Representation of space and time

The article continues a series of articles about our view of space and time .

It seems logical that we first considered the space, and then its dance in time, but the question is: why? Why can not we first consider the dance, and then consider the space? There are several reasons:

1. When modeling changes, we quickly roll into the area of ​​mythology, in which the printer prints pages, and the machine has a way to go. The world in which inanimate objects are responsible for changes is inherited by us from the distant past, when all objects were considered animate. The legacy of the mythological consciousness, which is sitting in our subconscious and confirmed by language, is almost irresistible. But in order to model the dance, we need to abandon ideas about the activities and the actors in particular.
2. We are engaged in self-deception when we say that we are looking at space beyond time. Of course, we look at space in time. Simply, we consider it so slowly changing that it allows us to walk in it in our imagination as if the space is frozen for a while.

Both space and its dance are different points of view on the same part of the Absolute Space-Time. But according to the rules of language, dance should be tied to space, but space should not. We cannot tell about dance without space, but about space without dance — supposedly we can. This is bad because it undermines symmetry. We need to constantly remind ourselves that the story about space always presupposes the presence of a dance, be it dynamic, or it is a frozen na. We must remember that there is no space without dance and no dance without space.

In this article we will learn to look at the dance of space in general.

General remarks

In any space selected for study there is a minimum spatial resolution (atomic point of space) and a maximum spatial volume (borders of the studied space). In any time selected for the study, there is a minimum time resolution (atomic instant) and a maximum time interval (the boundary of the time studied).

Is it possible to know the absolute space-time?

Suppose you see a surface, for example, from a height of 300 km you see the surface of the Earth in green. Going down below, you see that the surface is black, and green is the foliage on the trees, but it does not form a surface.

Suppose you look at a star and see that it is bright and flickers with blue light. By changing the time scale, you can register photons. You no longer see the light, you see flashes, and at different frequencies: from red to violet. Increasing the time scale yet, you will see darkness. Sometimes, very rarely there will be a flash of light, but there were about five of them during the whole observation period. But then you observe the dance of the imps (usually we do not notice them, because they dance very quickly).

Absolute Space-Time is indescribable. We can describe only the explored space-time. From the Absolute it is distinguished by the presence of a point of view, the minimum and maximum size of the investigated space, the minimum and maximum interval of the time studied.

What are we aware of?

Suppose that you have lost the ability to see part of the space. This can be imagined, because each of us has a blind spot. At first you are aware of it, but then you get used to it and cease to be aware of it. This is because our consciousness is able to smooth the visible image. Our consciousness does not work with a picture, but with a spline - functions that smooth it out. The same thing with time. If you show the 25th frame, you will not notice it. Only that space-time, which we perceived as continuous or homogeneous, is realized by us.

Only those parts of the space-time that are continuous, or, equivalently, homogeneous, can be endowed with meaning.

We work not with pictures or frames, but with smooth functions. Where functions break, we see boundaries in space and events in time. But there should not be many such breaks, otherwise our consciousness will not be able to cope with so much information. The restriction on the number of discontinuities in the space-time we study leads to a limited number of space-time structures that we are able to recognize and give meaning.

Uniformity formulation

How to formulate the presence of homogeneity in space-time? The first thing that comes to mind is an analogue of the definition of continuity from mathematical analysis: continuity is when the attribute values ​​differ slightly at two close points. Everything seems logical and beautiful, but the question arises, what is the point?

For example, you are holding a crystal. What is a point on its surface? You can answer that the dot is an atom. But it all depends on what attribute you are considering. If you are considering an attribute color, the atom does not possess color. Color has a surface of a huge number of atoms. Suppose that for the formation of ideas about color, there must be a million. This means that the point on the surface of the crystal, which is necessary to obtain color, contains a million atoms. This means that points can intersect, because neighboring points can have common atoms.

As Aleksey Borovskikh writes, “Such mathematical systems are known. The point is that the point without dimensions is a mathematical idealization, which is convenient and is used to solve a well-defined class of problems. This idealization is constructed by the canonical passage to the composition: the large object consists from a large number of small parts, and is carried out on the assumption that the internal structure of the constituent parts is insignificant for us. The techniques for solving such problems are called mathematical analysis, which are sometimes accompanied by the adjective “classical.” But there are situations when the internal structure of a part is significant. For example, this occurs in geological models where the macroscopic geological structure is one and microscopic - another (foliation is oriented in one way, and particles - in another), and this is essential (for example, in acoustics problems). It uses models in which the point is "thick." Mathematically, this corresponds to idealization, an alternative to the field of real numbers. Such idealizations can be quite different, they are called "non-Archimedean", and the corresponding means of solving problems are called "non-Archimedean analysis." The most famous are p-adic analysis and non-standard analysis. However, many, if not all, of the results obtained in these terms can be interpreted by the classical means of asymptotic methods, in which the game with different-scale limit transitions is performed. "

We assume that the size of a homogeneous simulated space-time is much larger than the resolution limit of the studied space and lasts much longer than one instant of the studied time.

Method one: first collect space

The concept of simulated space-time

In order to imagine a simulated space-time, which is part of the Absolute Space-Time, for each instant of the simulated time we will put in correspondence our simulated space. A stack of such simulated spaces will give us a simulated space-time. This is a rather primitive idea, but for now let's use it.

Empowering Space

Take a space endowed with meaning. Divide it into parts. Why can this be done? Because the space endowed with meaning is homogeneous and, therefore, divided into parts. Why do we consider part of the space endowed with meaning part of it? Because it is like the other part of it. Any part of the surface of the crystal is similar to another part of it. Whichever part of the space that is endowed with meaning is taken, the properties of this part are similar to the properties of another part of this space. This will be our formal way to define a homogeneous space.

The model of a homogeneous space looks quite impressive: for this you need to consider all possible parts of it, and there may be a lot of them.

The minimum size of a part of a homogeneous space is called the size of a homogeneous space. This size may be greater than the atomic point of the studied space, or it may be smaller. The homogeneity of a homogeneous space is very different from the points to which we are all used to. As I wrote earlier, unlike ordinary points, they intersect.

If the size of the homogeneity of a homogeneous space is less than the size of the atomic point of the studied space, we observe an absolutely smooth space. If the size of the homogeneous space is greater than the size of the atomic point of the investigated space, we observe a rough space.

Let me explain by example. Consider the surface of the carpet. He is fleecy, we see every flee. Any part of the carpet is also fleecy and looks like any other part. We will reduce the size of the parts. At some point in one part of the carpet there will be only one pile. Is it possible to call such a part fleecy? No, because one pile is not hairiness. Therefore, the size of the uniformity for the carpet is greater than the size of the atomic point of the studied space. Take two adjacent points of homogeneity of the carpet. In each let there be 20 lint, but 18 of them are common. This is what I called the intersection of the points of homogeneity. If the carpet consists of 30 fibers, is it possible to associate a homogeneous space model with it? Assume that the point of uniformity should include 20 fibers. This means that any part that includes more than 20 fibers will be considered as a lint. Suppose that you can select 100 different parts on a set of 30 villi. We see that the number of parts is large, each part can be called a fleecy. Consequently, a homogeneous space can be associated with such an object.

So, we give a definition of homogeneous space:

Homogeneous space is a set of parts of a single space, consisting of a large number of elements. For each atomic point of space we define the minimum size among those parts of it that include this point. This size will determine the size of the point of homogeneity for a given atomic point of space. The maximum size among all points of homogeneity will determine the size of the point of homogeneity of the entire homogeneous space as a whole.

Homogeneous space can be given meaning (interpretation). For example, the hatched area in the drawing models a homogeneous space that can be treated as a substance. It is modeled using all the possible parts that can be obtained from this space. There are incredibly many such parts, they intersect, the size of the point of homogeneity is comparable to the size of a group of a billion atoms.

Knowing the interpretation of a homogeneous space, one can introduce the notion of continuity of a property: for closely spaced points of homogeneity of a homogeneous space, the attribute values ​​should also be close.

Giving a sense of space in time

So that we can talk about the homogeneity of space in time, the space for each pair of consecutive atomic moments must be close together and be similar to each other. If spaces from moment to moment rapidly change their size, shape or position, we cannot speak of such a series of spaces as a homogeneous space in time.

The following statement seems obvious: if for each pair of atomic moments we can match two spaces, then we will assume that the space is uniform in time. But in fact, we are now appealing to the atomic moment. In fact, we have defined the absolutely smooth dance of space. But the dance may have rough edges. How to formulate the homogeneity of a rough dance without appealing to the moment?

Uniformity in time

Consider the shine of a star. We remember that it consists of individual photons arriving at a certain interval. If we consider moments that are longer than the interval between photons, we will get intervals during which we observe a uniform dance, if the instant is shorter, we will not see a uniform dance. If homogeneity in space was determined by our understanding of surface quality, then in time our understanding of homogeneity should be based on our understanding of the quality of the events.

Imagine a wave of the sea and try to describe its surface. It is very similar to the surface of a fleecy carpet, but now the villi are still moving. If we describe each wave surface every moment, we get a description of the dancing surface. Such a description makes sense if we need to follow every wave. However, let us assume that we do not need such a detailed description, which is so strongly dependent on time. How to get the right one? For this we need to increase the exposure. Those who shot the water through the gray filter will understand me: the surface will look like fog. And now this fog changes slowly over time. It is impossible to consider hairiness on the scale of the villus, it is impossible to consider the dance of the ocean surface during the dance of one wave.

Now we can formulate homogeneity in time. If we observe some kind of uniform dance of space, we can cut this dance into parts across time, take its temporary parts and see what they look like. This similarity of parts and makes the dance of space homogeneous. The set of temporal parts of a homogeneous dance, similar to each other, is a model of a homogeneous dance.

The star for us is a bright point, it is not uniform in space. But her light is uniform in time. Uniformity in time allows us to assign it some meaning. If we encounter something that does not have homogeneity either in space or in time, we simply will not notice it. However, it must be remembered that homogeneity manifests itself only within certain limits of time and space. Perhaps if we change the time scale, or the scale of space, homogeneity will manifest, and we will be able to see it.

A single view of a homogeneous space-time

We considered space-time as a stack of spaces ordered in time. It's time to abandon this view and learn to see space-time as a single whole block of meaning.

Let's go back to the ocean. We know that space-time, by which we model its surface in time, is uniform in time and space. Let's try to formulate it, without resorting to crutches in the form of frames from the film.

The surface of the ocean is the dance of space in time. We can divide it into parts both along time and across.

The division along gives us the result of observing some part of the ocean. Imagine a helicopter that moves (or rests) above the surface of the ocean. He has a sector of the review, which snatches under him some part of the ocean surface. This is the observation of what may be called part of the surface dance, carved along time. We will launch many helicopters, each of them will move independently of each other. The results of the observations are parts of the surface dance, cut along the time. They can intersect, diverge, they are incredibly many!

The division across time looks like the surface of the entire ocean, observed at some time interval within the simulated time.

Now combine these two divisions. Let's make the surface dance division both along time and across. This means that helicopters now begin to observe when they want, and finish when they want, have the opportunity to rise and fall above the surface, changing the field of view. This division of the surface dance gives us many pieces that still resemble each other. But at the same time, the rule is observed: the minimum time interval for which the homogeneity of space-time is maintained is greater than the atomic instant, and there is no sense in considering intervals less than this in this space-time. The minimum interval is called the time interval of homogeneity of a given space-time. The same applies to the spatial size of homogeneity: it is larger than the size of the atomic point. Therefore, space-time divisions cannot be less than the space-time size of homogeneity.

Homogeneous spacetime

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Uniform space-time is a set of parts of one space-time, consisting of a large number of elements. For each atomic point of space-time, we define the minimum size among those of its parts that include this point. This size will determine the size of the point of homogeneity for a given atomic point of space-time. The maximum size among all points of homogeneity will determine the size of the point of homogeneity of the entire homogeneous space-time as a whole.

We can interpret this homogeneous space-time in different ways. For example, as the surface of the ocean. I note that on this surface it is necessary to learn how to conduct measurements. None of the measurements lasting less than the time of homogeneity on a given surface makes sense. This means that it is impossible to measure the height of the wave, because its time of existence is much less than the time of homogeneity. No measurement of an area smaller than the size of spatial uniformity also makes sense. This means that you cannot measure the wavelength. At the same time, we can measure the thickness of the border and understand that it is non-zero: it is as thick as the height of the waves. And this boundary belongs both to the ocean and the atmosphere! Each thesis strongly beats on intuition, we see the waves! But everything should be formal. If you see waves and want to describe them, use other minimal spatial and temporal homogeneities and you will see waves. But other ideas will disappear, for example, the idea of ​​a stable ocean surface.

Anisotropy of homogeneous space-time

We talked about the homogeneity of space-time. But these homogeneities may not be just cubes in space-time. They may have elongated shapes. For example, velveteen. It has different uniformity in different directions. The same applies to the carpet with a thick pile. These are anisotropic homogeneous spaces. But the space-time anisotropy is much more interesting. It allows us to represent the business function as a set of scenarios of the same type! And this is exactly why I had to say so many words. To understand how a function is decomposed into an operation, we need to understand what the space-time anisotropy of a homogeneous space-time is.

The difference of space from time

The juxtaposition of space and time leads to errors that are very difficult to overcome. For example, we began consideration of our ideas with the concept of space. We somehow forgot that space does not exist outside of time and made it so (as if) there is no (time) in this view. But it is. In your imagination you wander through this space, you see parts of it from different sides. For such a movement and inspection takes time. You use time to inspect the space. This time is the one in which your space exists. Without it, you would not be able to see anything. And what you are watching is the dance of space in time, only the figure you are watching is frozen. In space-time, this figure corresponds to a cut along time. You put the helicopter over the object and what it received is a long clipping from space-time, made along the time. This creates the illusion that space can exist outside of time. Cutting the space-time along the time, in which the cut space does not change its position and properties, creates the illusion of space that exists outside of time. You will still say that this is not so, because time flows at one pace, and you study objects at a different pace. This is because our mind can get along with ideas of two different tempos at once. You see this reception in films when it stops at one time, and during the second, the camera bypasses the scene. And the time that you use to bypass the scene, you do not notice, thinking that it is not.

So, the idea of ​​space is impossible without the idea of ​​time. You are not observing space, but space-time. Therefore, when we speak of space, we always mean space-time.

We did not succeed in tearing the idea of ​​space from the idea of ​​time. But what about the time. Is it possible to create an idea of ​​it, ignoring the space? If I ask you what time is, you will say that there is a metronome that defines intervals, and time is the axis on which these intervals are plotted. But how can there be a metronome without space? Our consciousness plays hide and seek with us. There is no time without space, it is impossible to say what time is without appeal to the space-time structures! Therefore, I propose not to try to explain what space is without time, or what time is without space, but to accept as a fact that they are indivisible in order to spend all efforts to understand and imagine this.

The trick to which the mind went when trying to imagine a space outside of time, it was virtual time, which allowed to stop the moment and consider it in detail as if there were no changes. If we really would increase the sensitivity of our devices so that we could "stop" the moment, then at this level of detail new events like dance would surely surface in the vacuum. That is why the "stop" of an instant is an exclusively speculative operation, which has nothing to do with reality.

If you make the same mental act about space, what would it look like? An action similar to freezing time, but perfect in relation to space, looks like this: you must select a small area of ​​space and observe it throughout the entire period. We are able to observe homogeneous spaces, but we cannot concentrate on a small area of ​​space without highlighting areas of homogeneity in it. This is the difference - we can observe the local properties of the moment, but we are not able to observe the local properties of space. That is why we can easily imagine the frames of the film, but we cannot make a symmetric operation and watch a movie about the local part of space without selecting homogeneous sub-spaces in it.

To describe the four-dimensional space-time volume, we go on various tricks.

The first trick is to call space-time an object using a noun. This name is nothing more than the interpretation of space-time. We know perfectly well that an object does not stand still; it can jump and dance. Therefore, hearing that someone is talking about an object, we do not imagine it frozen, we imagine a film with his participation.

The second trick is to call a homogeneous space-time (or a composition of such spaces) an action. To do this, use the verb. This is a different interpretation of the same space-time. The same space-time is now called dance. Talking a dancer and talking a dance, we see the same movie.

The problem arises when we say that the dancer is dancing. In such a presentation, one of the interpretations makes a different interpretation of the same space-time, and this is nonsense from the point of view of space-time modeling. If we were able to separate the space-time model and the interpretation of this model from the point of view of the theory of activity, it would be much easier for us to understand how we imagine space-time.

Correlation between spatial and temporal dimensions of homogeneity

The basis of our view of the world is the conviction that the size of the spatial homogeneity of a homogeneous space and the duration of the temporal homogeneity have a correlation. That is, we believe that the larger the size of spatial homogeneity, the greater the duration of the temporal.

This law is the place to be, because the speed of light is limited. To detect homogeneity in space, we need to collect information from a large volume, while observing which different parts of space are observed at different times. This leads to the fact that homogeneity of a small scale, in principle, we are not able to perceive: the light does not bring us information about them. In other words, on homogeneities of a space of length L, it makes no sense to speak of a duration smaller than L / c.

Results

Any space-time, in order to make sense, must be homogeneous either in space or in time, or both there and there. This homogeneity manifests itself on a certain scale of space and time. For it, the size of the uniformity and the duration of the uniformity must be determined. A model of such a space will be many parts of a given space-time. This is a very big model! Usually we reduce it to: any part of the space-time is like any other part.

Reasoning does not depend on what kind of space we are modeling: either the one that we consider from the inside, or the one that we consider from the outside.

What makes the parts look like? Signals from our detectors. This means that the presence or absence of homogeneity depends on the type of detector used.