SELF

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S.B. Karavashkin and O.N. Karavashkina

3. The problem of coordinate transform between the IRFs moving arbitrarily on the plane from the point of classical formalism

To compare with the transform formulas yielded in the section 2, we in this section will find the transform equations from the resting frame to the moving frame in limits of classical formalism, using the same schemes.

1 VARIANT of solution

Let us first find the equations, using the intermediate frame S' moving relative to S₁  with the positive values of x₁ with the speed v_x1 . Given Fig. 2 , the passing formulas between S₁  and S' in classical formalism are

Relatively, the formulas to pass from S' to S₂ are

Joining (32) and (33), yield

2 VARIANT of solution

Now let us solve with the intermediate frame S" that will move with the positive values of y₁ with the speed v_y1 .

According to Fig. 4 , the transform formulas from S₁  to S" are

Relatively, the formulas to pass from S" to S₂ are

Substituting (35) into (36), yield

Comparing (34) with (37), we can make sure that in classical physics the transform formulas are independent of the way we yielded them, which is in full agreement with the condition that the space for which they were derived is homogeneous and isotropic.

4. On the unique-valued mapping necessary to describe correctly the processes of nature

In the previous sections we, basing on two phenomenological approaches to describe the phenomena of nature, yielded opposite results. From the view of classical physics, however we pass from one IRF to another, we yield the same regularity that interrelates the parameters of mutually moving frames. From the view of relativistic formalism, this is far from being obvious, as Lorentz transforms give different results, so they depend on the way, how we transformed.

This difference is the matter of principle; it makes questionable the non-unique passing between the IRFs when describing physical processes. If such ambiguity is admissible, the Einsteinian theory of relativity can be called a physical conception; if not, it cannot be called physical conception, irrespectively of any other facts.

To sort this matter out, let us first of all define the common properties usually attributed in physics to the homogeneous and isotropic space and time. “In a mechanical system enough remote from all other bodies, the motion occurs absolutely same, wherever we place this system. This means the following. Let we have two equal mechanical systems with equal initial conditions of motion. They both are well remote from all other bodies able to affect them. Then, if we consider them in one and the same frame, the motion in them will be exactly same. In other words, the motion will not change in transfer of all moving bodies by same distance along mutually parallel segments at the same moment of time. This statement is based, indeed, on very large experimental material accumulated by mechanics during all its history of development. In shorter appearance we call it the property of homogeneity of space.

We can choose two equivalent mechanical systems similar to the described not only superimposed but also turned by any angle relatively to each other. Again, if these two frames are enough remote from all bodies able to affect them, the motion in them occurs similarly. Speaking otherwise, all directions in space are equivalent. This property of space is also called isotropy. Isotropy of space is the same generalisation of experience as its homogeneity” [5, p. 19].

Isotropy of space has a direct relation to the conservation laws of energy and impulse in dynamics: “… the conservation laws have been yielded as a corollary of Newton’s equations of motion. So they are interrelated with the properties of space and time that are postulated in classical mechanics” [6, p. 102].

In these definitions of isotropy and homogeneity of space, it is silently admitted as obvious that the location and trajectories in each frame have to be mapped in unique values in each frame. It is proven in higher geometry: “If vectors e₁ and e₂  having common origin O are non-collinear to each other and fixed, then any radius-vector r can be expanded as

where the numbers x₁ and x₂  are determined in unique values.

And opposite, after the given pair of numbers x₁ and x₂  and vectors e₁ and e₂ , we can graph the only radius-vector

A pair of vectors e₁ and e₂  having a common origin and being non-collinear to each other is called the basis. The system of numbers x₁ and x₂  that takes part in the expansion

of an arbitrary radius-vector r   in the elements of basis e₁ and e₂  is called the affine coordinates of the vector r ” [7, p. 100].

In the light of this theorem of unique-valued mapping in the linear basis, it is clear that the location of any material point, as well as of their amount, has been determined in unique values in the chosen basis.

Given this, imagine that in some space some material point moves inertially and two observers observe its motion. They are located at the origins of their inertial frames; the frame S₂  moves with reference to the frame S₁  with some speed v whose vector is not in direction with the body’s motion, see Fig. 8.

Fig. 8. The motion of body B with reference to the inertial frames S₁  and S₂

In classical formalism, it is very easy to find the speeds of body in the frames S₁  and S₂ . Actually, according to the expressions (34) and (37) that we yielded for passing between the frames S₁  and S₂ , we have

Immediately from (39) we yield, irrespectively of the way in which we sought the transform equations,

It is easy to see that (40) shows the rule of parallelogram according to which the vectors are added through summation of their projections.

In the light of said above, it would be interesting to see the result of this operation for relativistic transform equations. In this case we already may not confine ourselves to one derivation, as dependently on the direction of passing from S₁  to S₂ , we yielded different formulas (6)–(8) and (13)–(15) relatively.

Determine first the projections of vector of motion of the body B in the transforms (6)–(8). Basing on (6), for the x-projection of displacement of the body B we have

for the y-projection, basing on (7),

and basing on (8), for the t-projection

It follows from (41)–(43) that

The system (44) describes the projections of body’s speed in the moving frame in the relativistic formalism. First of all, comparing (44) with the trivial expression of relativistic summation of speeds (2), we see that (44) is reduced to (2) if

On one hand, it shows that our derivation is correct, and on the other hand, it evidences that (44) is the generalisation of the known expression for the speeds summation.

Now consider the second variant of solution for the relativistic problem based on (13)–(15).

With (13), we have the x-projection for the displacement of body B as

with (14) for the y-projection we have

and for the t-projection we yield the expression exactly alike (43), as in both variants the time is transformed equally.

Now, with (43), (46) and (47), yield

The expression (48) is the second variant of solution of the relativistic problem, and it basically differs from (44). It is typical that if (45), which determines the motion of moving frame along the axis x, is true, both solutions are reduced to (2). This evidences that in 1-D motion of frame, the solution will not bifurcate. The bifurcation follows from the multidimensional mutual motion of frames and its cause is that the transforms are non-commutative in the relativistic formalism.

Thus, we see that the non-unique-valued transforms fall into discrepancy with the basic theorem of one-valued decomposition of vectors in the isotropic homogeneous space. And it touches not only the properties of isotropic space. We see that the laws of matrix transforms in application to the description of frame-to-frame transforms are incomplete.

Actually, when we describe the properties of algebraic ring, we usually give to their properties the utmost generality. And proceeding from this generality, “the product of two matrixes, generally speaking, depends on the arrangement of multipliers even in case when the ring K is commutative” [8, p. 13].

As we could see from this analysis, when we apply the matrixes to the problems related to the transform of coordinate systems, onto the general properties of matrixes we impose the requirement of one-valued mapping, which supplements the common requirements to an algebraic ring. In case of Galilean transforms we have no difficulty with it, as it satisfies both general properties of ring and additional requirement. Actually, the expressions of consequent transforms yielded in the section 3 can be written in the matrix form as follows. In passing to the frame moving in direction with the axis x, the transform matrix will have an appearance

In passing to the frame mowing with the axis y, the transform matrix will be

Multiplying these matrixes, we can easily make sure that

With it, the appearance of matrixes is so unique that least changes in any of them in (49) or (50) will give a non-commutative result.

At the same time, not only formal mathematical result of operation depends on the fact, how the law of frame-to-frame transform remains unchanged and independent of the way in which we pass from one inertial frame to another. The philosophical idea of cognition of the surrounding phenomena of nature depends on it, too. And we can easily understand it on a simple example.

Suppose, an observer located at the origin of resting frame S₁  cannot observe immediately the motion of some body B but has available some intermediate stations which observe this body and move relative to the resting observer with the speeds v'  and v" , see Fig. 9.

Fig. 9. The indirect observation of motion of the body B

If the results of observation depend on, in which path they were received, the determinacy of study will be violated, and we can take as the characteristic of B’s motion any arbitrary value, dependently on, which intermediate frame we use. So the observation loses any sense, as its reliability will be zero. Thus, violation of commutativity of transforms automatically violates the basic theorem of the unique way to decompose the vector in the affine basis and makes illegal the use of such decomposition in the formalism based on non-commutative transforms from one coordinate system to another. Namely because of violated basic theorem, the bifurcation is absent in case of 1-D motion of the moving frame. But as soon as the motion becomes multidimensional, the basis of relativistic formalism immediately comes to discrepancy. From this, the very decomposition of spatial vectors and vectors of speed is incorrect in the relativistic formalism. It is well seen in Fig. 9. For the intermediate frame we yield the same non-uniqueness, using, in their turn, the additional intermediate frames. So it will appear that in measurement through the intermediate frames we will yield one result and in direct measurement – other; as we said above, it violates the determinacy of our observation of processes. But this is irrelevant to practice.

Conclusions

In the course of this study we have established that the Lorentz transforms make the mapping non-unique, due to violation of the basic theorem of unique decomposition of vector in the orths of affine basis, which makes illegal to use the vector decomposition in the relativistic conception. So Einsteinian conception of relativity becomes to contradiction with the laws of description of processes in the nature, and, thus, loses its right to be called the physical conception.

We also showed that the only transform system satisfying the principle of determinacy of description of physical processes is the Galilean transform.

All other kinds of coordinate system transform have to be checked, besides correspondence to their postulational basis, for correspondence to the theorem of unique way to decompose the vectors in the affine basis. In case if these transforms are inconsistent with this requirement, we may not use them even as a hypothesis. This is just the criterion that allows to check the correspondence of a conception to the requirements of unique description of phenomena.

References:

1. Einstein, A. On the electrodynamics of moving bodies. - In: Collection of scientific works, vol. 1, p. 7. Nauka, Moscow, 1965 (Russian)

2. Pauli, W. Theory of relativity. Gostechizdat, Moscow- Leningrad, 1947 (Russian)

3. Einstein, A. The relativity principle and its corollaries in modern physics. - In: Collection of scientific works, vol. 1, p. 138- 164. Nauka, Moscow, 1965 (Russian)

4. Einstein, A. On the principle of relativity and its corollaries. - In: Collection of scientific works, vol. 1, p. 65- 114. Nauka, Moscow, 1965 (Russian)

5. Kompaneetz, A.S. The course of theoretical physics, vol. 1. Prosveschenie, Moscow, 1972 (Russian)

6. Olkhovsky, I.I. The course of theoretical mechanics for physicists. Nauka, 1970 (Russian)

7. Bakelman, I. Ya. Higher geometry. Prosveccenie, Moscow, 1967 (Russian)

8. Maltzev, A.I. Fundamentals of linear algebra. Nauka, Moscow, 1970 (Russian)