SELF

4

On conservation of energy and impulse

VLADIMIR: Going on carping, I would say, I also don’t understand exhaustively your consideration after (d1.24). In particular, why the right part has to be equal to ?²? Usually relativists write s². But the fact that the interval turns into infinity at b tending to zero relativists would have to explain to the folks.

SERGEY: I agree, but I took one of trivial variants. See, if I take, for example, b = kc, k > 1 and dimensional, it will happen even worse, as the acceleration, additionally, will be numerically more than the lightspeed . So relativists would have to explain not only, why the right part of (d1.24) turns to zero with uniform motion but also so large accelerations when uniform motion.

VLADIMIR: Second, the acceleration, from their view, has ‘to be adjusted’ to the situation in time, so that the speed of material body could not exceed c, i.e., they mean the acceleration not absolute, as classical physics does. In classical physics it is the master of situation, while in Relativity it is the servant of postulate of constant lightspeed. For example, in the relativistic limit their ‘acceleration’ will be in inverse proportion to the current moment of time in the resting frame.

SERGEY: Moreover, even if the body reaches the speed of light in the resting frame, in the actual frame its acceleration will remain, and will remain the same constant value, which is nonsense as such.

VLADIMIR: I understand what you would like to say, but I would mention, after relativistic opinion, the body never can reach the lightspeed. So, from their view, the body can be accelerated infinitely.

SERGEY: On one hand, I agree. And Pauli writes the same: “The speed does not grow unlimitedly but asymptotically approaches the lightspeed. The related world line is a hyperbola, due to which the uniformly accelerated motion in relativistic mechanics is called also hyperbolic, as opposite to the ‘parabolic motion’ of old mechanics” [W. Pauli. The theory of relativity, p. 114]. But let us look on this matter from other side. If we, following the Relativity in our problem in which we premise the translation between IRFs, establish the constant acceleration in the moving frame, then in the resting frame we will also yield the hyperbolic acceleration of body’s motion. Actually, at a' = b = const and from (d1.18)

basing on (d1.21), we yield

The plot of this regularity you can see in Fig. 4.

Fig. 4. The body’s acceleration against time in the resting frame with constant acceleration of the body in the moving frame at different speeds of moving frame and quantity of constant acceleration of the body in the moving frame b = 0,001c, ?/???²

This graph shows the same hyperbolic regularity that already is not limited by the condition when the body’s trajectory passes to the complex plane, and this motion can last infinitely, approaching the body’s speed to the lightspeed.

But let us ponder on a simple question. We have two IRFs whose observers watch the uneven motion of a body. In one frame the acceleration is seen constant, in another hyperbolic. Is not it a direct answer to our question of non-equivalence of IRFs in Relativity?

VLADIMIR: But in Relativity not (d1.13) but (d1.23) works. And formulas like (d1.11) and (d1.13) they call ‘the law-speed limit’ or something like this.

SERGEY: Agree, at least that relativists try to present their formulas in this way. But let us look at the numerator of (d1.23). We see there the product of some constant acceleration by time, and relativists write it not in the accelerated frame. In this frame the acceleration would be zero, true? This constant acceleration is written for the frame accompanying at the moment, it is really an inertial frame, but for a body, this IRF is new at each moment. But if at each new moment this frame is already new, relativists already may not find first, the more second differentials of displacement, within one frame. They have to translate coordinates and time from the new frame to that previous, and for the second differentials they have to use translation from two sequential frames. You see, with it nothing to say of constancy of acceleration in the actual frame, there will be total mess. None the less, relativists admit, in the IRF whose speed at some moment of time is same as the speed of body, it is the uniformly accelerated motion. Consequently, at the second and third moment of time this acceleration is observed from the same IRF. Only in this case we may speak of body’s acceleration relative to the particular frame. And if at these three moments of time we saw the body’s acceleration in one frame, we can state that we have detected the uniform acceleration of the body in the same IRF. Really, let the actual frame remains same during two intervals of time dt'₁ = dt'₂ (three specific moments of time), and during these two intervals of time the body has sequentially shifted by the distances dl'₁ and dl'₂ . If we have these data, we can find the acceleration b in this accompanying frame, true? And without these differentials Pauli, as well as all other relativists, might not write '_x ,  '_y , '_z  in (d1.27), true? But however little were the increments of time and coordinates of body’s displacement in this actual IRF, we always can take less and less infinitesimal increments, writing t'₁ = dt'₁/n and t'₂ = dt'₂/n, and with these increments find the body’s acceleration which will be b, too. This means, within the common infinitesimal increment of time, we can n times measure the body’s acceleration in this frame and show it constant in this frame, before we take a new actual frame. And if this constancy remains during an infinitesimal interval of time without transformation which, relativists say, causes the hyperbolic pattern of acceleration, we can state: as long as the body remains the law of its motion in this particular IRF (which was at the moment the actual frame of the body), the body’s motion remains uniformly accelerated, which means, at least in this frame the classical formula will remain. Proceeding from equivalence of all frames, this formula will remain also for all other frames. So we see that in the IRF the classical expression for uniformly accelerated motion will remain. All the rest only perverts the mathematical formalism for relativistic wishes.

Speaking of these discrepancies, I would like to draw your attention again to Fig. 4. You see in it how the acceleration in a resting frame changes at two values of speed of moving frame, given the acceleration in each frame is constant. But does not the acceleration in the resting frame mean that we measure just in this frame, not via other frames? Well, if so, then according to Relativity it appears that dependently on, which auxiliary virtual frame we take, such value of acceleration we will measure in the resting frame. As there is an infinite number of such virtual frames, including those moving with the sub-light speed, we appear unable to measure even a little acceleration, taking a frame moving with a sub-light speed as that virtual. Nonsense, absurd that follows from Relativity’s disability to describe the accelerated motions, due to which relativists have to measure and to give accelerations not immediately in some frame but via some intermediate frame relative to which they also cannot satisfy the conditions of their formalism with rigour required in mathematics.

VLADIMIR: None the less, I think, in this case the ‘admittance’ of relativists that the resting frame of classical mechanics is identical to the resting frame of relativistic mechanics – “Relativistic mechanics proceeds from supposition that in the coordinate system K’ in which the material point at the considered moment rests, the equations of motion of old mechanics are true” [W. Pauli. The theory of relativity, p. 170], – does not matter, because it has been done perhaps without enough thought. Most relativists probably don’t think so. For example, (d1.13) says of unlimited growth of speed, which is unacceptable for Relativity.

SERGEY: There is nothing new that relativists use to violate the conditions and limits of applicability of both their and borrowed formalism. But all formulas related to the acceleration and relativistic dynamics have been built on the principles and limitations which the founders of Relativity had to establish, seeing no possibility to agree the solutions with phenomenology in more general way, and all limitations in case of accelerated motion are reduced to little speeds, to have a possibility at least approximately to make an appearance as if it is obvious – to use the formalism of classical mechanics in relativistic kinematics. Should Einstein did not this limitation, he would have to repeat the work of many, many generations of scientists who developed dynamics, and in absence of experimental basis which would give him the grounds to introduce such or other relationship between the affecting force, body’s mass and acceleration. Not in vain all postulates of Relativity are substantiated so: ‘they are obvious from the view of classical physics and Maxwell theory’.

VLADIMIR: Your claim that in the resting frame, relativistic kinematics is taken identical to classical is absolutely true. But the classical physics says, dynamics is far from being kinematics, and it is groundless to extend here the ‘identity’ with (d1.12), even if relativists sometimes confuse dynamics and kinematics. Think, dear Sergey, maybe there is a sense to remain just in the low-speed region, to avoid such great contradiction to the formalism of SRT. The main regularities of figures will remain, I think.

SERGEY: The regularities of figures are not the matter, dear Vladimir, so I said before considering the problem, we have to be careful of indefiniteness connected with relativistic formalisation of the accelerated frame. While in our example, which we considered with you, factually only the trajectory is described from the point of two IRFs and relativists reduce the laws of kinematics in one frame to classical mechanics. Since, as I already showed you, having not given the laws at least in one frame, relativists are unable to write them in another frame, so the second point is only outwardly difficult. And they have to write namely in the form in which classical mechanics does it. Notice, describing the accelerated motion, relativists confine themselves to the relationship between the actual and resting frames. But we can observe the same accelerated motion from two inertial frames, cannot we? This situation happens in modelling more often. Relativists do not answer this question, because this translation is impossible with their actual frame. See yourself. Such relationship can form exceptionally through the actual frame that just gives the relativistic hyperbolic pattern of body’s motion. Above I have written this relationship between the 3-D accelerations of resting and actual frames (d1.25). Each expression includes the parameter   whose value depends on the momentary speed of the body related to the variation of speed of the sequence of actual frames. This means, in time, this parameter as such reflects the accelerated motion of the body. If the resting IRF is one, Pauli says, he had not much difficulty with the dependence of   on time, though he had to account the functional type of this parameter when found the prototypal curve. But if we have two IRFs moving with different speeds relative to each other and from them observe the accelerated motion of the body, then, to pass from one frame to another, we have to translate one value of time-dependent parameter   into another. I think, not only we don’t know such way, relativists also cannot prompt a proper way, as it does not exist, because this parameter varies in time. And trivial way to sum the speeds is inapplicable, as the parallelogram law does not work in Relativity even for constant speeds, nothing to say of summing the inertial and uniformly accelerated motions. Judge yourself. You observe this hyperbolic motion from the resting frame but would like to calculate not with the direct formula but with the expression yielded with some virtual frame which you introduced arbitrarily. As you well know, in classical formalism it will not cause problems. While here, as we showed in the subsection 7.1.5 of our work, dependently on the chosen speed of this virtual frame, you can yield any value of the studied speed, even faster-than-light. And if only direct measurements are true, due to ambiguity of speeds summing in relativistic kinematics, then it is illegal to interrelate IRFs in observing accelerated motion, just because the formula for speeds summing is untrue.

So, solving the problem through direct observation of body’s trajectories from two frames and giving the trajectory of one IRF, we, you and me, have avoided many difficulties of relativistic mechanics, and our consideration, as opposite to the relativistic way to solve, is not limited by v << c , because of applicability of Lorentz transforms up to the lightspeed. And we had a full right to study point by point any trajectories, in that number the trajectory that describes the uniformly accelerated motion. The fact that relativists use classical formulas to find the acceleration in inertial actual frames allows us to use these formulas in inertial frames. Yes, with it Relativity encounters difficulties – well, when accelerated motion, Relativity is short anyway.

VLADIMIR: To tell the truth, it took much my time to catch that you consider only the law-speed range (v << c).

SERGEY: As I showed you, I had to go a slightly seen path between discrepancies within Relativity.

VLADIMIR: I don’t know, how much correct is it – to take the coordinate transform in their exact form, and the expression for acceleration – in the limit. So when you wrote “substituting (d1.12) into (d1.14)”, I long time thought that you are doing ‘a mortal sin’, but then I decided: nothing terrible, I will see what you yield of it.

SERGEY: There was no sin, of course. See, should in (d1.12) there was the equation of uniform motion, neither you nor relativists would see a mortal sin in it, the more that the law of relativistic speeds summation is based just on the translation of body’s uniform motion in the moving frame into the corresponding motion relative to the resting frame. Actually, the statement of problem to find the expression for summation of relativistic speeds Einstein wrote so: “Let in the frame k moving with the speed v along the axis X of frame K, the point moves according to the equations

where w  and w  are constants.

Find the motion of point in the frame K. If to the equations of point’s motion we introduce the values x, y, z, t with the transform formulas introduced in the subsection 3 (trivial Lorentz transforms – Sergey), we will yield

[A. Einstein. On electrodynamics of moving bodies, vol. 1, p. 20].

You can see, dear Vladimir, the course of solution in case of body’s uniform motion is same as we applied in the problem of uniformly accelerated motion. The law of body’s motion in one IRF is found same and we substitute it to the Lorentz transforms to find the trajectory of body’s motion in other IRF.

Relativists encounter difficulties in connection with their postulate of constant speed of light. You could see, in this relation in the same IRF, the 4-D interval is changed only because the accelerated body is present in this frame. A very simple and natural question arises here – is the metric of space in Relativity dependent on the fact that some bodies move in this frame inertially and some other are accelerated? A "good" invariance of 4-D interval, I can say.    Let us think a while about the term ‘invariance’ – the absence of variants in change. Well, some body accelerates – and interval in the IRF gains another appearance: remains for inertial motions, while for accelerated motions we already have to use new, special measures. This itself evidences that relativistic laws are not equivalent, even for IRFs. We should not forget, when introducing relativity, relativists specially emphasise that they define the metric of space through measurability with measures resting in the related frame. If for some motions they have one measures and for other motions – other, then measure as such is absent in Relativity, and all their mathematics becomes senseless.

You also raised another important point – you mind that my computations concern the low speeds (v << c). On one hand, I already pointed you, as opposite to the relativistic version, in our problem we need not this limitation. But I would like to draw your attention to one more important aspect. When relativists considered the ‘uniformly accelerated’ motion, they undertook not the trivial way but changed the concept of uniformly accelerated motion. The answer is simple. See, in the Lorentz transforms, when they say that the body cannot exceed the lightspeed, they always operate with expressions yielded in Lorentz transforms, but in the initial frame there is no direct limitation. And if relativists operate with the concept of uniformly accelerated motion in the accompanying frame, the meaning of uniformly accelerated motion does not change because of it, and sooner or later the speed in that frame will exceed the speed of light. Namely so relativists delete the momentarily existing accompanying frames. It does not imply the limitation of speed in Relativity, as even in the expression for summation of relativistic speeds, in case of one-way motion of the body and moving frame, when the body reaches the lightspeed in this moving frame, we yield

As we see, in (d1.34) the speed of intermediate frame v has no limitations. The same we could find the speed of intermediate frame equal to the speed of light, and according to the computations similar to (d1.34), the speed u' or u could take any value faster than light. And, as we showed in our subsection 7.1.5, when the body and intermediate frame move oppositely, the speed can develop up to infinity and change the direction of body’s motion with respect to the speed of intermediate frame. Relativists impose the limitations, considering the speed of body in the initial resting frame from the point of moving frame, i.e. through the reverse operation. In this case the limitation is more likely same ‘obviousness’ than rigorous proof that the speed cannot exceed the lightspeed. Not in vain Einstein in the other his paper, “On the principle of relativity and its corollaries”, considering the possibility of relative faster-than-light motions, was already not so categorical: “The speed v can take any value less than c. But if W > c, as we premised, we always can choose v so that T < 0” [A. Einstein. On the principle of relativity and its corollaries, vol. 1, p. 76], where T is the time of transfer of signal between two points of frame. As we see, after Einstein, one has also to select the value of frame’s speed, the body’s speed to be slower-than-light. Einstein was sure, such selection is possible, but this is far from being obvious, as this speed is given in the statement of problem and is not the subject to select parameters arbitrarily.

Things look simpler with uniformly moving bodies, as their speed does not vary in time. With accelerated bodies this ‘obviousness’ already does not work, as the body varies its speed in time and we need not an illusory condition of limit but a strong mathematical substantiation, which factually does not exist in Relativity. It does not exist, because in the resting frame the relativistic kinematics is taken to be identical to that classical. So relativists had to give up again their postulates of invariant 4-D interval, in order to escape the faster-than-light speeds.

We see this feature in our case. The range in which the solution (d1.18) is true can be found as follows:

You see, there is no limitation, the speed to be little, and according to the Lorentz transforms, up to this limit the body can accelerate without any prohibition. The possible growth of mass can be the only relativistic argument. But here they fall into their own trap. They have reduced dynamics to kinematics, while the mass growth can be revealed in no other way than in dynamics. Before it, they yet have to prove that in translation between IRFs, the conservation laws are true in Relativity. Until they have not such proof, it appears that at the border defined by the condition (d1.35), both speed and acceleration of the body in the moving IRF turn to infinity. And this derivation is not limited by low speeds and accelerations of moving frame. To illustrate, I graphed in Fig. 5 the variation of acceleration in time in the moving frame for different values of speed of the frame itself.

Fig. 5. The acceleration against time in the moving frame for different values of frame’s speed and numerical value of constant acceleration in the resting frame a = 0,001 c, m/s²

You see, the accelerations reach their limit simply at different time, but the function fully remains its pattern.

VLADIMIR: However, attentively considering Fig. 3 and Fig. 5, I see that a = 0,001 c m/s², which is unclear. If c is just the c, the dimensionality is improper. Though it also is near the ‘little’ speeds, this obviously is not the Newtonian limit.

SERGEY: I wrote the acceleration so for convenience, meaning the dimensional coefficient of relation. Of course, this has no concern to the relativistic record of acceleration as a dimensionless scalar value. So I everywhere said of numerical value. I could write the full five-digit value of acceleration, but I thought, such form is more visual, noting that the acceleration 10-time exceeding that of free fall is critical for humans and instruments.

Concerning the Newtonian limit, this is a very indistinct issue, because of errors which we agree to omit in calculations. Sometimes it is important to be accurate to 5–7 decimal places, sometimes 1 % of error is very good, true? I have chosen little values only to show that relativistic distortions remain at small values of speeds and accelerations. At large speeds and accelerations all properties will remain and when reaching the time limit, in all cases the speed and acceleration will turn into infinity. Only the numerical values of parameters will change.

And if speaking of intervals within the lightspeed, the acceleration will change in time as shown in Fig. 6

Fig. 6. The joined graph of variation of speed and acceleration of the body in the moving frame, when uniformly accelerated motion in the moving frame, at v = 0,1 c, m/s, and numerical value of body’s constant acceleration in the resting frame a = 0,001 c, m/s²

As you can see from this graph, should the relativistic conception were true, we might not apply the concept of uniformly accelerated motion in the usual meaning even to thermal electrons being located, for example, in the field of plane condenser, which would be immediately revealed in the simplest electron microscopes and even in the cathode tubes.

VLADIMIR: You yielded quite paradoxical growth of acceleration a' with growing t'. Maybe, it happened because except the parameter (v << c), the parameter ? t'/v appeared in the solution. With it, because of first condition, the value of parameter cannot much exceed 1. Well, if we look at Fig. 3, we can see, the considered range for t' much exceeds this condition. Though, we see a monotonous function in the plot. I don’t know what to say of it. Is the classical limit true for your (d1.21)?

SERGEY: It is senseless, dear Vladimir, to speak of classical limit along with Relativity. The existing classical formalism, as you well know, has not such limit, and when developed for high speeds, the limitation will appear without any transformations of space and time and mythical growth of mass but because of correct account of field dynamics. Dynamics! Not ‘dynamical kinematics’. 

VLADIMIR: As to the ‘mass growth’, we would have to sort this matter out separately, dear Sergey. In SRT they obviously take the inertial meaning of mass. But further, in other sections of theory, they postulate the ‘equivalence’ of gravitational and inertial hypostasis of mass. And gravitational hypostasis reveals itself as the field, with all consequences. Though, it hardly is worthy to develop this matter in this dialogue. Perhaps there is a sense to consider it separately.

SERGEY: Indeed, and conclusions to which we will come in this present dialogue, considering relativistic ‘dynamical kinematic’, will be of great importance for this future dialogue. So let us go on, aren’t you against?