RELATIVITY: PARADOXES, ILLUSIONS AND REALITY

The everyday world we observe around us is only a tiny fraction of the physical reality our Universe offers us. In fact, it seems incredible that our everyday reality is the result of completely counterintuitive phenomena that occur at tiny scales, almost impossible to imagine. There are also other phenomena involving high energies and/or high speeds that we normally don't detect in our daily lives. However, very occasionally, our macroscopic world suddenly encounters these strange phenomena: when we try to precisely position the GPS coordinates of an object, when we try to explain the orbit of the planet closest to the Sun, when we try to explain why Mercury is liquid at room temperature... then "small and strange distortions" appear. These anomalies seem like shadows of another world, a world that behaves in strange and fascinating ways: the world of relativity.

This article does not intend to review the concepts and mathematical formulas of special relativity, but rather to offer brief but profound reflections on the physical significance of these fascinating phenomena. To do so, we will use the so-called "ladder and barn paradox."

The relativistic ladder and barn

A farmer has a 4-meter-long barn, and his son carries a 5-meter-long ladder. The son, who understands relativity, sets out to show his father that there is a way to store the ladder inside the barn: if he can achieve relativistic speed, the ladder will contract in length and fit inside the barn. Of course, the father, upon hearing this, thinks his son has lost his mind, but does it make any sense?

By the principle of relativity, every inertial reference frame (RF) (with constant velocity) is equivalent to any other and can therefore be considered at rest while considering the other RF to be moving. Because of this, for the farmer, it is the ladder that is moving toward him, while for his son, it is the barn that is moving. Thus, from the farmer's point of view:

While from your son's point of view we have:

Lo and lo are the rest length of the ladder and the barn respectively.

In the farmer's SR it is the ladder that contracts and therefore could fit in the barn but from the son's SR it is the barn that contracts which would make the situation much worse: Is it possible for the ladder to fit in the barn from the farmer's son's point of view?

Following the Lorentz transforms for a ladder of length Lo the relativistic contraction will be:

For it to fit in the barn it must be fulfilled that Lo'

For a 5-meter ladder, B must be greater than or equal to 0.6, or 60% of the speed of light. To better understand what's happening, let's consider just two reference events:

1) The front edge of the ladder reaches the back door of the barn:

2) The back edge of the ladder enters the barn:

Event 1: This event coincides with the time it takes the ladder to cross the barn. For the farmer, this time will be: tA = 10/v, while for his son, the time is:

Event 2: For the farmer's son the instant at which the back of the ladder passes through the front door of the barn coincides with the time it takes to enter the barn the entire length of the ladder, i.e.: t´B=Lo/v while for the farmer this happens at the instant:

The "relativistic correction factor" is the square root term and is always less than one, so comparing the above expressions, it's easy to see that for the farmer, event 2 happens first, while for his son, event 1 happens first. The conclusion from this is devastating to our common sense: the concept of simultaneity , which we take for granted in our everyday lives , doesn't exist when we consider relativistic speeds. The answer to the apparent paradox lies in the details of the concept of "fit": to know if the ladder fits in the barn, we have to compare several independent events: the front and back doors of the barn, and the front and back of the ladder. For the ladder to fit in the barn, there must be an instant in which both ends of the ladder are simultaneously in a certain position for both observers (within the space enclosed by the barn doors). However, as we have seen, this concept doesn't exist in relativity, so there is no paradox , since in relativity, each observer sees a different sequence of events. In fact, as we shall see, it makes no sense to ask whether the ladder fits in the space enclosed by the barn, but rather whether the ladder "fits" in four-dimensional space-time . We now proceed to list the profound insights that can be derived from the theory of special relativity:

Reflection 1: Dynamic space-time

Because the speed of light is finite, light will take more or less time to travel from one object to another depending on its position. This fact might lead us to think that relativistic effects are a kind of illusion, that is, that they are due only to the fact that light takes a while to travel from one place to another. However, the reality is more complex and fascinating: moving clocks actually move slower (relativistic time dilation has been verified by a multitude of experiments), and this is due to the dilation of space-time itself. In fact, in general relativity (derived from special relativity), the force of gravity is due exclusively to the motion of space-time itself. Of course, spatial contraction is also due to this phenomenon: it is not the distance between the atoms on the ladder that shortens (or the length between the marks on a ruler), but rather the space-time itself contained between the atoms on the ladder that contracts. It is true that this seems hard to believe, but it has been experimentally verified in a wide variety of experiments. In fact, to correctly calculate the distance to distant galaxies, one must not only take into account the time it takes for light to reach us, but also relativistic time dilation.

Reflection 2: Absolute simultaneity does not exist

Although the concept of simultaneity does not exist in relativity, it is evident that physics cannot be affected and physical phenomena must be coherent: at points where both SRs meet at the same point in spacetime, if we compare the measurements of both observers, they must both agree in their measurements since both observers cannot see different stories. If for example we try to close both doors just when the back end of the ladder has crossed the front door , both doors will not close simultaneously from the farmer's son's SR: first the back door will come down and then the front one so that the door will never hit the ladder and therefore it will never happen that both observers see different phenomena (such as a broken ladder and a whole ladder).

Reflection 3: Absolute rigidity does not exist

Accelerating macroscopic objects to velocities close to ac is practically impossible as it would require an immense amount of energy, which is why real relativistic phenomena involve tiny objects such as particles or atoms. Using macroscopic objects such as ladders often contributes to the emergence of supposed "paradoxes." As we know, solid objects are composed of atoms held together by electromagnetic forces. These forces are transmitted by the exchange of photons traveling at the speed of light. Because of this, no macroscopic object will respond instantaneously to any event. In fact, between the atoms at the front and back ends of the ladder, there exists a time Lo/c during which nothing that happens at one end can affect the other. In fact, if the back door of the barn were closed, from the son's SR, while the front end of the ladder has collided with the back door, the rear end would continue entering the barn at speed v for a time Lo/c (in fact, the son would also think that the ladder had fit into the barn) as if the ladder were flexible and made up of tiny parts. Reflection number 3 tells us that in relativity, macroscopic objects are not completely solid but have a certain "elasticity" because there is no instantaneous transmission of information between their parts.

Reflection 4: Space-time is four-dimensional

The world we inhabit is not the three-dimensional world we detect with our senses, but rather a four-dimensional world with three spatial dimensions and one temporal dimension that can be "blended" or "exchanged" depending on the observer's state of motion. This will be the subject of the next section.

Our four-dimensional world

Both the ladder and the barn can be considered physical objects in four-dimensional space-time. Different SRs measure different values of space and time for the same object or event in four dimensions. One will measure a greater value of time and a lesser value of space, and the other a smaller value of time and a greater value of space. Objects with mass can never reach the speed of light, and only very light objects, such as elementary particles, can travel at relativistic speeds. This implies that massive objects travel almost exclusively in the time dimension, and only very light objects can travel in the spatial dimensions.

When the relative velocity between two SRs is a significant fraction of c, the two observers see different three-dimensional "slices" or sections of the same four-dimensional object. As if we were in Plato's famous cave, we only observe the "projections" of a higher-dimensional entity and need physics and mathematics to deduce what that four-dimensional object is like. This can be seen in a simplified form in the following figure (only the time dimension is represented):

In the first SR, considered the SR at rest, the temporal "slices" are straight, the events in red are simultaneous, and the events in yellow occur at different times. In the second SR, considered the SR in motion, the "slices" are rotated, the events in red are no longer simultaneous, and the events in yellow now occur at the same time.

Let's return to the ladder paradox: consider the first SR above to be the farmer's SR, and the second SR to be the farmer's son's SR. Consider the red dots to be the two barn doors, and the yellow dots to be the two ends of the ladder. From the farmer's point of view, the barn and everything inside are at rest relative to it, and therefore everything that happens is simultaneous: both barn doors open or close simultaneously. However, the ladder is not at rest, and therefore what happens at the ends of the ladder is not simultaneous: first, he sees the back end enter the front door, and then he sees the front end of the ladder reach the back door. From the son's point of view, the two ends of the ladder are simultaneous, but the back door closes before the front one! This brings us to the last and perhaps most profound reflection: Both observers are seeing the same four-dimensional reality from different three-dimensional "angles" or "slices."

Now we are ready to know the true resolution of the "paradox": the farmer and his son see different portions of space and time of the same four-dimensional object , while the farmer observes a reduction in space of the ladder, the son observes a reduction (dilation) in time of the ladder. The ladder fits within the four-dimensional world map of the barn for both observers since what is space for one observer is time for another and vice versa:

The farmer and the son measure different amounts of space and time from the same four-dimensional object (green and blue segments). In this way, we only see different three-dimensional projections of the same four-dimensional object.

Sources: Beyond the pole-barn paradox: How the pole is caught , The theory of relativity