What the fourth dimensional dimension would look like.

The longest history of scientific discussion of all types of parallel universes boasts a parallel universe of higher dimensions. Common sense and our senses tell us that we live in three dimensions - length, width and height. No matter how we move an object in space, its position can always be described by these three coordinates. In general, with these three numbers, a person can determine the exact position of any object in the Universe, from the tip of his nose to the most distant galaxies.

At first glance, the fourth spatial dimension is contrary to common sense. For example, when smoke fills an entire room, we don't see it disappear into another dimension. Nowhere in our Universe do we see objects that would suddenly disappear or float away into another universe. This means that the higher dimensions, if any, must be smaller than an atom.

The three dimensions of space form the foundation, the foundation of Greek geometry. For example, Aristotle wrote in his treatise On Heaven:

"A quantity divisible in one dimension is a line, in two it is a plane, in three is a body, and apart from them there is no other quantity, since three measurements it's all measurements".

In 150 A.D. e. Ptolemy of Alexandria offered the first "proof" that higher dimensions were "impossible." In the treatise "On Distance" he argues as follows. Let's draw three mutually perpendicular straight lines (like the lines that form the corner of the room). Obviously, it is impossible to draw a fourth line perpendicular to the first three, therefore, the fourth dimension is impossible.

In fact, he was able to prove only one thing in this way: our brain is not able to visualize the fourth dimension. On the other hand, computers are constantly doing calculations in hyperspace.

For two millennia, any mathematician who dared to speak of the fourth dimension risked being ridiculed. In 1685, the mathematician John Wallis, in a controversy about the fourth dimension, called him "a monster in nature, no more possible than a chimera or a centaur." In the 19th century, the "king of mathematicians" Karl Gauss developed the mathematics of the fourth dimension to a large extent, but he was afraid to publish the results, fearing a negative reaction. He himself, however, conducted experiments and tried to determine whether purely three-dimensional Greek geometry really describes the universe correctly. In one experiment, he placed three assistants on tops of three neighboring hills. Each assistant had a lantern; the light of all the three lanterns formed a giant triangle in space. Gauss himself carefully measured all the angles of this triangle and, to his own disappointment, found that the sum inner corners the triangle is actually 180 °. From this, the scientist concluded that if deviations from standard Greek geometry exist, then they are so small that they cannot be detected in such ways.

As a result, the honor to describe and publish the foundations of higher-dimensional mathematics fell to Georg Bernhard Riemann, a student of Gauss. (Within a few decades, this mathematics became fully incorporated into Einstein's theory of general relativity.) In his famous lecture in 1854, Riemann overturned 2,000 years of Greek geometry in one fell swoop and established the foundations of mathematics for higher, curvilinear dimensions; we still use this mathematics today.

At the end of the XIX century. Riemann's remarkable discovery thundered throughout Europe and aroused the widest interest of the public; the fourth dimension has created a real sensation among artists, musicians, writers, philosophers and painters. For example, art historian Linda Dahlrymple Henderson believes that Picasso's cubism arose in part under the influence of fourth dimension... (Picasso's portraits of women with the eyes looking forward and the nose to the side represent an attempt to present a four-dimensional perspective, because when viewed from the fourth dimension, one can simultaneously see the face, nose and the back of the head of a woman.) Henderson writes: “Like a black hole, the fourth the dimension possessed mysterious properties that even the scientists themselves could not fully understand. Yet the fourth dimension was much more comprehensible and comprehensible than black holes or any other scientific hypotheses after 1919, with the exception of the theory of relativity. "

But historically, physicists have viewed the fourth dimension as just a funny curiosity. There was no evidence of higher dimensions. This began to change in 1919, when physicist Theodor Kaluza wrote a highly controversial article in which he hinted at the existence of higher dimensions. Starting with Einstein's general theory of relativity, he placed it in five-dimensional space (four spatial dimensions and the fifth - time; since time has already established itself as the fourth dimension of space-time, physicists now call the fourth spatial dimension the fifth). If you make the universe smaller and smaller along the fifth dimension, the equations magically split into two parts. One part describes Einstein's standard theory of relativity, but the other turns into Maxwell's theory of light!

This was a startling revelation. Perhaps the secret of light is hidden in the fifth dimension! This decision shocked even Einstein; it seemed to provide an elegant amalgamation of light and gravity. (Einstein was so shocked by Kaluza's suggestion that he hesitated for two years before agreeing to publish his article.) Einstein wrote to Kaluza: “The idea of ​​obtaining [the unified theory] by means of a five-dimensional cylinder would never have occurred to me ... At first glance I liked your idea very much ... The formal unity of your theory is striking. "

For many years, physicists have wondered: if light is a wave, then what, in fact, vibrates? Light can travel billions of light years of empty space, but empty space is a vacuum, there is no substance in it. So what vibrates in a vacuum? Kaluza's theory made it possible to put forward a specific assumption about this: light are real waves in the fifth dimension. Maxwell's equations, which accurately describe all the properties of light, are obtained in it simply as the equations of waves that move in the fifth dimension.

Imagine fish swimming in a shallow pond. Perhaps they are not even aware of the existence of the third dimension, because their eyes look to the sides, and they can only swim forward or backward, right or left. Perhaps the third dimension even seems impossible to them. But now imagine rain on the surface of a pond. Fish cannot see the third dimension, but they see shadows and ripples on the surface of the pond. Likewise, Kaluza's theory explains light as ripples that move in the fifth dimension.

Kaluza also answered the question of where the fifth dimension is. Since we do not see any signs of its existence around it, it must be "rolled up" to such a small size that it is impossible to notice it. (Take a two-dimensional sheet of paper and roll it tightly into a cylinder. From a distance, the cylinder will appear to be a one-dimensional line. So you have folded a two-dimensional object and made it one-dimensional.)

For several decades, Einstein began to work on this theory from time to time. But after his death in 1955, the theory was quickly forgotten, it turned into a funny note on the pages of the history of physics.

Fragment from the book of Peter D. Uspensky " New model universe ":

The idea of ​​the existence of hidden knowledge, superior to the knowledge that a person can achieve by his own efforts, grows and strengthens in the minds of people when they understand the insolubility of many questions and problems facing them.

A person can deceive himself, he can think that his knowledge is growing and increasing, that he knows and understands more than he knew and understood before; however, sometimes he becomes sincere with himself and sees that in relation to the basic problems of existence, he is as helpless as a savage or a child, although he has invented many clever machines and tools that have made his life difficult, but did not make it clearer.
Speaking with himself even more frankly, a person may admit that all his scientific and philosophical systems and theories are similar to these machines and tools, because they only complicate problems without explaining anything.

Among the unsolvable problems surrounding a person, two occupy a special position - the problem of the invisible world and the problem of death.

Without exception, all religious systems, from such theologically developed to the smallest details like Christianity, Buddhism, Judaism, to completely degenerated religions of "savages" that seem to modern knowledge to be "primitive" - ​​they all invariably divide the world into visible and invisible. In Christianity: God, angels, devils, demons, souls of the living and the dead, heaven and hell. In paganism: deities personifying the forces of nature - thunder, sun, fire, spirits of mountains, forests, lakes, spirits of waters, spirits of houses - all this belongs to the invisible world.
Philosophy recognizes the world of phenomena and the world of causes, the world of things and the world of ideas, the world of phenomena and the world of noumena. In Indian philosophy (especially in some of its schools) the visible, or phenomenal, world, Maya, an illusion, which means a false notion of not visible world is generally considered non-existent.

In science, the invisible world is a world of very small quantities, as well as, oddly enough, very large quantities. The visibility of the world is determined by its scale. The invisible world is, on the one hand, the world of microorganisms, cells, the microscopic and ultramicroscopic world; then it is followed by the world of molecules, atoms, electrons, "vibrations"; on the other hand, it is the world of invisible stars, distant solar systems, unknown universes.

The microscope expands the boundaries of our vision in one direction, the telescope in the other, but both are very insignificant compared to what remains invisible.

Physics and chemistry give us the opportunity to explore phenomena in such small particles and in such distant worlds that will never be accessible to our eyes. But this only strengthens the idea of ​​the existence of a huge invisible world around a small visible one.
Mathematics goes even further. As already indicated, she calculates such ratios between quantities and such ratios between these ratios that have no analogies in the visible world around us. And we are forced to admit that the invisible world differs from the visible not only in size, but also in some other qualities that we are unable to define or understand and which show us that the laws found in physical world, cannot relate to the invisible world.
Thus, the invisible worlds of religious, philosophical and scientific systems are in the end more closely related to each other than it seems at first glance. And such invisible worlds of various categories have the same properties common to all. These properties are as follows. First, they are incomprehensible to us, i.e. incomprehensible from an ordinary point of view or from ordinary means of knowledge; secondly, they contain the causes of the phenomena of the visible world.

The idea of ​​causes is always associated with the invisible world. In the invisible world of religious systems, invisible forces govern people and visible phenomena. In the invisible world of science, the causes of visible phenomena stem from the invisible world of small quantities and "vibrations".
In philosophical systems, the phenomenon is only our concept of the noumenon, i.e. an illusion, the true cause of which remains hidden and inaccessible to us.

Thus, at all levels of his development, a person understood that the causes of visible and observable phenomena are beyond the scope of his observation. He found that among observable phenomena, some facts can be regarded as causes of other facts; but these conclusions were insufficient to understand everything that happens to him and around him. To explain the reasons, an invisible world is needed, consisting of "spirits", "ideas" or "vibrations".

Reasoning by analogy with existing dimensions, it should be assumed that if the fourth dimension existed, it would mean that here, next to us, there is some other space, which we do not know, do not see and cannot go into. In this "area of ​​the fourth dimension" from any point of our space it would be possible to draw a line in a direction unknown to us, which we can neither define nor comprehend. If we could imagine the direction of this line coming from our space, then we would see the "region of the fourth dimension".

Geometrically, this means the following. You can imagine three mutually perpendicular lines to each other. With these three lines we measure our space, which is therefore called three-dimensional. If there is a "region of the fourth dimension" lying outside our space, then, in addition to the three perpendiculars known to us, which determine the length, width and height of objects, there must be a fourth perpendicular, which determines some kind of incomprehensible to us, a new extension. The space measured by these four perpendiculars will be four-dimensional.

It is impossible to define geometrically or to imagine this fourth perpendicular, and the fourth dimension remains extremely mysterious for us. There is an opinion that mathematicians know something about the fourth dimension that is inaccessible to mere mortals. It is sometimes said, and this can be found even in the press, that Lobachevsky "discovered" the fourth dimension. In the past twenty years, the discovery of the "fourth" dimension has often been attributed to Einstein or Minkowski.

In fact, mathematics has very little to say about the fourth dimension. There is nothing in the hypothesis of the fourth dimension that would make it unacceptable from a mathematical point of view. It does not contradict any of the accepted axioms and therefore does not encounter special opposition from mathematics. Mathematics quite allows the possibility of establishing the relationship that should exist between four-dimensional and three-dimensional space, i.e. some properties of the fourth dimension. But she does all this in the most general and indefinite form. Precise definition there is no fourth dimension in mathematics.

The fourth dimension can be considered proven geometrically only when the direction of an unknown line going from any point of our space to the region of the fourth dimension is determined, i.e. found a way to construct the fourth perpendicular.

It is difficult to even roughly outline what significance the opening of the fourth perpendicular in the universe would have for our entire life. The conquest of air, the ability to see and hear at a distance, the establishment of communications with other planets and star systems - all this would be nothing compared to the discovery of a new dimension. But this is not the case yet. We must admit that we are powerless in the face of the riddle of the fourth dimension - and try to consider the issue within the limits that are available to us.

With a closer and more precise study of the problem, we come to the conclusion that for existing conditions it is impossible to solve it. At first glance, purely geometrical, the problem of the fourth dimension cannot be solved geometrically. Our geometry of three dimensions is not enough to study the issue of the fourth dimension, just as planimetry alone is not enough to study the issues of stereometry. We must discover the fourth dimension, if it exists, purely empirically, - and also find a way to represent it in three-dimensional space. Only then can we create the geometry of the four dimensions.

The most superficial acquaintance with the problem of the fourth dimension shows that it must be studied from the side of psychology and physics.

The fourth dimension is incomprehensible. If it exists and if, nevertheless, we are not able to cognize it, then, obviously, something is missing in our psyche, in our perceiving apparatus, in other words, the phenomena of the fourth dimension are not reflected in our sense organs. We must figure out why this is so, what defects cause our immunity, and find conditions (at least theoretical) under which the fourth dimension becomes understandable and accessible. All of these questions are related to psychology, or perhaps to the theory of knowledge.

We know that the area of ​​the fourth dimension (again, if it exists) is not only unknowable for our mental apparatus, but is inaccessible purely physically. It no longer depends on our defects, but on special properties and conditions of the region of the fourth dimension. We need to figure out what kind of conditions make the area of ​​the fourth dimension inaccessible to us, find the relationship between the physical conditions of the area of ​​the fourth dimension of our world and, having established this, see if there is anything similar to these conditions in the world around us, whether there are relationships similar to relationships between 3D and 4D regions.

Generally speaking, before constructing the geometry of four dimensions, you need to create a physics of four dimensions, i.e. find and define the physical laws and conditions that exist in the space of four dimensions.

"We cannot solve problems using the same thinking approaches that we used to create problems." (Albert Einstein)

via quantum-tech. ru and blogs.mail.ru/ chudatrella.

I will describe it in mathematical language.

Consider the usual three-dimensional space in which we live. We perfectly understand what a point, a line and a plane are in this space. The intersection of two planes gives us a straight line, the intersection of two straight lines gives us a point. Each point in this space can be described by three coordinates: (x, y, z). The first coordinate usually means the length, the second is width, third - the height a given point relative to the origin. All this can be easily illustrated and imagined.

However, four-dimensional space is not that simple. Any point in this space can now be described by four coordinates: (x, y, z, t), where a new coordinate t is added, which in physics is often called time... This means that in addition to the length, width and height of a point, its position in time is also indicated, that is, where it is located: in the past, in the present or in the future.

But let's move away from physics. It turns out that mathematically, a new axiomatic object is added in this space, called hyperplane... It can be conditionally represented as one whole "three-dimensional space". By analogy in three-dimensional space, the intersection of two hyperplanes gives us the plane... Various combinations of this thing with 4D shapes give us unexpected results. For example, in 3D space, the intersection of a plane with a ball gives us a circle. By this analogy, in four-dimensional space the intersection of a four-dimensional ball with a hyperplane gives us a three-dimensional ball. It becomes obvious that it is almost impossible to mentally imagine and draw a four-dimensional space: biologically, our senses are adapted only to a three-dimensional case and below. Therefore, the four-dimensional space can be clearly described only in mathematical language, mainly with the help of actions with the coordinates of points.

However, it can be described less accurately in other language as well. Consider the concept of parallel worlds: in addition to our world, there are other worlds, in which some events went differently. Let's designate our world through the letter A, and some other world through the letter B. From the point of view four-dimensional space we can say that world A and world B are different "three-dimensional spaces" that do not intersect. That's what it is parallel hyperplanes... And there are infinitely many of them. If it happens that if at a certain moment in time in world A "grandfather died", and in world B "grandfather is still alive", then worlds A and B intersect in some four-dimensional figure, in which all events went the same way up to a certain moment in time , and then the figure seemed to be "divided" into non-intersecting three-dimensional parts, each of which describes the state of the grandfather, whether he is alive or not. It could be described in two-dimensional format: there was one straight line, which then split into two non-intersecting lines.

»We will touch upon the well-known problem of the number of dimensions in general and the transition to them in particular. We will try to consider this issue not from a traditionally mystical point of view, but from a practical point of view (with the help of practical exercises and training videos).

The transition to the fourth dimension has been of interest to people for a very, very long time. However, there are still two groups of views that relate differently to the fourth dimension. One of the groups is the spatial fourth dimension, and the second is the temporal O e fourth dimension.

The spatial fourth dimension is very well illustrated in one of the issues of the Tram magazine, where an article was published about a four-dimensional mouse (if anything, it is called "Mouse WHAT-YOU-ROOH-MER-NAYA" and you can read it here http: //tramwaj.narod .ru / Archive / LJ_archive_2.htm). There was such an analogy: for the inhabitants of one dimension (line), any two-dimensional creatures will be perceived only as components of one dimension. Anything that goes beyond this dimension will not be noticed (because there is nothing to look with).

In the same way, inhabitants of two-dimensional space (planes) can see inhabitants of three-dimensional space only as their two-dimensional imprints-projections. They simply have nothing to see the third dimension with. That is, if a person got into this two-dimensional space, then in best case the local inhabitants of the plane got acquainted with the imprints of its soles. And at worst - a cross section 🙂

Likewise, the inhabitants of the third dimension (that is, you and I) can see four-dimensional beings only as their three-dimensional projections. That is, ordinary bodies having length, width and height.

The higher dimension has one important advantage over the lower dimension: beings from more high dimensions can violate the laws of physics of lower dimensions. So, if in a two-dimensional universe, on a plane, you put a resident in a prison, then he will not be able to get out of it, surrounded on all two sides (since there are only two dimensions) by walls. But if you put a three-dimensional creature in such a prison (or rather, only its projection), then it easily leaves two dimensions, say, upward - and finds itself outside the two-dimensional prison.

Exactly the same buns are available to four-dimensional creatures in our three-dimensional universe. Agree, all this sounds very tempting, mystical, and when mastering the fourth dimension it promises to bring a lot of bonuses such as peeping in women's locker rooms 🙂 Perhaps that is why there is high ethics among the requirements for those passing into this dimension.

But let's not delve into the mystical jungle - after all, we promised practice, not mysticism. For this we generalize. So, one normal measurement perpendicular to the other and the third, forming the familiar coordinate axes:

Whereas, according to this logic, the fourth spatial dimension should be perpendicular to these three.

The transition to the fourth spatial dimension is carried out with the help of the development of a special organ of perception of this dimension. This organ is usually called the "third eye". Since this phrase does not mean anything, we will not use it. Moreover, the fourth spatial dimension is perceived by no means through the eyes. As an advice on the development of the organ of perception of the fourth spatial dimension, we present an exercise from the book by P.D. Uspensky (student of Gurdjieff, if anything) "TERTIUM ORGANUM" (third organ, if translated):

Train to see (for a start - in the imagination) volumetric figures (cubes, pyramids, spheres, etc.) from all sides at once.

Here is a simple description for a difficult exercise. Hopefully everything is clear: we can usually see a maximum of 3 sides of a cube. And we must imagine the cube as if we saw it from all six sides at once. Puzzle, huh? 🙂

In order to get more mass about the fourth spatial dimension, you can use these videos:

The first part of the video about the fourth dimension:

The second part of the video about the fourth dimension

Having considered the practical training for the transition to the spatial fourth dimension, there is one more point to consider. Oddly enough, the fourth (as well as the fifth, sixth ... eleventh) spatial dimensions is by no means an empty phrase. At least in light of the latest advances in superstring theory.

So, in order for the laws of physics to work equally at the micro and macro levels (from a level thousands of times smaller than the size of a molecule to intergalactic distances), the formulas need eleven spatial dimensions. Three of these dimensions are unfolded, and the rest are curled up, and that is why we do not perceive them. Although the vibrations of the constituent subatomic particles are very much dependent on these rolled-up dimensions.

Unfortunately, the ancient magicians did not even suspect about these rolled-up dimensions, so the transition to these rolled-up dimensions remains completely occult, that is, secret. For if anyone had figured out how to do it, he did not say how.

Now is the time to move on to the fourth dimension in terms of time. This approach is widely developed by physicists, so there is not much to tell here. The only apparent difference is the timing O th dimension is that it is impossible to move backward along it, as in three spatial dimensions. Just forward. However, this is not entirely true - and it is this nuance that gives the key to the transition to the fourth time O e measurement.

Moreover, if in order to perceive the fourth spatial dimension, you need to train a special organ to work with the fourth temporal dimension. s m measuring organ is already there. Moreover, with the help of this organ, people can move along this dimension both back to the past and forward to the future.

Have you already guessed what kind of thing it is that allows you to travel in time?

Quite right, this is the human mind.

Therefore, the transition to the fourth time O e dimension is just a figurative expression. We are all already in this fourth time O m measurement. However, not everything is the same. There are people who remember only yesterday and do not look beyond tomorrow. Their fourth dimension is scanty, and life is hard (although from the outside it may seem fun and carefree).

And, on the contrary, there are people who are able to look far, far into the past, compare the data obtained with observations from the present and make practical conclusions both about the near and the distant future. As you can see, these people have mastered the fourth dimension to a very large extent. As a result, the lives of such people are much more stable, calm and happy.

Therefore, the question is not about the transition to time O not the fourth dimension, but in the deepening of this dimension. Well, for this you need to train your mind. How to do it? It's very simple. The main thing is to work out the main activity of the mind: to compare data from the past with data from the present and draw the right conclusions. Well, there are just a huge number of methods.

Another nuance is the data that the mind uses to work. After all, if the data received for processing is erroneous (from the past or from the present), then the conclusions will be erroneous. And then not the fourth dimension will turn out, but some kind of garbage.

Why are the received data from the past and the present erroneous? It's very simple: because it is misjudged data due to painful experiences. Example: a person has been bitten by a dog, and now whenever he sees dogs, he receives data not about their real intentions or their appearance, but a glitch from the past associated with pain. Therefore, conclusions for the future (for example, "all dogs are dangerous") will be false. And the fourth dimension is with a wormhole.

How can you avoid such mistakes? Naturally, by correctly assessing the data obtained in the presence of pain, collision or loss. How to do it? There are far fewer of these ways than ways to improve thinking. But they are, and you can find them if you want 🙂

Thus, the transition to the fourth dimension depends on where you want to go.

Happy transitions!

If anything - write in the comments!

• Translation

Surely you know that the planets move around the sun in elliptical orbits. But why? In fact, they move in circles in four-dimensional space. And if you project these circles into three-dimensional space, they turn into ellipses.

In the figure, the plane represents 2 of the 3 dimensions of our space. The vertical direction is the fourth dimension. The planet moves in a circle in four-dimensional space, and its "shadow" in three-dimensional space moves along an ellipse.

What is this 4th dimension? It looks like time, but it is not really time. This is such a special time that flows at a speed inversely proportional to the distance between the planet and the sun. And relative to this time, the planet moves at a constant speed in a circle in 4 dimensions. And in normal time, its shadow in three dimensions moves faster when it is closer to the sun.

Sounds strange - but it's simple unusual way representations of ordinary Newtonian physics. This method has been known since at least 1980 thanks to the work of the mathematical physicist Jurgen Moser. And I found out about this by emailing a paper by Jesper Goranson entitled "Symmetries in Kepler's Problem" (March 8, 2015).

The most interesting thing about this work is that this approach explains one interesting fact... If we take any elliptical orbit and rotate it in 4-dimensional space, then we get another valid orbit.

Of course, it is possible to rotate an elliptical orbit around the sun and in ordinary space, getting a valid orbit. The interesting thing is that this can be done in 4-dimensional space, for example, by narrowing or widening the ellipse.

In general, any elliptical orbit can be turned into any other. All orbits with the same energy are circular orbits on the same sphere in 4-dimensional space.

Kepler's problem

Let's say we have a particle that moves according to the inverse square law. The equation of its motion will be

Where r- position as a function of time, r is the distance from the center, m is the mass, and k determines the force. From this we can derive the law of conservation of energy

For some constant E, depending on the orbit, but not changing over time. If this force is an attraction, then k> 0, and on an elliptical orbit E< 0. Будем звать частицу планетой. Планета двигается вокруг солнца, которое настолько тяжело, что его колебаниями можно пренебречь.

We will investigate orbits with the same energy E. Therefore, the units of mass, length and time can be taken as arbitrary. We put

M = 1, k = 1, E = -1/2

This will save us from unnecessary letters. Now the equation of motion looks like

And the conservation law says

Now, following Moser's idea, let's move from ordinary time to new. Let's call it s and require that

This time passes more slowly with distance from the sun. Therefore, the speed of the planet increases with distance from the sun. This compensates for the tendency of the planets to move more slowly as they move away from the sun in normal time.

Now let's rewrite the conservation law using new time. Since I used a dot for the time derivatives, let's use a prime for the time derivatives s. Then for example:

Using such a derivative, Goranson shows that the conservation of energy can be written in the form

And this is nothing more than the equation of the four-dimensional sphere. The proof will come later. Now let's talk about what this means for us. To do this, we need to combine the normal time coordinate t and the spatial coordinates (x, y, z). Dot

Moves in four-dimensional space as the parameter s changes. That is, the speed of this point, namely

Moves along a four-dimensional sphere. This is a sphere of radius 1 centered at the point

Additional calculations show other interesting facts:

T "" "= - (t" - 1)

These are the usual harmonic oscillator equations, but with an additional derivative. The proof will come later, but for now let's think about what that means. In words, it can be described as follows: 4-dimensional speed v performs simple harmonic oscillations around the point (1,0,0,0).

But since v at the same time remains on a sphere centered at this point, then we can conclude that v moves with constant speed in a circle on this sphere. This implies that the average value of the spatial components of the 4-dimensional velocity is 0, and the average t is 1.

The first part is clear: our planet, on average, does not fly away from the Sun, so its average speed is zero. The second part is more complicated: the usual time t moves forward with average speed 1 relative to the new time s, but the rate of its change fluctuates sinusoidally.

By integrating both parts

We'll get

a... The equation says that the position r oscillates harmoniously around the point a... Insofar as a does not change over time, it is a conserved quantity. This is called the Laplace-Runge-Lenz vector.

Often people start with the inverse square law, show that angular momentum and the Laplace-Runge-Lenz vector are conserved, and use these conserved quantities and Noether's theorem to show that there is a 6-dimensional symmetry group. For solutions with negative energy, this translates into a group of turns in 4 dimensions, SO (4). With a little more work, you can see how the Kepler problem is coupled with a harmonic oscillator in 4 dimensions. This is done through reparameterization of time.

I liked Gorasnon's approach better because it starts with reparameterizing time. This allows us to effectively show that the elliptical orbit of a planet is the projection of a circular orbit in four-dimensional space onto three-dimensional space. Thus the 4D rotational symmetry becomes apparent.

Goranson carries over this approach to the inverse square law in n-dimensional space... It turns out that elliptical orbits in n dimensions are projections of circular orbits from n + 1 dimensions.

He also applies this approach to positive-energy orbits, which are hyperbolas, and to zero-energy orbits (parabolas). For hyperbolas, the symmetry of the Lorentz groups is obtained, and for parabolas, the symmetry of the Euclidean groups. This known fact however, it is remarkable how easy it is to deduce with the new approach.

Mathematical details

Because of the abundance of equations, I'll put boxes around the important equations. The basic equations are energy conservation, strength and change in variables, which give:

We start by conserving energy:

Then we use

To obtain

A little algebra - and we get

This shows that 4D speed

Remains on a sphere of unit radius centered at (1,0,0,0).

The next step is to take the equation of motion

And rewrite it using dashes (derivatives with respect to s) rather than dots (derivatives with respect to t). We start with

And differentiate to get

Now we use a different equation for

And we get

Now it would be nice to get a formula for r "" as well. Let's count first

And then we differentiate

Let's connect the formula for r ", something will be reduced, and we get

Recall that the conservation law says

And we know that t "= r. Therefore,

We get

Since t "= r, it turns out

As we need.

Now we get a similar formula for r "" "... Let's start with

And we will differentiate

Let's connect the formulas for r "" and r """. Something is reduced, and remains

We integrate both sides and obtain

For some constant vector a... It means that r oscillates harmoniously with respect to a... Interestingly, the vector r and its norm r oscillate harmoniously.

The quantum version of a planetary orbit is a hydrogen atom. Everything that we calculated can be used in the quantum version as well. See Greg Egan for details.

If we compare a flat sheet of paper and a box, we will see that a sheet of paper has a length and a width, but has no depth. The box has length, width and depth.

The world we are accustomed to consists of three dimensions, but let's imagine existence in two-dimensional space. In this case, everything will look like drawings on a sheet of paper. Objects will be able to move in any direction on the surface of this paper, but it will be impossible to rise or fall on the surface of this very paper.

Imagine a square drawn in two-dimensional space - no object can get out of the square, unless there is a hole or hole in it. Moving under and over the square will be impossible.

What is the fourth dimension

It's another matter in the three-dimensional world - having drawn a square around any object, it costs nothing for that very object to step over it or crawl over it. Now imagine that the object is placed inside a cube, or, for example, in a room with a ceiling, floor and four solid walls. No object will be able to get out of the room, provided that there are no holes in it.

Of course, all this is clear and understandable enough. It is also clear that almost all phenomena can be explained from the perspective of the three-dimensional world. For example, it is simple and clear why liquid can be placed in a jug or why a dog can live in a kennel.

It is now worth considering the paranormal phenomena - materialization and dematerialization. A famous psychic, Charles Bailey could materialize hundreds of objects in an iron cage in the presence of numerous, skeptical witnesses. It is quite possible that objects passed between the bars of an iron cage, and this is absolutely inexplicable from the point of view of the three-dimensional world.

To explain such phenomena, it was hypothesized that there is a fourth dimension of space, which is inaccessible under normal circumstances. However, from time to time, objects get the opportunity to enter and leave the fourth dimension.

Transcendental physics

Exists special work entitled "Transcendental Physics", dedicated to the study of the concept of the fourth dimension and written by Johann Karl Friedrich Zellner. In his work, the author took as an example the phenomena created by the psychic Henry Slade. Tom managed to make some object disappear completely, and then make this very object appear somewhere else. In addition, he could materialize two solid rings around the table leg.

After some time, Slade was jailed for fraud, and this caused irreparable damage to Dr. Zellner's reputation. However, this seems irrelevant today, as Zellner was able to offer the world a carefully formulated theory. In addition, Slade's fraud remains in question.

Excerpt from "Transcendental Physics":

“Among the evidence, there is nothing more convincing and significant than the transfer of material bodies from enclosed space... Although our third-dimensional intuition cannot allow an intangible exit to open in a confined space, four-dimensional space provides such an opportunity. Thus, the transfer of the body in this direction can be carried out without affecting the three-dimensional material walls. Since we, three-dimensional beings, lack the so-called intuition of four-dimensional space, we can only form its concept by analogy from the lower region of space. Imagine a two-dimensional shape on a surface: a line is drawn on each side, and an object is placed inside. By moving only on the surface, the object will not be able to get out of this two-dimensional enclosed space, unless there is a break in the line. "