What is relativity? Can a description of the world not be relative?? How the coordinate system is built? What is the Difference Between Coordinate Systems and Reference Systems, and is it?

Relativity. This word became very famous after the creation of the Special Theory of Relativity., which struck the imagination of contemporaries. And now it is firmly associated with the name of Einstein.. And for this reason alone, the idea of, what does this word mean for many is far from clear. One side, the word is quite ordinary. On the other hand, two theories in physics, associated with revolutionary changes in it, directly called theories of relativity.

In physics, first of all, the concept of relativity is associated with the concept of motion. Then, that movement as such always occurs in relation to something, obvious enough. You can be motionless at the same time (regarding the car, eg) and move (together with this car relatively road). This is the intuitive concept of the relativity of motion.. With the simplest formalization of this concept, physics came to the idea of ​​the addition of the velocities of two relative motions (example – a man walks on a carriage, and the carriage goes on the rails), which is also called Galileo's principle of relativity. In the most precise terms, the principle of relativity says, how to calculate physical quantities in some uniformly and rectilinearly moving frame of reference from their known values ​​in a stationary. It is the method of calculation that distinguishes SRT from its preceding mechanics.. In fact, the role of the SRT is so important precisely because, that for the first time really focused on this issue. And the General Theory of Relativity also included reference frames, moving rapidly relative to each other. At the same time, the calculation method turned out to be unexpectedly different from the usual one exactly from that method, taken to calculate the speeds of movement, which for almost all people seems to be self-evident.

but, relativity, as a concept, much wider than the relativity of motion. Though, of course, includes the relativity of motion too. What kind of relativity, then, do I want to talk about here?? And I want to discuss here relativity is much more general., than that, which is spoken of in general relativity. First, a little about everyday things, then I will try to translate everything said into the formal language of mathematics. In this form, so that it is clear that we are still talking about the same everyday things.

Let's start, eg, from trade. Good invention of money. Equivalent to any product. Going to another place, take some money with you, and you have everything you want there, any product. And we all understand things like:

• The price of the product. May be different for different goods. May be different for the same product, but in different markets (in different places). Expressed by some number. Number maybe too different, depending on currency, from chosen monetary units. Units value of goods. Respectively, next item on our list –
• unit of measurement. May be selected in the same place different – ruble, euros, dollars. In different places, all the more. There is a conversion factor from one unit to another –
• Exchange rate. The ratio of one unit of measure to another. It can be different in different places. Conversion factors between different units do not have to change equally either., neither agreed upon when going from place to place.

What follows from all this? Things very clear to everyone. The price of the product, as net number, makes little sense. it number, labeled, unit name, dimensional. It is relative to the unit of measurement. And the unit itself is not the only one. There are a lot of such units.. You can have one. But for this you need to agree with all market participants.. And such a treaty will not cancel the potential existence of all other possible units.. Let's call the whole set of all kinds of currencies a group of currencies. In mathematics, the choice of currency would be called the choice of a particular coordinate system. And the whole group of possible currencies would be called group permissible coordinates. In this place, in this market, some limited part of currencies may be in circulation – subgroup general group. Between them, certain courses are established here.. The price of the product, as a number, calculated, based on its price in any one, arbitrary currency, multiplying this dimensional number by the course. Price changes as a number, dimension changes (corresponds to the currency, unit of measure). The price of each specific product in a given market is not one number, and the whole set of these numbers. We call this set the vector. But the goods are often piece. One loaf. The price of a roll in different currencies is different, vector. But it's still the price of the same roll. amount bullock invariantly, does not depend on the choice currencies (units). Let's call this number “scalar“. Scalar concept is also relative. Brides – scalar with respect to such units, as currency. And the vector in relation to such units of measurement (unrelated to price), like loaves of bread. You can also cut a loaf in different ways., slice thickness (unit) may be different and may vary along the roll. Respectively, the same roll will produce a different number of slices. And the number of slices will be a vector with respect to the size of the slice, internal unit on a bun. But in relation to the price of the roll, to the market it will be a scalar. Not quite the same as the bun itself. The selection of a slice occurs only inside the roll., this is the parameter on the bun. Scalar parameter for the price market. But you can sell a bun in slices. And each slice will have a price – price brides, divided by the number of slices. And this will be a vector in the price market.

And next to another market. And on it they agreed on everything differently – chose another subgroup of currencies, and they have their own exchange rates, and prices for the same products may differ. Are you a merchant, go to different markets and trade. What do you need for this? Need to, so that at least some of the currencies in different markets are the same. In mathematics, this requirement means, that both places (both markets, both points) covered by one coordinate system. Then you can take them with you. And already in a new place to change into those, which is not on the old one. By doing this, you simply expand the used subgroup of currencies., units of measure. There is also something very important.. Here you drove off a little bit, lo and behold, the same product (the bride), in the same currency (ruble) there is already a little different – more expensive. And in dollars, on the contrary, cheaper. Well, how to get income on it, I will not speak. But what conclusion will you make for yourself – ruble changed (water)? – dollar has changed (got heavier)? – the product has changed (lighter or heavier)? It doesn't matter to you. Easier and more correct to accept, what is the relative cost, and hence unit values, any, may vary by location. Knowing all these dependencies, you will always trade successfully in all markets.. Knowing and accepting the relativity of the price of the goods, as numbers. Relativity in relation to units her (prices) measurements. Relativity in relation to place (point, market). There she is, true, most general relativity.

A specific group of traders can agree, which will only use one group of currencies everywhere. And apply the same courses everywhere. And they will call their set of currencies a group of especially correct ones, the most important (inertial) frames of reference. Within this group, some relativity still remained., special. Is there anything here familiar to you from physics?

Physicists in relation to the quantities they measure (but only such are directly related to the real world) these are the same merchants. Well, may be, more honest, and if they are smart, then not by malicious intent.

Measurement is the cornerstone, the foundation of all physics. And not only physics, as, hope, showed the above example of trading. Any theories, no matter how abstract they are, in the end they talk about the relationship between the measured values. Sometimes, measured in very strange ways, but still measured. That is why any of our descriptions of the world are always relative., depend on those procedures, which are used to measure, to map numbers to objects in the world. So that relativity, which Einstein's two theories talk about, contained in this relativity, quite obvious. Below I will try to show it..

I'll start with “naive” coordinate system constructions in space-time. For simplicity, we restrict ourselves to just one distance, one coordinate to describe space. There is nowhere else to simplify the coordinate for describing time. – she is alone. Consequently, if you want clarity, we are talking, eg, about two point masses, the distance between which changes over time. A very common case, I would say the simplest for physics. We use a sheet of paper to depict this little world.. Yet again, to make the imaging process (motion descriptions) as clear as possible.

I choose one of the points as the reference body. At some point in time, I fix its position and put a point on a sheet of paper – mark start countdown. Note – I am actually building two descriptions at once – in a two-point system, and their image on paper.

I need a watch to measure time. Hours people came up with a great many. Without further ado, take whatever are at hand. But I remember: many hours, good and bad, different. I, of course, sure, what are mine – the best of the best. Go evenly. Second, time unit, reproduced by them always and everywhere the same. I put this clock on (in) reference body. Time has gone! What's on paper? The choice of time at one of the material points allows you to describe it Existence. While out of touch with anything else. On paper I can draw from the origin line, which will depict the existence of the reference body at all times of its existence. Which line? AND any, if only continuous. Can straight. She will be the best for me – more beautiful, at least. But also more convenient. And that's why. How can I mark different points in time, elapsed from the origin, on this line of existence? Right, I'll take a ruler with centimeter divisions (another unit of measurement, already in the image, on a sheet of paper), I will name this centimeter in a second – I will match seconds, measured by hours, centimeters, deposited along the line of existence. And so that the ruler is convenient to move (she's perfect for me, straight), line of existence, temporary coordinate line, I will draw a straight line. I have the right! The choice is mine. But I remember: this line could be any curve. In accordance with a second, I can not put a centimeter, and, let's say, inch (the Englishman will do so). And I could even place dots on the line as I like (why not? eg, logarithmic scale is often used) and mark them – 1 second, 2 seconds, etc.

On paper, I determined the coordinate line of time. AND in the very space-time of two material points, what is she, time coordinate line? To me it is convenient to count, what's straight, with a uniform scale. But this is only mine opinion, no more. But no less. I think so, and point. I have the right. Other observers, which can use completely different units of measurement to describe the same material points, can claim, based on your own measurements, what's mine the line is not straight, and its scale is not uniform in their dimensions. And they'll be just as right, how right i am. Every observer, by definition, considers the units chosen by him to be ideal – equal to ourselves everywhere, where are they used. Existing and immutable. This statement is relative in the most general sense. – it is true for everyone from his point of view and may be wrong from the point of view of anyone else.

so, unit of time (notice, so far only for the reference body!) we introduced. We now introduce a unit for measuring distance, to ascribe the second coordinate of space-time our simple, two-point little world. Everybody knows, what are the rulers for this?. Good ones, hard, straight rulers with divisions. Let one division be called a meter. Or maybe it could be an inch, yardstick, elbow, foot? Maybe, why not. Remember: there are also many units for measuring distance, like clock. Our specific coordinate system is based on one of many possible units.. So, chose a ruler with one. So, what is next?

And then I measure the distance from my reference body at the initial moment of time to the second point of my world. how? Give ruler and packing (move) her along the line, connecting point, until I get to the second point. I place marks on the way – 1 meter, 2 meters, etc. On paper, I draw another coordinate line of distance from the origin., Where I assign one centimeter to each meter in my world (like a second!) on paper. What line should I draw on paper? Any, if only continuous. Because I like it simpler, yes prettier, take a straight line, and not anyhow, and perpendicular coordinate time line. I have the right! But remember: same way, as for the coordinate time line, distance coordinate line could be any curve. And the meter could be matched not to the centimeter, a inch, foot, elbow… And labels 1,2, 3…, can be placed on a line on paper as needed, not evenly.

And what is the coordinate line of distance in space-time of two material points? Same way, as in the case of the coordinate time line, it is convenient to count, what is it straight, uniformly scaled and perpendicular to the time line. And that's it, what is said about the time coordinates, introduced by other observers above, is completely true with regard to distance measurement.

And this is how I do for every moment in time. If distance between my dots does not change, I think, what they rest relative friendand. If changes – then are moving. Coordinate system available. Then the physics of movement begins, his description. truth? With the coordinate system, everything is already absolutely clear to you? And I did not mention many important things.…

Where did the clock come from in the world of two material points?? Where do the rulers come from? Okay, let them be there, everywhere, and not only on these two massive points. And if not? Then another question. Like this – move the ruler along the line connecting the point without changing the moment in time, marked with a clock on the reference body? You can measure distances everywhere instantly? Me not. Even on paper. It could be considered, that at every point in space-time there is a ruler for measuring distance. And this is the best option. But there remains the problem of the identity of the rulers at all points. Another question. And time elsewhere, what clock is measured? The same, as on the reference body? Then they need to be transferred there., So? And again, instantly! And if others, which are again at every point, then are they all identical to each other? There are no clear answers to all these questions yet., not exactly described yet the behavior of the units themselves when moving from point to point, further description of the movement whether, whether other physical phenomena, remains rather arbitrary, contingent upon adopted agreements.

Here are some examples. Prior to the special theory of relativity, it was tacitly believed, what is the clock, which in space-time go everywhere in the same way and show for all observers the same time at once at all points. There are rulers, exactly the same always and everywhere. You can forget about all the others, “bad” measurement procedures, using the wrong watch and ruler. Observers, using them – wrong observers. And physics (relationship between measured values) they have the wrong, artificially complicated. Correct, good physics is described correctly, good observers, which use only good scales (watches and rulers). Coordinate systems, they built, are called reference frames. And not just frames of reference, but special, special reference systems, to distinguish them from others – inertial reference frames. Sets of scales, hours and rulers, allowed to move relative to each other evenly and in a straight line. At the same time, they remain all the same good. Others, not such reference systems named non-inertial. The revolution of special relativity was, what questioned existence of one of the components procedures for constructing this very inertial reference system. It turned out, what the ability to establish absolute simultaneity for different points in space is fictitious. She simply does not exist. The nicest clocks show different times., if they move relative to each other. Facts are a stubborn thing, and that they agree with each other, I had to realize it. So separate space, existing in time, and time, current at every point in space is the same, ceased to exist. They are merged into a single space-time. Here is just an idea of ​​the presence of some special group of the best reference systems., inertial survived. You already understand, what does it actually mean assumption about the existence in all space-time of special clocks and rulers, the truth is now not separate from each other, but together, united the same everywhere metric. Moreover, a very specific, pseudo-Euclidean.

The next step was taken in General Theory of Relativity. Yet Galilee it was known, that in a gravitational field all bodies fall with the same acceleration, regardless of mass. It became clear to Einstein, what does this fact mean the impossibility of finding the difference between the most important inertial frame of reference and the frame of reference accelerated relative to it in the case, if this system falls freely in a gravitational field. It turned out, that such frames of reference are not worse than inertial. Physics, formulated in such coordinate systems, constructed by observers arbitrarily moving relative to each other, no worse, than that, for which only inertial reference frames were suitable. Even better, since it naturally describes the gravitational field. truth, only the wording, the same for all coordinate systems. That is why this theory bears the proud name of the General Theory of Relativity.. Unfortunately, relativity in this theory is still not so general, as supposed. After all, as we saw above, freedom in the choice of coordinate systems is much wider, than a set of arbitrarily moving relative to each other once and for all selected sets of rulers and clocks, related by some metric condition. And the condition is, albeit locally, remained unchanged. General relativity is still present the assumption of the existence of among all proclaimed equal procedures for measuring the most equal, special, for which the metric exists and is the same at all points of space-time. But existence of such a metric – this is a condition for units of measurement, no more. After all physics must still be convinced, that such good sets of watches and rulers actually exist!

And if they don't exist? Then how to be? It's very simple. Actually, for a correct description of the world it is not at all important, are there any special sets of scales with some given, the properties we need. If you are democrats, so consistent democrats. If we just build a description, valid for any selected set of scales, for any market participant, will it be a mistake? Never. Even if among your currencies (units of measure), as it seems today, there is one, well, so stable and reliable (on the dollar market, in physics units, realized, eg, laser standards), what is breathtaking. We are all witnessing a breathtaking drop in the dollar against the euro today. Do you think laser standards are much more stable? May be, may be… And about the effect Doppler heard? Or about gravitational redshift? But if your description of the world is really relative to any reasonable choice of scales, it doesn't matter to you.

How to build such a description? On the one hand, simply. Need to, so that the ratio between numbers, measured in different coordinate systems (different measurement procedures), recorded this way, to be valid in all admissible coordinate systems without exception. Such relations in mathematics and physics are called covariant.. By the way, this is one of the postulates of general relativity. But there are pitfalls here., namely, those questions that I asked above – how to equip spacetime with the necessary sets of scales for measurements everywhere, and how to compare results, obtained at different points

There are two sides to this problem.. One of them, obviously, suitable mathematical formalism. This formalism already exists. This is the mathematics of spaces affine connection. Second side, of course, is the physical interpretation obtained using mathematical methods, relations expressed in the language of mathematics. You can get acquainted with her in the book “Dimension and Properties of Space-Time”

Let's dwell on one more question, given at the very beginning. What is the Difference Between Coordinate Systems and Reference Systems? And is it? I used both names along the way. And it seems to me, which should already be clear to you, that any frame of reference is nothing else, as some coordinate system. In physics, the frame of reference is traditionally associated with physical scales., which allow measurements. But the coordinate system is often attributed to some arbitrariness., isolation from any units of measurement. The coordinates are then simply considered arbitrary, easy-to-use parameters. For, in order to show the groundlessness of such a point of view, I carried out an emphasized parallel construction of two coordinate systems simultaneously in the simplest world of two material points, and on a piece of paper. Hope, it's perfectly clear to you, as in the coordinate system, drawn on a sheet of paper, units of measurement are also necessary components. Their choice, no matter how strange he is, also has a physical meaning and the right to exist. As a matter of fact, i built two different images of the same world, the world of two points. And built in parallel display one image to another. The physical relations in the description of the mutual motion of two material points can be equally correctly judged by the mathematical relations, aligned lines, depicting the existence of these points in any of these images.

In this way, the difference between coordinate systems and reference systems is no more than, than the result of the agreement. With the most general view of the relativity of the description of the world, it disappears.

© Gavryusev V.G.
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