V.7 No 1

3

Non-unique-valued transforms between inertial frames in SRT

2.3. The relationship of angles in mutually moving frames from the view of relativistic conception

  When we solved the problem of coordinate transform, introducing the intermediate frames, we saw that different ways give different solutions, which is not inherent in Euclidean spaces. As one of supposed causes of such inconsistency, relativists use to think the distortion of angles in transforms either turn of axes by an imaginary angle.

At the same time, one of conditions of Einstein’s statement of problem in finding the Lorentz transforms is just that the coordinates have to be parallel: “Let now to the coordinate origin of one of these frames (k) a (constant) speed v is given in direction with the increasing values x of another, resting frame (K); this speed is also passed to the coordinate axes, as well as to the related scales and clocks. Then to each moment of time t of the resting frame (K), a definite location of the axes of moving frame relates, and we, thinking of symmetry, may assume that the motion of frame k can be such that the axes of moving frame at the moment t (through t we always will denote the time of resting frame) will be parallel to the axes of resting frame” [1, p. 13].

On this causation of mutual parallelism of axes, Einstein based his initial transform equations in general form which in the course of his derivation take the form of Lorentz transforms: “we can choose both coordinate systems so that the axis x of the frame S and axis x'  of frame S'   coincided and the axis y'  of frame S'  related to S was parallel to the axis y of the frame S … From the known now location of the coordinate planes of frame S'  relative to S, it immediately follows that each of the following equations are equivalent in their couples:

[4, p. 71].

As we can see from this citation, the system (16) has been written namely for the parallelism of axes of mutually moving frames. It implies no turns, as well as it does not imply the transforms in curvilinear spaces.

The references of relativists to the turns in transforms are same ungrounded. Such turns in linear spaces are given by expressions basically different from (16), and the space of SRT, by the condition given by Einstein, is strongly linear: “We can immediately say, these equations have to be linear with reference to the given variables, as the properties of homogeneous space and time require it” [4, p. 71]. It follows from the same linearity that the 3-D space of SRT (spatial axes) is Euclidean: “The tensor of curvature is obviously equal to zero. However, already Riemann in his dissertation pointed that the inverse theorem is also true: if zero tensor of curvature, the space is Euclidean, i.e., then we can find such coordinate system in which g_ik are constant. It was Lipschiz who first gave a sound proof of this statement. Very elegant and visual consideration has been made by Veil” [2, p. 78].

Thus, nothing to say of nonlinear space or of intentional turns of axes in the Einsteinian theorem. This means, in both variants given in the subsection 2.2, all axes of main and auxiliary frames remain their mutual parallelism and in both cases the transform goes from the frame S₁   to the same frame S₂ .

None the less, in order to lift the extraneous objections of relativists, consider, how the angles transform in the mutually moving frames. From the very start we will point that in this case we may not use the trivial relativistic formula of the angle transform [2, p. 32]

In Pauli’s symbols, (17) has been written for the transform of angle of inclination of speed in passing from the moving (dotted) frame to the resting (non-dotted) frame. Just in this connection, in the right part of (17) we see not only the reciprocal speed of frames v but the speed u' with reference to the moving frame. In the relativistic formalism it means that for bodies resting relative to the moving frame, to which we usually relate the coordinate origin, (17) loses its sense, and to transform the spatial angles, we have to use other relationship derived namely for the transform of angle of the spatial vector resting with reference to the moving frame. We would emphasise, in classical physics this difference would be absurd, but space-time in its formalism never transforms and the correspondence between the inclination angles of vectors remains in all IRFs. In SRT this correspondence is intentionally violated, so we have to seek the regularities of angular transforms separately for the vectors of speed and separately for spatial vectors. And, as we will see below, these regularities are basically different.

To find the regularity of transform of spatial angles in passing from one IRF to another within the relativistic conception, consider two IRFs K and K' ; this last moves in direction of +x of the frame K, while the axes x and x'  remain full coincidence. Also, let in the frame K'   the spatial vector '  is located so that its angle of inclination to the axis x'  is '  and its origin coincides with the coordinate origin of K' , see Fig. 6. Let us find the inclination angle of this vector in the resting frame K.

Fig. 6. Motion of the spatial vector   relative to the resting frame K

To find correctly the relationship for angles, we have, as opposite to the classical formalism, to account not only spatial but also temporal characteristics, because, as we showed in our previous works, the Lorentz transforms incline the plane of events, which produces an imaginary non-simultaneity. And if we premise that the inclination angle of the spatial vector is made in the physical time of each frame (which corresponds to the Einsteinian condition of Lorentz transforms), we have to make additional transforms to agree the results of our transform with the physical time in each frame.

With account of the said, to transform the vector ' , we have to transform the coordinates of start and finish of this vector. According to the statement of problem and with the trivial Lorentz transforms,

for the starting point of vector x'₁ = 0 , y'₁ = 0 , t' = t'₀   and

for the finishing point of vector x'₂ = N' cos ' , y'₂ = N' sin '  , t' = t'₀ and

We see from (18) and (19) that the starting and finishing coordinates of vector in the resting frame are mapped at different moments of time, as expected. To put the yielded values into agreement with the physical time in the resting frame, we have to account that all points of this spatial vector uniformly shift in time along the axis x of the frame K. So, if the location of the vector’s origin (x₁ , y₁) corresponds to the time t₁ , the end of vector is at this moment at the point whose x-projection is

and the y-projection of the end of vector corresponds to (19), as it shifts along the axis x .

  With account of (20) and (18), the difference of x-projections in the resting frame is

From (18) and (19) we have for the y-projections

Hence,

The yielded expression (23) describes the sought relation between the angles of spatial vector inclination in the mutually moving IRFs in the relativistic conception. Comparing this expression with the law of transform of vectors of velocity (17), we see the basic difference in the pattern. It evidences that the vectors of different nature are transformed differently in the relativistic conception.

Further, we see from (23) that the inclination angles of spatial vectors directed along the axes of moving frame do not transform, i.e. at ' = 0  we have = 0   and at ' = /2  yield that = /2 , too. Thus, we have rigorously proven that no turn of spatial axes of which relativists say occurs in the motion of frame K' . But the angles differing in their value from those pointed are transformed in correspondence with the relativistic multiplier.

2.4. Proof that the axes remain parallel in passing from the resting to the moving frame through the intermediate frame moving in direction of one of projections

In the previous problem we considered the regularities of transform of inclined spatial vector moving with the origin of moving frame and showed that with the horizontal and vertical arrangement of this vector the angle does not transform. In the new problem we will complicate this spatial vector making it moving also transversely.

Let we have a resting IRF S₁ and a frame S' moving along the axis x₁ of the resting frame with the speed v_x1  in direction of positive values of x₁ , remaining the axes parallel. And let we have some segment AB moving with the frame S' and at the same time in direction of the axis y' . With it, it moves so that the point A of the segment moves strongly along the axis y' , see Fig. 7.

Fig. 7. The segment AB moving relative to the frames S₁ and S'

Suppose, the length of segment is l, the longitudinal speed of segment as the whole from the point of resting frame is v_x1 , its transverse speed, again from the point of resting frame, is v_y1 , and the inclination angle of the segment to the axis x₁ is ₁ . Find the inclination angle in the moving frame S' . To simplify the derivation, suppose that at the initial moment of time t₀ = 0  the origins of both frames and the point A coincide.

First of all, let us write the equations of motion of the studied segment in the resting frame S₁ , which means to write the equations of motion for its start and finish. Given the initial conditions, these equations will be

As the next step, let us find the laws of motion of the ends of rod in the moving frame. For it, just as in the previous problem, we have to extend the definition and to account the physical time. To do it correctly, recall, in which connection the problem of rod arose when we considered the problem of passing from the frame S₁ to the frame S₂ moving in the parallel plane relative to S₁ under an angle to it. This problem arose, as we had to prove that the axes of S₂ do not turn relative to S₁ when we use the intermediate frames S' and S" moving in the direction with projections of the speed of frame S₂ . As in the main problem we consider the passing between S₁ and S₂ , the orientation of axes of S₂ in the physical time S' does not matter. The question is of importance, whether the frame S₂ turns with reference to S₁ in passing through S' or S" . Proceeding from this, and with account that in the Lorentz transforms we use, as a rule, the physical time of the frame from which we make the transform, i.e. in this case of the frame S₁ , we are interested to see, whether the axes of S₂ are parallel namely with reference to S₁ , in the physical time of the initial frame S₁ . While we may think the rod as some generalised model of the axes of frame S₂ .

Having cleared this issue, we can go on with our study. Proceeding from the additional condition, and given that the moving frame moves along the axis x₁ and is parallel to it, we can write the equations of motion of the ends of rod in the moving frame as

We see from the yielded expressions that the end A of the rod in the moving frame has strongly vertical motion. We will not do the time transform, as we are interested in the transform of the rod inclination angle in the physical time of S₁ .

For the end B we have

We see from (27) that the second end of rod moves vertically, too, as x' is independent of time, and in S' also is involved only in y' -projections. But it also contracts, because to find the rod inclination angle in the moving frame, we are interested not in the absolute values of projections but in the difference in coordinates of rod ends at each moment of time. So

and 

With account of (29) and (30) we need not to find the time transform, while we factually yielded the simultaneity from the point of S₁ when substituted (26) and (28) into (30).

With (29) and (30), we can find the rod inclination angle with reference to the moving frame S' in the physical time of S₁ . And this inclination of rod, as we already said, at definite orientation will model the location of axes of S₂ relatively to S₁ :

It follows from (31) that the position of directed segment AB that imitates the axes of S₂ when orientation is multiple of k/2 , k = 0, 1, 2, 3   remains its value; so the axes of S₂ with reference to axes of S₁ remain parallel. We can prove the same for passing from S₁ to S₂ through the intermediate frame S" moving in direction with v_y1 relative to S₁ .

It also follows from the said that the ambiguity arising in different ways of passing from S₁ to S₂ relates not to the fact that dependently on the way we come to different frames – we saw that passing through the intermediate frames S' or S" moving in direction with the projections of speed we do not change the inclination of S₂ with reference to S₁ but, using in both cases the same physical point A₂ to build S₂ , we in both cases come to the frame at whose origin the same physical point A₂ is located, and axes in both cases remain parallel to those of S₁ . The non-unique value is caused merely by the non-unique result that we yielded, using the Lorentz transform.

This is an important conclusion. It shows that the cause of difference between the first and second variants in the subsection 2.2 was not that, going different ways, we as if come to different frames S₂ . The calculation shows that if we pass step by step through an arbitrary number of intermediate frames, using the motion of next intermediate frames along the axes of previous frames, then, independently of the number of those intermediate frames, the axes of all frames will remain parallel. This means, going different ways from the initial frame S₁ to the final frame S₂ through the intermediate frames that move along the axes of initial frame S₁ , we will yield the frame S₂ whose orientation is same but the degree of length transformation in axes will be different, which evidences merely of non-unique valued passing with the Lorentz transforms in the arbitrary inertial motion of frames.