v.7 No 1

3

  On conservation of energy and impulse

VLADIMIR: I’m not against your approach, dear Sergey, only I’m afraid, is not it the case ‘to swat a fly with a sledgehammer’ . And see, why.

You are writing, for relativists the very fact of translation is of no importance, they are satisfied with the fact that the laws of dynamics are true in the frame freely falling in the homogeneous gravity field.

I agree, this is of no importance for them, but, as I can judge, ‘of no importance’ is too delicately said. The task which Einstein pursued when started his GRT was, as far as I can judge, to lift, to deny, to consign to oblivion the statement of classical physics that free motion of bodies in space and time and their same motion under affection of forces are basically different motions. According to their logic, there is, and can be, no translation between them. 

In classical mechanics we consider kinematics and dynamics. Classical physics cries out “danger!” seeing such theory as SRT – mere kinematic theory, or rather such which mere mathematically and so ruthlessly approach the problem of space-time and chisels from them, for beloved itself, the 4-continuum (!) after definite rules (!), to please its basic postulates. It cries out “danger!” also because these postulates, as you correctly underline, are kinematic, not dynamic, though the basic work had the name “On electrodynamics of moving bodies”, – but just “On…”. .

I can be wrong, but it seems to me, here also, breaking into the gravity fields, they cared more of kinematics than of dynamics, they did not think of dynamics and, truly speaking, lost it already in SRT. So they had to postulate such eccentric things as ‘energy-mass translation’ and other ‘wonders’. .

SERGEY: I absolutely agree with you.

VLADIMIR: I mean, don’t you think of relativists too good supposing that they cared, whether the laws of dynamics are true in non-IRFs? Then why don’t they care that these laws don’t work in SRT? I think, relativists simply have not the problem of translation from IRF to non-IRF, as their point is the equivalence of these frames. Having claimed so loudly, they by volition freed themselves from the necessity to think, how to translate from frame to frame. Everything for people = everything for me! 

SERGEY: Thus, dear Vladimir, just as I said from the beginning, the issue, how much legal is the Relativity, is not in physics but in the plane of clan interests and psychology. You now are corroborating it.

You are right, of course, saying, it’s impossible to prove something to relativists. I would add, impossibility follows from the fact that they long ago made sure, how absurd is their conception. In fact, the situation is like in the novel “Golden calf” by Ilf and Petrov, when they sawed ‘golden’ dumbell (for English-speaking readers: they have stolen from one millionaire a 16-kg dumbell, thinking, it is gold):

– Don’t come up to me with this iron! – squealed Panikovsky, running off. – I despise you!” [p. 191].

Is not it a very common behaviour of relativists? And now, when people gradually return from the absurd imposed on the science by relativists, a part of them, not so militant and not having lost the limits in defending of dogma as such, disappears from the battles without good-bye and finds some other activities. But most insistent will require, of course, to saw to the end.

Not in them is the problem. This all will gradually calm down. It is important to understand, how mistaken are the relativistic approaches, lest, developing the classical formalism for the high speeds and energies, to replicate their mistakes, as the adherents of aethereal conception do now, thinking that we may partially change most absurd discrepancies of Relativity and retain the fantasies like photons, time transforms, wrapping space-time in the aethereal theory, may substitute the dynamical description of processes by that kinematic, – you say it absolutely right, – and the crisis of Relativity will be got over. No, until we have not revealed all mistakes and have not taken off from models all absurd statements and corollaries based on them, the crisis will go on, producing new and new monsters in new post-relativistic incarnation.

So, with your permission, I would return to the promised analysis, how legal is it to postulate the equivalence of laws of kinematics and dynamics in frames of relativistic conception.

Before we start analysing, whether it is legal to think equivalent the inertial and freely falling frames in the aspect, how the laws of dynamics work in them, let us reveal the reefs which we will encounter and which we would want to avoid. The main reef is that in frames of SRT the translation from inertial to non-inertial frame is not defined formally.

As you remember, in classical formalism we easily made it kinematically, as neither time nor spatial parameters transformed in translations. And the main, the lightspeed was not strongly related to each frame and its value we yielded as the corollary of Galilean transforms. The value of lightspeed could remain same or become anisotropic, the distribution of equiphase surfaces could change, but this all did not affect the rest phenomena and material bodies. In other words, though we detect the phenomena with help of light, the properties of light do not affect physical properties of other material bodies and the more geometry of space and properties of time. In Relativity, Einstein’s postulate of constant speed of light is true exceptionally in IRFs and properties of light have a direct effect on the properties of time, space and geometrical size of bodies. In case when the bodies in the moving frame are accelerated differently, to prove this postulate true is impossible. From the classical view, it is clear, why – because even at small speeds and accelerations the lightspeed in non-inertial frames will not be constant and Einstein never found ‘evidences’ (when classical ‘obviousness’ is minded but it is realised relativistically). And he could not find, as in classical formalism there also is no way to this ‘obviousness’ – nothing to copy. Just so, when Einstein encountered the necessity of accelerated motion, especially with necessity to pass to the accelerated frame, he ‘distracted’ to some approximations, reducing the accelerated frame to the frame ‘locally inertial in time’, which, basically, is the trick he multiply used to interrelate GRT with SRT. Already in his first paper “On electrodynamics of moving body” Einstein defined the statement of problem of electron’s acceleration in the external field so: “Let a point particle with an electric charge   (called further ‘electron’) move in the electromagnetic field; of the law of its motion we will premise only the following.

If the electron rests during some interval of time, in the nearest next moment of time the motion of electron, as it is slow (! – Sergey) will be described by equations

where x, y, z are the coordinates of electron and   is the mass of electron.

Then, let the electron during some interval of time to have the speed v (note, in this statement of problem, the electron has not the momentary speed v, as it is done in accelerated motion, but some constant speed during though little but interval of time – Sergey). Find the law according to which the electron moves in the immediately next element of time after this interval.

Doing not limiting the generality of considerations, we can assume, and assume actually, that at the moment when we start the observation, our electron is in the coordinate origin and moves along the axis X of the frame K with the speed v. In this case it is clear that at the indicated moment of time (t = 0), the electron rests relative to the coordinate system k moving in parallel to the axis X with the constant speed v ” [A. Einstein. On electrodynamics of moving body. – In: Collection of scientific works, vol. 1, p. 32 (Russian)].

Having read this statement of problem, you hopefully will agree with me, Einstein’s statement of substitution of momentary speed by some speed constant in the time interval, as well as confinement to the small speeds and accelerations, essentially limit the generality. The more they disable to substitute the accelerated frame by some inertial frame, even at small time interval. And this makes illegal to apply solutions yielded in this approximation to the sub-light speeds. Even if solutions are correct, they are applicable only to small speeds. So, when relativists attempt solving electrodynamic problems for charged bodies at sub-light speeds, they automatically exceed the limits in which Einstein’s mathematical tool is true and automatically make their solutions incorrect.

I also think, you already understand, why Einstein dared to make such step, having substituted the non-inertial frame by that inertial. Yes, because of vagueness, whether the postulate of constant speed of light remains true in the accelerated frame. And accelerated not in the homogeneous gravity field but in some arbitrary electromagnetic field.

Perhaps we should not be caught by this vagueness again, the more, below we will see, we have no necessity in it.

We can first check the conservation of laws of dynamics in translation not from IRF to non-IRF but from IRF to IRF. Surely, if Einstein wrote (d.1.11) in the initial inertial frame as the Newton law and, naturally, premised some arbitrarily given frame, in this way he premised that in Relativity this law is true in all relativistic IRFs. Consequently, as the initial point of check, how much correct the laws of dynamics are in Relativity, we can consider a simple case – a lumped force of constant magnitude affecting some body with the mass m.

It can outwardly seem that we repeat the problem of electron’s acceleration, but it is not so. First of all, in the Einsteinian problem, the electron was affected by the field in which additional fields could arise and disappear. Here we consider a constant lumped force, and in translation from one IRF to another, also inertial frame, additional fields would be absurd, they just would violate the conservation laws, true? Second, the moving frame in case of electron was quasi-inertial only in a small interval of time. In our case the inertia of resting and moving frames is not limited in time, therefore we can follow the pattern of body’s trajectory in a considerable time interval, doing not exceeding the limits of model. And third, Einstein premised from the beginning that the Newton second law is true in the moving frame, and proceeded from it. We do not dare to premise beforehand that the laws of classical physics are true in relativistic translations. To avoid the necessity to transform the Newton law with Lorentz transforms, presenting the force vector as the speed vector, we can make use of the circumstance that, as relativists admit, the resting frame of classical mechanics is identical to that in relativity: “Relativistic mechanics proceeds from the premise that in the coordinate system K’, in which the material point rests at the considered moment, the laws of motion of old mechanics are true” [W. Pauli. The theory of relativity, p. 170]. We can add, according to the above cited Einstein’s statement of problem, Einstein thought the Newton law true also in the frame relative to which the material point (in that case electron) moves; so on the grounds of Newton second law we can immediately write the body’s trajectory and then transform the trajectory, not forces, in premise that there is no difference, whether we observe the accelerated motion of body in the moving IRF by the measures and clocks related to it or we observe the uniformly moving body. Basing on this technique, we get the possibility to compare the body’s motion in the resting and moving frames, doing not premising beforehand, which is the law of body’s acceleration in the moving frame under affection of force, and we can analyse, whether the basic law of dynamics is true and the patterns of trajectories of motion are comparable. If the pattern of variation of speed and acceleration in frame-to-frame translation conserves, the pattern of force’s affection is conserved, too. If the pattern of variation of speed and acceleration in frame-to-frame translation does not conserve, this will naturally mean that the pattern of force’s affection changes. To reveal it clearer, we will not premise that in the resting frame the mass of body changes with speed.

Thus, with account of premature conditions, suppose, in some resting (unprimed) frame, the body A having mass m moves with the constant acceleration a along the axis x. From a definite arbitrariness of choice of resting frame, suppose that at the initial moment of time t₀ = 0 the body was at the coordinate origin, i.e. x₀ = y₀ = z₀ = 0, and the speed of body at this moment is zero, too, i.e. u₀ = 0. As the body moves along x, we can further disregard the coordinates y and z, they will not vary.

Proceeding from the statement of problem, the body’s trajectory will be

and its momentary speed

To be able to study the pattern of body’s motion in the moving frame, suppose, this (primed) frame moves respective to the resting frame with some constant speed v with +x , at the initial moment of time the coordinate origins of both frames coincide and t'₀ = t₀ = 0, which is in a full agreement with the initial conditions usually used by Einstein to simplify the computations..

We will translate from the unprimed frame to that primed through the trivial Lorentz transforms

Substituting (d1.12) into (d1.14), yield

In the yielded equations (d1.15) the parameters of primed frame depend on the same parameter of time in unprimed frame. And if we eliminate this last parameter of resting frame, we will yield one equation describing the motion of body A in the physical time of moving frame. As we are interested in the trivial appearance of regularity for coordinates in time, let us represent the second equation of (d1.15) as

The solution of this quadratic equation is

The equation (d1.17) to provide the equality t'₀ = t₀ = 0 at the initial moment, choose minus before the root, so the solution will take the appearance

Now, substituting (d1.18) to the first equation of (d1.15), yield

where

The yielded expression (d1.19) describes the sought pattern of motion of the studied body in the moving frame from the view of Relativity. From the solution we immediately see that time during which the body can accelerate is limited from the point of moving frame; it is determined by the interval in which the value B remains constant, while according to (d1.12), the body can move in the resting frame unlimited time. We could speak here of impossibility to move with the speeds exceeding the lightspeed, but then the coordinate of body would not become imaginary, it would remain some constant speed when reached the lightspeed. While here we deal with a clearly illogical transformation which the Lorentz transform suggests.

VLADIMIR: As I see, the expression (d1.19) can be effectively simplified. Opening brackets, yield

SERGEY: I agree, and this is important for further operation with the yielded solution. We are interested in the law of body’s acceleration from the point of moving frame. The second derivative of coordinate in time in (d1.20) is

The graph plotted with (d1.20) is shown in Fig. 3.

Fig. 3. The body’s acceleration against time in the moving frame with the speed of moving frame v = 0,1 c, m/s, and constant acceleration of body in the resting frame a = 0,001 c, m/s²

And what is your impression of this regularity, dear Vladimir?

VLADIMIR: Before I answer, can I, dear Sergey, tell my impression of the problem you showed? Accelerations are surely very painful point of Relativity, real ‘nail in a shoe’, and quite large nail. Of course, they pretend that there is no ‘nail’. For example, their ‘Talmud’ – L.D. Landau and Ye.M. Lifshiz, vol. 2, “Field theory” – mentions acceleration in passing, in few lines in the end of section 7 which whole is less than a page; this means, accelerations are such trifle that nothing to say about, nothing unclear with them. And then they give the problems on ‘relativistic uniformly accelerated motion’. What follows from it? First, that they think c invariant in all cases. Without any proof. So to say, ‘the wife of Caezar is above any suspicion’. .

SERGEY: I would say, dear Vladimir, that basically, the uniformly accelerated motion is not such in Relativity. In the book you mentioned, acceleration is taken constant relative to its actual frame: “Relativistically invariant condition of uniform acceleration has to be represented as the constancy of 4-scalar which coincides with w² (where w is the 4-acceleration – Sergey) in its own (! – Sergey) reference frame:

… In the ‘resting’ frame

[ L.D. Landau and Ye.M. Lifshiz, vol. 2, Field theory, p. 39]. As you can see, in relativistic calculations also, with all that they think the acceleration already not a vector but a scalar and they don’t solve the problem straight, as they do it with uniform speeds (though, I already said, nothing differs the technique to observe the accelerated motion from that to observe uniform motion), and they use the pseudo-accelerated frame through the permanently changed inertial frames, having not substantiated the validity of such use, – with all that, relativists yield non-uniformly-accelerated motion in the stationary frame.

And this is not the feature of Landau’s representation. For example, W. Pauli gives same definition of uniformly accelerated motion: “As the uniformly accelerated motion in relativistic kinematics (! – Sergey) we may naturally (?! – Sergey) think such motion for which the acceleration has permanently same value b in the frame K' accompanying the body at the moment (! – Sergey) or the material point. The frame K' for each moment of time is other; in one definite Galilean frame K the acceleration of such motion is inconstant in time (just what you and me yielded in calculation – Sergey). Such are things with rectilinear uniformly accelerated motion” [W. Pauli. The theory of relativity, p. 114].

Furthermore, for this acceleration, the 4-interval also is changed in Relativity. In case of uniformly accelerated motion in the resting frame, it has already other appearance: “A special interest there have the formulas for translation of acceleration from the frame K' that accompanies the body at the moment to the frame K relative to which the body moves with the speed u. If we direct the axis x with the speed, we will yield for this case… From (194) we easily yield through integration:

[W. Pauli. The theory of relativity, p. 113–114].

With it, the very fact that the 4-interval changes in case of accelerated motion evidences, relativists might not expand the regularities of uniformly accelerated motion onto the accelerated on the whole and the more use them, translating into so-called actual frames, if on the whole, even in non-accelerated frames, their 4-interval does not remain in its initial appearance. We see from (d1.24) that at b = 0, i.e. in absence of acceleration in our understanding, the right part of (d1.24) does not become equal to c² , neither to ds², but turns into infinity. This means, in Relativity, the condition b = c corresponds to the absence of acceleration, which is nonsense, as, I can repeat, according to Relativity, we have to think the uniformly accelerated motion “such motion for which the acceleration has permanently same value b in the frame K' accompanying the body … at the moment”.

As you see, here they also play on the ‘obviousness’ of classical concepts. When any physicist is said of uniformly accelerated motion, he understands this term in the exact meaning and interrelates it with the stationary (lab) frame, not with the frame accompanying to the accelerated body at the moment.

VLADIMIR: Indeed, for kinematic theory that describes translations between IRFs (!), the translation to the accelerated ‘accompanying frame’ is a prohibited operation. I understand so: it is something like ‘retroactive law’ or, as the humorist said, ‘seven write, three keep in mind’. These are some tentacles of GRT seen from the spine of SRT. But however they present it, it seems ridiculous. Only the formula (d1.24) looks really strange, as the negative left part is equalised to the positive right part. Is it possible, Pauli has mistaken?

SERGEY: This is shrouded in mystery, dear Vladimir. See yourself, the system of equations (194) to which Pauli refers in the citation is

In its turn, this equation Pauli yielded, joining two systems of equations for each frame (Pauli’s p. 113):

and the equation of interrelation

The substitution of (d1.26) and (d1.27) into (d1.28) basically has to give the following expression for the first expression of the system (d1.25):

not to the expression which Pauli wrote. But even if the first expression of Pauli’s (d1.25) is true, – and Pauli says, “these relationships ((d1.25) – Sergey) are already in the first work by Einstein” (I can mention, I did not find it in the first either few first works by Einstein – Sergey) [W. Pauli. The theory of relativity, p. 114]. Then, given the statement of problem

the prototypal curve has to be found from the expression

Perhaps you will agree, it is not so simple to find the prototype of the left part of (d1.30), and we have to find the second interval, which will be more difficult. So I approach this matter very simply: several generations of relativists have been educated on this book, and we still heard no claims. The more that according to Born, the left part of (d1.24) is not negative: “We can call

‘the 4-D distance’, but we have to bear in mind, we use this expression only symbolically. Proceeding from our invariant F, we can easily interpret the real meaning of the value s. Let us confine ourselves to the plane xt (or xu); then

Further, for any space-like world line, F is positive, which means, s as the square root of positive number is a real value. Then we can make the world point (event) x, t simultaneous with the origin, by way of proper choice of the frame S. At t = 0 we have s = (x²)^1/2 = x as the spatial distance from the world point to the origin” [M. Born. The Einsteinian theory of relativity, p. 298–299]. Note, both in (d1.31) and in (d1.32) the 4-interval has been written as in Pauli’s (d1.24). Though you are right, of course, for the trajectory of material body the difference in the left part of this expression has to be negative at v < c . Such arithmetic, dear Vladimir. So the most reliable is to analyse the results shown by relativists themselves, and this is enough to show their approach to the physical problems untrue.