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THEORY OF RELATIVITY, NEW APPROACHES, NEW IDEAS. PRINCIPLE OF RELATIVITY POSTULATION OF RELATIVITY + COVARIANT SYSTEMS CARDINATE

THEORY OF RELATIVITY, NEW APPROACHES, NEW IDEAS

To the 100th anniversary of the theory of relativity

Author of the article: VM Myasnikov

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PRINCIPLE OF RELATIVITY
POSTULATION RELATIVITY + COORDINATE COVARIANT SYSTEMS

On coordinate systems . Next, for simplicity, we call the coordinate system only the Cartesian rectangular coordinate system. And for simplicity , the reference frames are referred to as inertial frames of reference. We believe that the reader is well aware of the "construction" of the Cartesian coordinate system.

The coordinate system is built exclusively in the reference frame, physical or virtual, and, by definition, is stationary relative to this frame of reference. The motion of a coordinate system is the motion of the corresponding reference frame (as a rule, virtual if it is a motion of the coordinate system ). We also assume that the time is determined in some way at each point of the coordinate system. The coordinates of the point + time are called the event in the given coordinate system.

Let us return to the postulate of relativity. Above, I specifically emphasized that the physical equality of frames of reference means that the observer who makes experiments and measurements in a stationary or moving frame of reference is physically located in these frames of reference. While it is a question of the Galilean postulate of relativity with low speeds of reference frames (remember, for example, the Galilean ship sailing in still water, a modern plane on a cruising track or a spacecraft), the situation seems quite understandable and obvious - the observer can physically move from one Frame of reference to another (this need not necessarily be done at full speed), or to place another observer there with the possibility of exchanging information and to make sure that the laws of physics are really the same in these frames of reference.

In the Einstein postulate of relativity, the situation is fundamentally different. (Imagine that the moving frame is connected with an electron flying at a near-light speed!). The observer is physically in the same fixed frame of reference (physical laboratory) and observes a certain physical phenomenon both in his "own" fixed frame of reference and in the moving frame, while in principle he can not physically move into a moving frame of reference and See how the physical laws behave there. But nothing prevents the observer (and you, the reader, too) mentally "go" into the moving frame of reference and "do there" what he can do , and so that there " everything happens the same way " as in a fixed system Reference (see my formulation of the postulate of relativity).

And here the main question arises - what and how is "the same"? I will first formulate the answer to this question, and then comment.

So, the requirement of the postulate of relativity, that in the mobile reference system " everything was the same ", should be understood as the need to determine the coordinate system in the mobile reference system ("mobile" coordinates) so that the formulation of laws (equations) in these coordinates was in the form of The same as in the coordinates of the fixed reference frame ("fixed" coordinates). Such "movable" coordinates are called covariant "fixed" (and vice versa), the noted property of laws and equations - covariance ( Galilean , Lorentz or other depending on the type of transformation), and the very requirement of covariance - the principle of covariance .

The principle of relativity of Einstein (Galileo) is still called the postulate of Einstein's (Galileo) relativity plus the principle of covariance, but now the principle of covariance has a completely different meaning , below I will explain in more detail.

But first a little history. Until the end of the 19th century, the so-called "physics" predominated in physics. A mechanistic picture of the world, the basic idea of ​​which was that all physics "comes down to mechanics", and mechanics is Newton's theory. And when the Galilean covariance of Newton's laws was discovered, this was naturally perceived simply as another descriptive aspect (property) of Galileo's relativity postulate. These two sides (Galileo's relativity postulate plus Galilean covariance ) are called Galileo's relativity principle . Often the Galilean principle of relativity is Galilean covariance, which, in itself, is not true, but you can agree with this name if the postulate of relativity is implied "by default").

At the turn of the 19-20 centuries, the so-called. The electromagnetic picture of the world, Electromagnetic theory is Maxwell's theory, Maxwell's theory is Maxwell's equations. There was a problem: on the one hand, Maxwell's theory perfectly describes all the electrical, magnetic, electromagnetic (including light, radio, etc.) phenomena, on the other hand - is not Galilean covariant. It was necessary to choose - either Maxwell's theory, or covariance. Science has followed the path of generalization of covariance. Efforts of many scientists (I.Fogt, D.Fitzgerald, G.Lorenz, A.Puankare, A.Einstein) were used to derive new transformations, called (from A.Puankare's submission) Lorentz transformations that ensure the Lorentz covariance of Maxwell's equations. Eventually, the electromagnetic picture of the world was abandoned, but the Lorentz covariance remained the most important component of Einstein's relativity principle (the postulate of Einstein's relativity plus the Lorentz covariance), considering that the Lorentz transformations included Galileo's transformations as a limiting case at low speeds. (Further, for brevity, I will only talk about the principle of Einstein's relativity, but all of this is true for Galileo's principle of relativity, with the exception of specific formulas, of course).

Thus, traditional covariance is interpreted as follows: covariant coordinates (including time as one of the coordinates) are determined in a fixed and moving reference frame in some way, and then a relationship is established (coordinates are derived) between the coordinates, providing Lorentz covariance. In this case, no definition of "moving" coordinates is made! The case, as a rule, is limited to adding a stroke to the notation of "moving" coordinates in comparison with "fixed" coordinates. Schematically, the traditional principle of covariance can be represented as follows:

{" Fixed " co-ord . + " Mobile " coordinates } => { Lorentz transformations }

Our new interpretation of the covariance principle is schematically represented as follows:

{" Fixed " co-ord . + Lorentz transformations } => {" mobile " coordinates }

First of all, we note that there are no problems with the definition of "fixed" coordinates (they were not in the traditional approach.) Recall that this is a Cartesian rectangular coordinate system + time). Further, the Lorentz transformations that ensure the covariance of the Maxwell equations can be taken in a ready-made form from mathematics, where they are defined, with all mathematical rigor, as purely mathematical transformations (see, for example, Chapter III, where the Lorentz transformations are defined as spinor hyperbolic rotations , And Appendix A (A-I), where the Lorentz transformations are defined as the orthogonal transformations in Minkowski space obtained by the orthogonalization procedure from any linear transformation, in particular from the Galileo transformation.),

Here, the covariance of Lorentz transformations is considered as a primary property, already known from mathematics and does not require proof here, and which is used here to define covariant "moving" coordinates.

What is the problem of determining the "moving" coordinates? Yes, a stationary observer can not physically move into a mobile reference system, but he can "send there" a virtual observer, who will determine there "mobile" coordinates "as well" as in a fixed reference frame (I think that approximately so, explicitly or Implicitly, reasoning is carried out under the traditional approach, that is why even the notation of coordinates is preserved, only marking their difference from the "fixed" coordinates).

However, if the mobile reference frame is physical (see definition of the physical reference system), then in this system its standards (lengths, times, etc.) are not necessarily the same as in a fixed system, and if so (a This is the case, see below), it is impossible to determine the "same" "moving" coordinates without knowing how the "mobile" standards have changed in comparison with the "fixed" ones. Here it is, the problem!

Let us consider this problem in more detail. Let - a fixed frame in which the Cartesian coordinate system Oxyz is defined and the reference frame Moves with a constant velocity V along the axis Ox of the fixed frame of reference. In the mobile reference frame there is a coordinate system ("mobile"), its definition is our goal. A stationary observer determines the "primed" coordinates , Using the Lorentz transformations known to him,

. (4)

"Shaded" coordinates Are not the sought "mobile" coordinates in the mobile reference frame , Since they are determined with the help of "fixed" standards (the mobile observer does not know the "mobile" standards). These coordinates can be called "movable in a fixed frame of reference". (Strictly, it should be formulated as follows: along with the mobile reference system The virtual reference system is also considered , With which the coordinate system is rigidly connected, and which moves like . Then the motion Can be interpreted as the motion of a coordinate system. These coordinates are defined in (4)).

Further, the motionless observer argues as follows: " if there is an observer in the mobile reference frame (let it be virtual), then for him his reference frame And my movable "dashed" coordinates for it are fixed, and my reference frame Moves with the speed -V relative to its system. " And then (according to the postulate of relativity), the mobile observer" as "(ie, by (4), replacing only V by -V ) determines it "Mobile coordinates in its fixed reference frame":

(5)

On the other hand, in a fixed frame of reference , Reversing formulas (4) with respect to "fixed" time and coordinates, the fixed observer receives

. (6)

From the point of view of the stationary observer, formulas (5) and (6) describe the same event, in the same coordinate systems, but in different frames of reference and coincide up to the notation of the coordinates belonging to reference frames. Can a fixed observer conclude that these formulas really coincide? There are some grounds for asserting this only for the last two equalities in (5) and (6), i.e. believe and . The first two equalities (5) can differ from the corresponding equalities in (6) by a constant factor that does not depend on time and coordinates, but depends, perhaps, on the velocity V. In other words, the time and the corresponding coordinates in the mobile and fixed reference frames can be proportional, and if the proportionality coefficients are different from 1, this is only due to the fact that the standards of time and length (in the direction of motion) in the moving and fixed frames from the point of view The fixed observer do not coincide. In this case, obviously, there are

and , (7)

Since Standards depend on the motion of frames of reference, but not on the motion of coordinate systems in these frames of reference. From a comparison of (5) and (6) we conclude that And finally

. (8)

Thus, the problem of determining the "movable" Lorentz-covariant coordinates from the given "fixed" coordinates (or vice versa) is reduced to finding the proportionality coefficient of the "mobile and" fixed standards. "(In the case of the Galilean covariance principle, all arguments are carried out in a similar way, but the last one There is no problem, since in Newtonian physics absolute time is postulated, that is, in (8) and ) In this case, the Lorentz transformations, by themselves, can not solve this problem. It requires some additional and independent condition. Looking ahead, we note that one of these conditions is the condition of simultaneity of spatially separated events, more precisely, the condition of preserving the simultaneity of events when passing from one reference frame to another (more precisely, the condition for preserving the reality of events.) Simultaneity is only a necessary condition of reality.

Thus, the Lorentz transformations, on the one hand, define new coordinates covariant to the old , on the other hand, they are independently determined by one parameter - the "angle" (spinor) of the hyperbolic rotation, uniquely determined from the definition of the new reference frame relative to the old one. In addition, this interpretation of the Lorentz transformations makes it possible to extend the scope of their application from inertial systems (SRT *) to centrally symmetric gravitational fields (SOTO and the Quaternary Universe) and, possibly, others.

print version
Author: V. M. Myasnikov
PS The material is protected.
Date of publication 09.02.2005гг