SELF

2

  On conservation of energy and impulse

VLADIMIR: It seems, now I have a prompt where to are you driving. But if I caught it correctly, you even more have to improve your text in the page 73.

But return to the items of your post.

To the item 1.

“Just in this meaning of balance of momentary values of impulses and energies ‘before’ and ‘after’ we write the expressions for impulse and energy in interactions within the set of n bodies that raised your protest”.

It does not follow from the text of your paper that the “balance” of energies and impulses has been written for some “interactions within the set”. On the contrary, we have there such statements: “where 1 and 2 denote the sum of impulses before and after some process occurring in this frame, then in the another frame”, “Now, to yield from (6.43) and (6.44) the energy equation for the moving frame, we only have to account that the second summands in the right parts of these expressions are equal”.

Thus, we find in the text no word about the “interactions within the set”. So you have to admit that the summands mentioned in the second citation are not equal and that ‘the process’ is usually understood as the evolution in time. Well, if so, it is necessary to correct it immediately.

Now about the items 2 and 3.

3. Another thing that interrelation between the inertial and non-inertial frames does not keep, for example, the condition of conservatism, as in gravity field in IRF, the total impulse and total energy will permanently change, while in freely falling frame these parameters will remain constant”.

Just after these your notions I began to ‘catch’ the meaning of all previous constructions, but in the following way. You are thinking that ‘having escaped’ into the local freely falling frames, relativists believe that they secured themselves from every side and already need not IRFs. However, they are mistaken, as their ‘barrel-organ’ works only in strongly homogeneous fields which do not and cannot exist in nature. This is true. But I would say, ‘conservatism’ and work of conservation laws are ‘imaginary’ in Relativity. So I strongly disagree with your “in a homogeneous gravity field in freely falling frame, the laws of conservatism are true, i.e. the conservation laws for energy and impulse”. It temporarily seems to relativists that the conservation laws remain true, but when their ‘freely falling frame’ bangs down onto the surface of gravitating body, all their illusions will be dispelled as a haze.

Planets move only in inhomogeneous gravity field. But this is already another thing. Here they cannot even dream (temporarily) of such joy as conservation laws, I agree with you. I’m strongly sure, there is no and cannot be any equivalence between IRF and non-IRF, in any meaning, these are basically different motions. Basically different. The references that at some conditions something can ‘seem’ to the observer cannot serve as an adequate excuse.

But here we have to thoroughly put things on shelves, to sort out finally with such a wonder of nature as revolutionary motions which, to my mind, are an ‘edge’ between uniformly and rectilinearly moving bodies and bodies rectilinearly accelerating under affection of external force. The motion which is rather inherent only in material bodies, not in fields. Such motion with which all parts of a body are accelerated but the body as the whole does not change in time its energy and impulse.

SERGEY: Not simple, dear Vladimir, you see, not so simple task is to grasp these matters. And you caught well, not only in Relativity but in classical formalism also we see indefiniteness in modelling, as they both looked at many processes not from the point of scrupulous analysis but from the point of “it’s obvious”. Sometimes this so-called obviousness was taken instead the proof and analysis. But let us go sequentially.

You are saying, our work does not point that the balance of energies and impulses has been written for some interactions inside the system. At the same time you are citing our paper where we said, “where 1 and 2 denote the sum of impulses before and after some process occurring in this frame, then in the another frame”. Not as a result of external interaction but because of processes occurring in this system of material bodies. This means, the conservation law has been written for internal processes. And here reveals itself not an indefiniteness of proof in the paper but just the inertia of ‘obviousness’ which prevents to see the sense of written literally in the meaning put to the text. I can add, if you open any of our previous papers on this subject, you will see, we then were also under the load of ‘obvious’ and expected that in passing from inertial frame to non-inertial frame, the conservation laws have to fail even in homogeneous gravity fields. Moreover, even in the first version of discussed paper, in the calculation related to the above Fig. 1 we still thought so, and you were right when pointed us and helped us to throw it off, just as I from my side help you now to get free of it.

To reveal this load, I will try to show you that even if besides the gravity field some external force affects the studied set of bodies and does not affect the freely falling frame but only the bodies in it, in this case also, the laws of dynamics in the freely falling frame will remain true.

Consider the same pair of frames as in Fig. 1, but now suppose some external force affecting the studied body (see Fig. 2); the direction of this force does not depend on the position of freely falling frame. We can easily provide this last premising that the source of force also freely falls in the homogeneous gravity field but is not included into the considered model, it remains the outer source affecting the studied body; we often do so in IRFs.

Fig. 2. A body freely falling together with the primed frame in the outer gravity field that is under additional affection of the external force

From the point of resting frame we can write the second Newton law for this model as

where we can express the vector  in classical formalism through the respective vector of trajectory '  in the primed (freely falling) frame. So we yield

And as far as in the homogeneous gravity field

we yield the resulting

So to say, as was to be shown.

Thus we see, dear Vladimir: in the frame freely falling in a homogeneous field, actually, the same laws of dynamics work as in IRFs. If we don’t go into details of translation from IRF to non-IRF and back, if with it we omit the issue of translation between the open and closed frames, we will find no problem in, how the laws of dynamics work. And I showed you above, relativists have no intention to interrelate IRFs and non-IRFs. They feel well that the laws of kinematics and dynamics work in a freely falling frame, to introduce in this frame the 4-D interval and Lorentz transforms. And they don’t care of possibility to limit the motion by some surface in the time of freely falling frame. In the field of mythical black holes their frame can fall infinitely long time, and in it, as relativists say, the same laws of mechanics that they took from IRFs have to be locally true.

Here is another difficulty, dear Vladimir. Relativists accounted a very interesting feature of laws of dynamics in a freely falling frame which was disregarded by supporters of classical formalism. But they did not account that this correspondence of laws is true in classical mechanics in absence of transformation of space and time. Notice, we have analysed different aspects of translation and always premised as obvious (oh, again this ‘obvious’!) that the time goes similarly in both frames, that '  is measured in the same measure units as , that the mass of a body is independent of speed. And we yielded a good correspondence; it differed only because it is unaccustomed from the usual view in classical physics. At the same time we never exceeded the limits of classical formalism and yielded unexpected solutions within the usual mathematical tool and usual regularities. Relativists have not such advantages which allowed us to prove so easily the correspondence of laws in IRFs and non-IRFs. In their formalism, both spatial and temporal parameters are transformed even in passing from one IRF to another. This means, to prove the correspondence of laws, relativists might not rely on the basis which they essentially changed. They had to prove the correspondence within their phenomenology and their mathematical formalism, hadn’t they?

VLADIMIR: Our dialogue about understanding of formulas (6.38)–(6.44) clearly came to a deadlock, as you are saying, “the conservation law has been written for internal processes. And here reveals itself not an indefiniteness of proof in the paper but just the inertia of ‘obviousness’ which prevents to see the sense of written literally in the meaning put to the text”. I am ready to stop protesting at the point that when reading the text, I as a reader inadequately understood what the authors would like to say; and you authors can decide for yourselves, whether you will correct your text with my comments.

Now, after your explanation, I understood what you would like to say, comparing the bodies’ motion from the inertial and freely falling frames. You would like to say that “in the frame freely falling in a homogeneous field, actually, the same laws of dynamics work as in IRFs”. With this I agree, but I would add that the ‘homogeneous field of gravity forces’ is a mere mathematical model that roughly idealises the reality. The physicist has to say – yes, physical effect of likeness of laws is present, but they are not exhaustively same, as it is present in a short interval of time and at physically unrealisable conditions.

Special thanks for the considered problem with the force, it is pretty good. None the less, I believe, the expectation “that in passing from inertial frame to non-inertial frame, the conservation laws have to fail even in homogeneous gravity fields” did not mislead you, as in classical physics we always think that when the observer passes from IRF to non-IRF, it is same as to introduce an acceleration, i.e. the external affection, that appeared here from nowhere. The fact that when considering some models of interaction in non-IRFs, conservation laws for internal interactions within the set work, does not exclude the fact that on the whole, the interacting bodies of set in IRF – non-IRF translation gain acceleration which you had to compensate in the initial problem by a ‘homogeneous gravity field’. But then the conservation laws became true in a freely falling frame and untrue in IRF, i.e. the result of translation is same. While the form of dynamic laws remains only in particular cases and under some limitations of which we said above. In general case of non-IRF, neither laws of dynamics nor conservation laws remain true. However, maybe we differently understand the word ‘translation’.

SERGEY: However it would sound strange for you, dear Vladimir, despite your well grounded conclusions, I am far from the thought that we came to a deadlock. I’m even sure, we are in a good progress to understand the main target – to reveal, how much possible and legal it was when relativists introduced the equivalence between the laws in IRFs and non-IRFs. And we do agree with each other. The same as you, I believe that the laws of dynamics are highly limited in a freely falling frame, they are true just in a homogeneous field and exceptionally for material bodies. For fields the equivalence is broken, and we showed it in the subsection 6.4.2.4 of our work. Furthermore, both in the work and now, the translation of laws between IRFs and non-IRFs has been shown true, as opposite to relativists who postulated this equivalence, referring only to the acceleration of free fall that is same for all bodies located in a homogeneous gravity field.

But just as I pointed you in the beginning of our dialogue, now I would like to draw your attention again: we not simply analyse the possibility to introduce the equivalence between the inertial and freely falling frames. In classical physics we, generally, have not a great necessity in it, as well as the proofs that we gave in the paper and that I gave now in this dialogue are not complicated. Yes, this is an interesting regularity worth for attention of classical physics, the scientists to be not repeating from generation to generation same computations and, so to say, to do not invent a bike again. But with this all, classical physics has no necessity to establish the equivalence which would considerably change something in the formalism, because all translations are derivable in frames of same classical formalism. With the proof we, you and me, automatically defined the limitations in which the translations are correct.

In the relativistic formalism, – and I already said of it, – the equivalence is the matter of principle. If in a freely falling frame the laws of dynamics are true, relativists say that SRT can be locally translated into GRT and postulates of SRT generalised onto gravity fields. If such translation of laws is impossible, all constructions of GRT based on 4-D metric of Poincare – Minkowski – Einstein appear senseless, and tensor representation of curvature, geodetics introduced by them or kinematical limitation of laws of dynamics are of no help. The question factually is, whether relativists may use the joined representation of space-time, doing not limiting themselves to IRFs and uniform motions. This first of all is the question, whether all tensor representations are true in the Riemann space, as the matter is of dimensionality and properties of metric which they will then wrap by the gravity field.

We also don’t touch here the issue of approximate idealisation of mechanics. We compare and oppose not real but ideal mechanics. Perhaps you will not argue that Newton laws, as well as conservation laws that follow from them, in a definite meaning also are approximate, because of non-ideal interaction, energy dissipation, imperfect surface etc. The same, two bodies in the field of Earth will not fall simultaneously, and the difference will be caused by the degree of air resistance. Moreover, you will not argue that in some cases one of bodies or both can fall not down but upwards, against the gravity force, and without visible external affection. This does not cause you thinking that the gravity laws are wrong for such bodies. You find the regularities which do not distort the gravity law but simply complicate the model by additional interactions. In this way we introduce for ourselves a definite difference between the ideal model and real and versatile processes in nature. With it we use the laws of dynamics thinking them true in the very ideal system where dissipation is absent, all interactions are absolutely elastic and surfaces have a shape given by the mathematical equations, true? So we may not, using some idealisation in classical representation, disprove Relativity, proceeding from the non-ideality in real processes, may we?

The same with the issue that relativists are interested not in the very fact of translation but in the fact that the laws of dynamics are true in the frame freely falling in the homogeneous gravity field. I will try to explain you it on a simple example. Suppose, living in Moscow, you studied theoretically and experimentally the work of laws of dynamics, with account of a definite idealisation. Now you forward a task to see how these laws work in Vladivostok, thousands miles away from Moscow. You pack up your experimental devices, take a train and go to the Pacific ocean. In some time you arrive Vladivostok, unpack your devices and make sure, in this town the laws of dynamics are same workable. Are you interesting during this journey, how many points and stations have you passed, how these points work and which rules of railway make your journey secure? Hardly. You are interested in the very target – to transfer your devices to Vladivostok.

I would immediately point, agreeing with your objection: this approach is fragmentary; to build a good theory, we have not only to point the start and finish but to prove our way legal, i.e. to build yourself the railway, advancing from Moscow to Vladivostok. Basically, this is what we did, proving our transfers legal. But you know, relativists do not sequentially prove in frames of formal logic – they simply use the results which classical physics has developed directly or indirectly. So they don’t build the railway, they use the way of classical physics and forget, classical physics built the ways of its formalism in frames of its phenomenology. Relativists have other formalism, and if we approach the issue with rigour, they would have to prove the possibility to travel safely their own way – and you see, they avoid this issue silently. This is why I say, we are not in a deadlock. We long ago and safely have reached Vladivostok and showed, the laws of dynamics in frames of classical conception are there same, and we showed the limits in which this equivalence will be true. What will be interesting for everyone – to fill in the gap and to show which way relativists have built with their postulates and why they don’t want go their way, – and this is what I would like to consider with you, being sure that all other arguments, until we are discussing in premise that relativists travel the way of classical physics, are fruitless. Aren’t you against such approach?