What could have happened if there were more than three dimensions in our world? How could an “extra”, additional dimension affect the course of various physical processes? Let's approach the answer to this question from a distance …
Nowadays, in science fiction literature, it is quite often possible to meet with almost instantaneous overcoming of large cosmic distances using the so-called zero-transportation or crossing through "hyperspace", or "subspace", or "superspace". What do science fiction writers mean in this case?
It is generally accepted that the maximum speed with which any real body can move in space is, according to the theory of relativity, the speed of light in a void, which is 300,000 km / sec. Moreover, this speed is practically unattainable! What kind of lightning "jumps" through millions and hundreds of millions of light years can talk about? Of course, the idea of this kind of "transitions" is fantastic. But it is based on very curious physical and mathematical considerations.
Imagine a "one-dimensional being" - a point located in one-dimensional space, that is, on a straight line. In this "small" world, there is only one dimension - length and only two possible directions of movement - forward and backward.
The imaginary two-dimensional creature - "flat" - has much more possibilities. They are able to move in two dimensions: in their world, in addition to length, there is also width. But in the same way they cannot go into the third dimension, just as creatures-points cannot "jump out" beyond their straight line. One-dimensional and two-dimensional inhabitants, in principle, are able to come to a theoretical conclusion about the probability of the existence of more dimensions than in their worlds, but the paths to subsequent dimensions are practically closed for them!
On both sides of the plane there is a three-dimensional space, we live in it - three-dimensional creatures that are not visible to two-dimensional inhabitants, enclosed in their flat world: after all, they can even see only within their space. Two-dimensional creatures could practically collide with the three-dimensional world and its inhabitants only if some person, for example, pierced their plane with a nail or a needle. But even then a two-dimensional creature could observe only a two-dimensional area of intersection of the plane and the nail. It is unlikely that this was enough to draw some conclusions about the "otherworldly", from the point of view of a two-dimensional inhabitant, three-dimensional space and its "mysterious" inhabitants.
However, exactly the same reasoning can be applied to our three-dimensional space, if it were enclosed in a more "vast" four-dimensional space, just as the two-dimensional plane is enclosed in itself.
But let us first try to find out what exactly four-dimensional space is. In our three-dimensional world, as noted above, there are three mutually perpendicular directions - length, width and height - three mutually perpendicular coordinate axes. If it was possible to add to these three directions a fourth, also perpendicular to each of them, then we would get a space with four dimensions - a four-dimensional world!
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From the point of view of mathematical logic, our reasoning about the construction of four-dimensional space is absolutely flawless. But by themselves they still do not prove anything, because logical consistency is not proof of "existence" in the physical sense. Only experience can provide such proof. And experience shows that in our space through one point only three mutually perpendicular straight lines can be drawn.
Let us turn to the help of the "flatheads" again. For them, the third dimension, into which they cannot go, is the same as the fourth for us. But there is a significant difference between imaginary flat beings and us, the inhabitants of the three-dimensional world. While the plane is a two-dimensional part of the real-world three-dimensional world, all the scientific evidence at our disposal strongly suggests that the space in which we live is geometrically three-dimensional and is not part of any four-dimensional world! If such a four-dimensional world really existed, then rather strange events and phenomena could occur in our three-dimensional world.
Let's return again to the two-dimensional, "flat" world. Although its inhabitants are not able to “go out” of their plane, nevertheless, due to the presence of the external three-dimensional world, it is in principle possible to imagine some phenomena that imply an exit into the third dimension. This circumstance makes possible such processes that could not occur in two-dimensional space in itself. Imagine, for example, a clock face drawn in a plane. No matter how we rotate and move this dial, remaining in the plane, we will never be able to change the position of the numbers so that they follow each other in a counterclockwise direction. This can be achieved only by "removing" the dial from the plane into three-dimensional space, turning it over, and then returning it to the plane again.
In three-dimensional space, this operation would correspond, for example, this. Is it possible to transform a glove intended for the right hand into a glove for the left hand by simply moving it in our three-dimensional space (that is, without turning it inside out)? You can easily see that such an operation is not feasible! But given four-dimensional space, it could be as easy to achieve as it is with a dial. But we do not know the way out into four-dimensional space. Apparently, nature does not know him either. At least, no phenomena that could be explained by the existence of a four-dimensional world, covering our three-dimensional, have never been registered! It's a pity. If four-dimensional space and the exit into it actually existed,then truly incredible opportunities and prospects would open up before us.
Let us turn again to the two-dimensional world and imagine a "flat plane", which needs to overcome the distance between two points of the flat world, which are 50 km apart from each other, for example. If the "flat" moves at a speed of one meter per day, then this kind of journey will take no less than 50,000 years. But imagine that a two-dimensional surface is folded or, more precisely, "bent" in three-dimensional space in such a way that the points of the beginning and end of the route are only one meter apart from each other. Now they are separated by a distance equal to only one meter. That is, the distance that the "flat" could cover in just one day. But this meter is in the third dimension! This would be "nulltransportation" or "hypertransport".
A similar situation could arise in a curved three-dimensional world. As we already know, our three-dimensional world, according to the ideas of the general theory of relativity, is curved. And since the curvature depends on the magnitude of gravitational forces, then if there were an enveloping four-dimensional space, in principle this curvature could be controlled. Decrease or increase it. And it would be possible to "bend" the three-dimensional space in such a way that the start and end points of our "space route" are separated by a very small distance. In order to get from one to the other, it would be enough to "jump" through the "four-dimensional gap" separating them. This is what science fiction writers mean. Another question: how can this be done?
These are the seductive advantages of the four-dimensional world … However, like other multidimensional worlds, it also has “disadvantages”. It turns out that with an increase in the number of dimensions, the stability of motion decreases. Numerous studies have shown that in two-dimensional space no disturbances can disturb the equilibrium and remove a body in a closed orbit around another body to infinity. In the space of three dimensions, that is, in our real world, the limitations are already much weaker. But here, too, the trajectory of a body moving in a closed orbit can go to infinity only if the perturbing force is very large.
But already in four-dimensional space, all circular trajectories turn out to be unstable. In such a space, the planets, for example, would not be able to revolve around the Sun - they would either fall on it, or fly away into infinity!
Using the equations of quantum mechanics, it is possible to show that in a world with more than three dimensions, the hydrogen atom could not exist as a stable entity. An inevitable fall of the electron onto the nucleus would take place.
Thus, in the world of four or more dimensions, neither various chemical elements nor planetary systems could exist …
The "addition" of the fourth dimension would also change some of the purely geometric properties of the three-dimensional world. One of the important branches of geometry, which is not only of theoretical, but also of great practical interest, is the so-called theory of transformations. It is about how different geometric shapes change when moving from one coordinate system to another. One of these types of geometric transformations is called "conformal". This is what angle-preserving transformations are called.
Imagine a simple geometric shape such as a square or a polygon. Let's put an arbitrary grid of lines on it, a kind of "skeleton". Then "conformal" we will call such transformations of the coordinate system, in which our square or rectangle goes into any other figure, but so that the angles between the lines of the "skeleton" are preserved. An illustrative example of "conformal" transformation is the transfer of images from the surface of a globe (and in general from any spherical surface) to a plane - this is how geographic maps are constructed.
Back in the 19th century, the outstanding mathematician Bernhard Riemann showed that any flat solid (that is, without "holes", or, as mathematicians say, "simply connected") figure can be conformally transformed into a circle. Riemann's contemporary Georges Liouville proved another important theorem that not every three-dimensional body can be conformally transformed into a ball!
Thus, in three-dimensional space, the possibilities of conformal transformations are far from being as wide as in the plane. Adding only one coordinate axis imposes rather strict additional restrictions on the geometric properties of space.
Isn't that why our real space is precisely three-dimensional, and not two-dimensional or, for example, five-dimensional? Perhaps the whole point is that the two-dimensional space is too free, and the geometry of the five-dimensional world, on the contrary, is too rigidly "fixed"?
And really - why? Why is the space we live in three-dimensional, and not four-dimensional or five-dimensional?
Some of the scholars have tried to answer this question based on fairly general philosophical considerations. The world must be perfect, argued, for example, Aristotle, and only three dimensions are able to provide this perfection.
The next step was for Galileo, who noted the fact that in our world there can be only three mutually perpendicular directions. But Galileo was not engaged in clarifying the reasons for this state of affairs.
Leibniz tried to do this, however, with the help of purely geometric proofs. But these proofs were constructed speculatively, out of connection with the really existing world and its properties.
Meanwhile, this or that number of dimensions is a physical property of real space, and it must be a consequence of quite definite physical reasons: some deep physical laws.
The answer to this question was obtained only in the second half of the 20th century, when the so-called anthropic principle was formulated, which reflected the deepest connection between the very existence of man and the fundamental properties of the Universe.
And finally, one more question. The theory of relativity speaks of the four-dimensional space of the universe. But this is not exactly the four-dimensional space mentioned above: the fourth dimension in it is time. As you know, the theory of relativity has established a close connection between space and matter. But not only. It turned out that matter and time are also directly related! And, as a result, space and time!
Bearing in mind this dependence, the famous mathematician G. Minkowski, whose works formed the basis of the theory of relativity, asserted: "From now on, space itself and time in itself should become shadows, and only a special kind of their combination will retain independence." It was Minkowski who suggested using a conditional geometric model - the four-dimensional "space-time" for the mathematical expression of the interdependence of space and time. In this conditional space, along the three main axes, as usual, length intervals are plotted, while along the fourth axis, time intervals.
Thus, the four-dimensional "space-time" of the theory of relativity is just a mathematical device, an auxiliary mathematical construction that makes it possible to describe various physical processes in a convenient form. Therefore, to assert that we live in four-dimensional space is possible only in the sense that all events occurring in the world take place not only in space, but also in time.
Of course, any mathematical constructions, even the most abstract ones, reflect some aspects of reality, some relations between really existing objects and phenomena. But it would be a gross mistake to equate the auxiliary mathematical apparatus, as well as the specific conventional terminology used in mathematics and objective reality.
In this regard, it is worth mentioning that in mathematical physics a technique is often used, which is called the construction of "phase spaces". We are talking about conditional physical and mathematical constructions, in which certain physical parameters, for example, mass, momentum, energy, speed of movement, angular momentum, etc., are considered as quantities deposited along purely conditional "coordinate axes". In such "phase spaces" the behavior of a physical object or system looks like its movement along a certain conditional "trajectory". And although this technique is purely arbitrary, it allows - which is quite convenient - to get a visual representation of the state and behavior of the object under study.
In the light of these considerations, it becomes clear that to assert, while referring to the theory of relativity, that our world is actually four-dimensional is approximately the same as defending the idea that dark spots on the Moon or Mars are filled with water, on the grounds that astronomers call them seas.
V. Komarov