the garage, the ladder at any specific time is represented by a horizontal set of points, parallel to the x axis, in the red band. One example is the bold blue line segment, which lies inside the blue band representing the garage, and which represents the ladder at a time when it is fully inside the garage. In the frame of the ladder, however, sets of simultaneous events lie on lines parallel to the x' axis; the ladder at any specific time is therefore represented by a cross section of such a line with the red band. One such example is the bold red line segment. We see that such line segments never lie fully inside the blue band; that is, the ladder never lies fully inside the garage.
Figure 6: A Minkowski diagram of the ladder paradox. The garage is shown in light blue, the ladder in light red. The diagram is in the rest frame of the garage, with x and t being the garage space and time axes, respectively. The ladder frame is for a person sitting on the front of the ladder, with x′ and t′ being the ladder space and time axes respectively. The blue and red lines, AB and AC, depict the ladder at the time when its front end meets the garage's exit door, in the frame of reference of the garage and the ladder, respectively. Event D is the rear end of the ladder reaching the garage's entrance.
Shutting the ladder in the garage
Figure 7: A ladder contracting under acceleration to fit into a length contracted garage
In a more complicated version of the paradox, we can physically trap the ladder once it is fully inside the garage. This could be done, for instance, by not opening the exit door again after we close it. In the frame of the garage, we assume the exit door is immovable, and so when the ladder hits it, we say that it instantaneously stops.[1][2] By this time, the entrance door has also closed, and so the ladder is stuck inside the garage. As its relative velocity is now zero, it is not length contracted, and is now longer than the garage; it will have to bend, snap, or explode.
Again, the puzzle comes from considering the situation from the frame of the ladder. In the above analysis, in its own frame, the ladder was always longer than the garage. So how did we ever close the doors and trap it inside?
It is worth noting here a general feature of relativity: we have deduced, by considering the frame of the garage, that we do indeed trap the ladder inside the garage. This must therefore be true in any frame - it cannot be the case that the ladder snaps in one frame but not in another. From the ladder's frame, then, we know that there must be some explanation for how the ladder came to be trapped; we must simply find the explanation.
The explanation is that, although all parts of the ladder simultaneously decelerate to zero in the garage's frame, because simultaneity is relative, the corresponding decelerations in the frame of the ladder are not simultaneous. Instead, each part of the ladder decelerates sequentially,[1][3] from front to back, until finally the back of the ladder decelerates, by which time it is already within the garage.
As length contraction and time dilation are both controlled by the Lorentz transformations, the ladder paradox can be seen as a physical correlate of the twin paradox, in which instance one of a set of twins leaves earth, travels at speed for a period, and returns to earth a bit younger than the earthbound twin. As in the case of the ladder trapped inside the barn, if neither frame of reference is privileged — each is moving only relative to the other — how can it be that it's the traveling twin and not the stationary one who is younger (just as it's the ladder rather than the barn which is shorter)? In both instances it is the acceleration-deceleration that differentiates the phenomena: it's the twin, not the earth (or the ladder, not the barn) that undergoes the force of deceleration in returning to the temporal (or physical, in the case of the ladder-barn) inertial frame.
Figure 8: A Minkowski diagram of the case where the ladder is stopped all along its length, simultaneously in the garage frame. When this occurs, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. When the back of the ladder enters the garage at point D, it has not yet felt the effects of the acceleration of its front end. At this time, according to someone at rest with respect to the back of the ladder, the front of the ladder will be at point E and will see the ladder as DE. It is seen that this length in the ladder frame is not the same as CA, the rest length of the ladder before the deceleration.
Ladder paradox and transmission of force
Figure 1: A Minkowski diagram of the case where the ladder is stopped by impact with the back wall of the garage. The impact is event A. At impact, the garage frame sees the ladder as AB, but the ladder frame sees the ladder as AC. The ladder does not move out of the garage, so its front end now goes directly upward, through point E. The back of the ladder will not change its trajectory in spacetime until it feels the effects of the impact. The effect of the impact can propagate outward from A no faster than the speed of light, so the back of the ladder will never feel the effects of the impact until point F (note the 45° angle of the line A-F, corresponding to the speed of light transmission of information) or later, at which time the ladder is well within the garage in both frames. Note that when the diagram is drawn in the frame of the ladder, the speed of light is the same, but the ladder is longer, so it takes more time for the force to reach the back end; this gives enough time for the back of the ladder to move inside the garage.
What if the back door (the door the ladder exits out of) is closed permanently and does not open? Suppose that the door is so solid that the ladder will not penetrate it when it collides, so it must stop. Then, as in the scenario described above, in the frame of reference of the garage, there is a moment when the ladder is completely within the garage (i.e., the back of the ladder is inside the front door), before it collides with the back door and stops. However, from the frame of reference of the ladder, the ladder is too big to fit in the garage, so by the time it collides with the back door and stops, the back of the ladder still has not reached the front door. This seems to be a paradox. The question is, does the back of the ladder cross the front door or not?
The difficulty arises mostly from the assumption that the ladder is rigid (i.e., maintains the same shape). Ladders seem rigid in everyday life. But being completely rigid requires that it can transfer force at infinite speed (i.e., when you push one end the other end must react immediately, otherwise the ladder will deform). This contradicts special relativity, which states that information can travel no faster than the speed of light (which is too fast for us to notice in real life, but is significant in the ladder scenario). So objects cannot be perfectly rigid under special relativity.
In this case, by the time the front of the ladder collides with the back door, the back of the ladder does not know it yet, so it keeps moving forwards (and the ladder "compresses"). In both the frame of the garage and the inertial frame of the ladder, the back end keeps moving at the time of the collision, until at least the point where the back of the ladder comes into the light cone of the collision (i.e., a point where force moving backwards at the speed of light from the point of the collision will reach it). At this point the ladder is actually shorter than the original contracted length, so the back end is well inside the garage. Calculations in both frames of reference 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 the physics, the ladder could break; or, if it were sufficiently elastic, it could bend and re-expand to its original length. At sufficiently high speeds, any realistic material would violently explode into a plasma.
Man falling into grate variation
A man (represented by a segmented rod) falling into a grate
This early version of the paradox was originally proposed and solved by Wolfgang Rindler[1] and involved a fast walking man, represented by a rod, falling into a grate.[4] It is assumed that the rod is entirely over the grate in the grate frame of reference before the downward acceleration begins simultaneously and equally applied to each point in the rod.
From the perspective of the grate, the rod undergoes a length contraction and fits into the grate. However, from the perspective of the rod, it is the grate undergoing a length contraction, through which it seems the rod is then too long to fall.
The downward acceleration of the rod, which is simultaneous in the grate's frame of reference, is not simultaneous in the rod's frame of reference. In the rod's frame of reference, the front of the rod is first accelerated downward (shown in cell 3 of the drawing), and as time goes by, more and more of the rod is subjected to the downward acceleration, until finally the back of the rod is accelerated downward. This results in a bending of the rod in the rod's frame of reference. Since this bending occurs in the rod's rest frame, it is a true physical distortion of the rod which will cause stresses to occur in the rod.
For this non-rigid behaviour of the rod to become apparent, both the rod itself and the grate must be of such a scale that the traversal time is measurable.
Bar and ring paradox
The diagram on the left illustrates a bar and a ring in the rest frame of the ring at the instant that their centers coincide. The bar is Lorentz-contracted and moving upward and to the right while the ring is stationary and uncontracted. The diagram on the right illustrates the situation at the same instant, but in the rest frame of the bar. The ring is now Lorentz-contracted and rotated with respect to the bar, and the bar is uncontracted. Again, the ring passes over the bar without touching it.
A problem very similar but simpler than the rod and grate paradox, involving only inertial frames, is the "bar and ring" paradox (Ferraro 2007). The rod and grate paradox is complicated: it involves non-inertial frames of reference since at one moment the man is walking horizontally, and a moment later he is falling downward; and it involves a physical deformation of the man (or segmented rod), since the rod is bent in one frame of reference and straight in another. These aspects of the problem introduce complications involving the stiffness of the rod which tends to obscure the real nature of the "paradox". The "bar and ring" paradox is free of these complications: a bar, which is slightly larger in length than the diameter of a ring, is moving upward and to the right with its long axis horizontal, while the ring is stationary and the plane of the ring is also horizontal. If the motion of the bar is such that the center of the bar coincides with the center of the ring at some point in time, then the bar will be Lorentz-contracted due to the forward component of its motion, and it will pass through the ring. The paradox occurs when the problem is considered in the rest frame of the bar. The ring is now moving downward and to the left, and will be Lorentz-contracted along its horizontal length, while the bar will not be contracted at all. How can the bar pass through the ring?
The resolution of the paradox 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 since simultaneity is relative, so is this length. This variability in length is just the Lorentz contraction. Similarly, a physical angle is defined as the angle formed by three simultaneous events, and this angle will also be a relative quantity. In the above paradox, although the rod and the plane of the ring are parallel in the rest frame of the ring, they are not parallel in the rest frame of the rod. The uncontracted rod passes through the Lorentz-contracted ring because the plane of the ring is rotated relative to the rod by an amount sufficient to let the rod pass through.
In mathematical terms, a Lorentz transformation can be separated into the product of a spatial rotation and a "proper" Lorentz transformation which involves no spatial rotation. The mathematical resolution of the bar and ring paradox is based on the fact that the product of two proper Lorentz transformations (horizontal and vertical) may produce a Lorentz transformation which is not proper (diagonal) but rather includes a spatial rotation component.
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 treatment, the theory is presented as being based on just two postulates:
The laws of physics are invariant (identical) in all inertial frames of reference.
The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer.
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur.
In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, as a consequence of an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey. Another way of looking at it is to realize the travelling twin is undergoing acceleration, which makes them a non-inertial observer. In both views there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not actually a paradox in the sense of a logical contradiction. There is still debate as to the resolution of the twin paradox.
The Trouton–Noble experiment was an attempt to detect motion of the Earth through the luminiferous aether, and was conducted in 1901–1903 by Frederick Thomas Trouton and H. R. Noble. It was based on a suggestion by George FitzGerald that a charged parallel-plate capacitor moving through the aether should orient itself perpendicular to the motion. Like the earlier Michelson–Morley experiment, Trouton and Noble obtained a null result: no motion relative to the aether could be detected. This null result was reproduced, with increasing sensitivity, by Rudolf Tomaschek, Chase and Hayden in 1994. Such experimental results are now seen, consistent with special relativity, to reflect the validity of the principle of relativity and the absence of any absolute rest frame. The experiment is a test of special relativity.
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald contraction and is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.
Proper length or rest length is the length of an object in the object's rest frame.
Terrell rotation or Terrell effect is the visual distortion that a passing object would appear to undergo, according to the special theory of relativity if it were travelling at a significant fraction of the speed of light. This behaviour was described independently by both Roger Penrose and James Edward Terrell. Penrose's article was submitted 29 July 1958 and published in January 1959. Terrell's article was submitted 22 June 1959 and published 15 November 1959. The general phenomenon was noted already in 1924 by Austrian physicist Anton Lampa.
Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this hyperbolically accelerated frame can be compared to effects arising in a homogeneous gravitational field. For general overview of accelerations in flat spacetime, see Acceleration and Proper reference frame.
In relativistic physics, Supplee's paradox is a physical paradox that arises when considering the buoyant force exerted on a relativistic bullet immersed in a fluid subject to an ambient gravitational field. If a bullet has neutral buoyancy when it is at rest in a perfect fluid and then it is launched with a relativistic speed, observers at rest within the fluid would conclude that the bullet should sink, since its density will increase due to the length contraction effect. On the other hand, in the bullet's proper frame it is the moving fluid that becomes denser and hence the bullet would float. But the bullet cannot sink in one frame and float in another, so there is a paradox situation.
Special relativity is a physical theory that plays a fundamental role in the description of all physical phenomena, as long as gravitation is not significant. Many experiments played an important role in its development and justification. The strength of the theory lies in its unique ability to correctly predict to high precision the outcome of an extremely diverse range of experiments. Repeats of many of those experiments are still being conducted with steadily increased precision, with modern experiments focusing on effects such as at the Planck scale and in the neutrino sector. Their results are consistent with the predictions of special relativity. Collections of various tests were given by Jakob Laub, Zhang, Mattingly, Clifford Will, and Roberts/Schleif.
The history of special relativity consists of many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. It culminated in the theory of special relativity proposed by Albert Einstein and subsequent work of Max Planck, Hermann Minkowski and others.
What is now often called Lorentz ether theory (LET) has its roots in Hendrik Lorentz's "theory of electrons", which marked the end of the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century.
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.
Bell's spaceship paradox is a thought experiment in special relativity. It was first described by E. Dewan and M. Beran in 1959 but became more widely known after John Stewart Bell elaborated the idea further in 1976. A delicate thread hangs between two spaceships headed in the same direction. They start accelerating simultaneously and equally as measured in the inertial frame S, thus having the same velocity at all times as viewed from S. Therefore, they are all subject to the same Lorentz contraction, so the entire assembly seems to be equally contracted in the S frame with respect to the length at the start. At first sight, it might appear that the thread will not break during acceleration.
In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possibility was raised by mathematician Henri Poincaré in 1900, and thereafter became a central idea in the special theory of relativity.
Albert Einstein presented the theories of special relativity and general relativity in publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most notably to the work of Henri Poincaré and Hendrik Lorentz for special relativity, and to the work of David Hilbert, Carl F. Gauss, Bernhard Riemann, and Ernst Mach for general relativity. Subsequently, claims have been put forward about both theories, asserting that they were formulated, either wholly or in part, by others before Einstein. At issue is the extent to which Einstein and various other individuals should be credited for the formulation of these theories, based on priority considerations.
Wolfgang Rindler was a physicist working in the field of general relativity where he is known for introducing the term "event horizon", Rindler coordinates, and for the use of spinors in general relativity. An honorary member of the Austrian Academy of Sciences and foreign member of the Accademia delle Scienze di Torino, he was also a prolific textbook author.
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When using the term "the speed of light" it is sometimes necessary to make the distinction between its one-way speed and its two-way speed. The "one-way" speed of light, from a source to a detector, cannot be measured independently of a convention as to how to synchronize the clocks at the source and the detector. What can however be experimentally measured is the round-trip speed from the source to a mirror and back again to detector. Albert Einstein chose a synchronization convention that made the one-way speed equal to the two-way speed. The constancy of the one-way speed in any given inertial frame is the basis of his special theory of relativity, although all experimentally verifiable predictions of this theory do not depend on that convention.
This timeline describes the major developments, both experimental and theoretical, of:
Pierce, Evan (2007). "The lock and key paradox and the limits of rigidity in special relativity". American Journal of Physics. 75 (7): 610–614. Bibcode:2007AmJPh..75..610P. doi:10.1119/1.2711827.
McGlynn, Enda; van Kampen, Paul (2008). "A note on linking electric current, magnetic fields, charges and the pole in a barn paradox in special relativity". European Journal of Physics. 29 (6): N63–N67. Bibcode:2008EJPh...29...63M. doi:10.1088/0143-0807/29/6/N03. S2CID121939564.
Further reading
Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (2nd ed) (Freeman, NY, 1992)
- discusses various apparent SR paradoxes and their solutions
Rindler, Wolfgang (2001). Relativity: Special, General and Cosmological. Oxford University Press. ISBN0-19-850836-0.
Special Relativity Animationsfrom John de Pillis.This inter-active animated train-and-tunnel paradox is an analog of the pole (train) and barn (tunnel) paradox.
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