V.5 No 2
|Supplement 2. The answers to Sergey Khartikovs questions
3. Citation: Sorry, in this item I see you confining yourself to some Brief course which you initially did not show. The issue of linear element of metric is reduced in this case to the fact that in passing to GR, Einstein had to retain the basis of SR in the local sense, i.e. in constancy of gravity potential, and namely in this sense Einstein required, the elements of metric to be linear, as in SR the metric of 4-D interval is linear. Moreover, in this local metric Lorentz transformations have to be true.
Sergey, in presence of gravitation, all solutions of GR have a non-linear metric. In Schwarzschilds work he did not state his metric linear (it only in the limit, at the infinity, tends to become linear). The words linear element which Schwarzschild uses mean in mathematical analysis only one thing: the differential of length of the arc ds. You saw the discrepancy that Schwarzschilds solution is nonlinear, while Schwarzschild said, linear element . Only in this meaning I objected: Schwarzschild did not state the metric linear and used this word so as it is done in mathematical analysis. Of Einsteins requirements I will tell below.
Nothing new is that relativists write one, say second and mean third. They say of the equivalence of inertial and gravity masses and mean the equivalence of laws in IRF and NRF. They say of intrinsic RF and use it as an accompanying one to study the non-inertial motions. They say of full identity of IRF in SR and separate RFs into the accelerated and being at rest they say and mean many things. Einstein also said much that the metric of SR and GR has to be locally linear. You surely well know it, and we said it in our Supplement 1. If the metric was linear, the postulate of constant speed of light is violated. Due to this, the 4-D interval and its invariance lose their sense. Judge for yourself, which sense can be included in the interval
No sense. This interval has no invariance. It does not determine a real trajectory of a body. The Lorentz transformations are not true for (20). It only comes to paradoxes of zero interval in a near-light-speed motion of the body. But on the other hand, without the 4-D interval, GR is impossible. Einstein has found the only chance - to retain this interval locally. For it, he needed, the metric with which the interval was postulated in SR to be linear. So Einstein said of local metric, and we showed it in the Supplement 1. While the nonlinear metric is senseless in GR, as in GR only local operations in the metric are admissible. And generalisation of relativity principle onto a NRF has brought Einstein to a deadlock: Last years I tried to build the general theory of relativity, proceeding from relativity of non-uniform motions also. I thought, I actually have found the unique law of gravity, which relates to the general postulate of relativity understandable in its sense, and tried to prove the necessity of namely this solution in the work that appeared past year in this journal . However, the re-analysis showed that, following the suggested way, it is absolutely impossible to prove somewhat; this what seemed to be done nonetheless, was based on misunderstanding. The postulate of relativity is true to the measure I required always when we take the Hamilton principle as the basis; but factually it disables us to determine the Hamiltonian function H of gravity field. In reality, the relationship (77) of the cited work that limits the choice of H expresses nothing else than the fact that H has to be invariant with respect to linear transformations, and this requirement has nothing in common with the relativity of acceleration [4, p. 425]. To the point, this citation is a good additional confirmation that the principle of equivalence of physical laws in IRF and NRF (which we discussed above and will discuss below in connection with your confidence that it is true) is absurd. In frames of the answer to your question of requirement, the local metric to be linear, it remains to mark, namely this Schwarzschild took as the basis to model his metric. He took completely the Einsteins statement of problem. And as you have his paper, you can read in it the following: Mr Einstein showed that in the first approximation the solution of this problem leads us to the Newton law, and in that second it correctly describes the known anomaly in motion of Mercury perihelion. And the below calculation gives the exact solution of the problem The following lines will add more accuracy to the brilliant result of Mr Einstein [2, p. 201]. Thus, taking the metric as (1), Schwarzschild had to check its local linearity. He did not, and you omitted this important point when formed your question. While you had to check. It was incorrect to solve the problem without this check, which we did and which showed the Schwarzschilds initial modelling metric to be fully illegal.
4. Citation: You, dear Namesake, are neglecting the basic aspect pointed in our paper - namely, that all computations in the BH conception premise a free fall of substance onto the centre of gravitating body. This is the matter of principle.
In fact, Schwarzschilds solution and Landau - Lifshiz solution are not the BH theory. Schwarzschild has distinctively said, which problem and in which approximations (the gravitating body taken as the point) he solved. Landau and Lifshiz said clearly that they neglect the pressure of substance (page 406). If you want to criticise the theory of black holes, you have to address to other works, not to these. I would point again, in GR for a distant observer the body never goes under the event horizon.
Well, such your position is a surprise for us. Before, you expressed a basically other opinion: Then you are writing, the radius of gravitating body which in Schwarzschilds problem was infinitesimal. While in fact, in Schwarzschild problem it is considered the centrally symmetrical gravity field created by the spherically symmetrical distribution of substance. So it is absolutely unclear, why it seems to you as if conditions to apply (53) (of our paper - Authors) have not been satisfied. In fact, all conditions have been.
Unfortunately for you, we have to mark, neither your previous opinion nor this position which you are trying to take after our argumentation is valid. First of all, you agreed that Schwarzschild solved just the problem of point gravitating body. Well, how could you disagree when the very name of his paper - On the gravity field of point mass in Einsteinian theory - says of it? You also lifted your question and doubt, whether the relativistic problem is static. After this you admitted an unavoidable change of metric inside the gravitating body, as well as decrease of gravity potential to the centre of body: But his solution is true for any centrally symmetrical distribution of substance. We can consider, say, a case of a homogeneous sphere. Then out of its surface the metric will be determined by the clean Schwarzschilds solution, and within the sphere it will be determined by the same equations but with other coefficients - just as in Newtonian case we have to account the lessening gravitating mass as we deepen into the sphere. But in Newtonian case also, for the point source there appears the singularity in the centre! The fact that you, having admitted the lessening of gravity potential, at the same time claim of singularity in the centre, can follow only from an incomplete understanding of the whole amount of information which has fallen onto your head. Pity that you are omitting the fact on which we accented your attention in the first supplement to the paper: the singularity in Newton law is only a consequence of approximation which is true at a considerable distance from the gravitating body.
Additionally to this, you have admitted that we have to account the counter-pressure, and even said, Landau - Lifshiz solutions are not the BH theory. On one hand this shows that you agreed with our arguments whose account fully disproves the possibility of unlimited collapse of dust sphere, but on the other hand, we can find alike approach in other literature on BH. For example, Penrose wrote so: General pattern is well known for spherically symmetrical body. If the mass was large enough, the finite balanced state does not exist. After the radiation carries away enough of thermal energy, the body begins to shrink, and this goes on up to achieving the physical singularity located at r = 0 [9, p. 390]. Furthermore, in the problem of dust sphere Landau uses the same Schwarzschilds metric without any changes in passing from the outer to inner space of the dust sphere; this means, he uses the same statement of problem which is applied in the relativistic modelling of BH. With a great part of stretch we could imagine that Landau considers the metric of usual gravitating body, but it was namely Landau who wrote: Let us draw our attention that in all cases the moment of passing the surface of collapsing body under the Schwarzschilds sphere (r (, R0 = rg )) is of no significance for its inner dynamics (described by the metric in the related reference frame) [3, p. 405]. From your point it looks like the body freely shrinks into a point, the surface goes under the Schwarzschild sphere - and does not become a black hole? Its interesting and even intriguing.
Speaking of other works, we would like to notice, in the paper we gave not only the analysis of collapsing dust sphere but also the Oppenheimers solution for the case of collapse of dying star. We showed that the condition of zero pressure within the sphere, as well as a non-decelerated fall of substance to the very centre, is determining for this problem. Developers of the BH conception widely use Oppenheimers solution and they used to refer to it. Moreover, in limits of BH conception and keeping at least outward formality, you cannot show us other conditions, as these conditions are objective in limits of BH modelling. In presence of counter-pressure, the metric inside the sphere will not be singular at all and its appearance will not be limited by the change of coefficients. Thus, the grounds on which Schwarzschild selected his metric disappear. So, even if some today author writes that he accounts the counter-pressure but uses the Schwarzschilds metric, he will only show himself to be not objective and, truly speaking, ignorant in physical processes. The problem described by Landau directly relates to the BH conception and you have no grounds to say the opposite, even if you want much. Although we understand, this leads straight to deleting a part of material in the popular course of theoretical physics. Pity.
At the same time, a considerable transformation of your opinion that you are denying now Landaus concern to the BH conception speaks only that our analysis is correct and is deeper than relativists understand these physical processes. And you have no counter-arguments to our analysis.
In this item, there was one more issue, which we have to enlighten separately.
Citation: I asked you, do you understand the difference in R and r?. Now you have a case to compare your knowledge with the standard mathematical formalism, in accordance with which in the end of problem you have to return to the initial coordinates, to put the solution in correspondence with the conditions of modelling.
I am substituting the solution into the initial equation, in the initial coordinates: the metric appears nonlinear (and I did not deny it), the metric tends at the infinity to become linear (as it was required). I asked you: what namely is not true? In your work you wrote on this subject, the conditions (25) are not true. I reminded you and am reminding again, (25) is that to which the coefficients have to tend at the infinity - and you wrote in the end of (25): at x1 = . Please explain, what are you meaning under the untrue conditions (25).
If we compare our citation with your answer/question, we will see a clear inconsistency. We said in our citation of necessity to return to the initial metric after having found the solution in the artificially introduced coordinates. Your question relates to the inconsistency in coefficients of final solution and conditions for the determinant that we touched in our paper by the way.
It is seen in the form of your respond, you well understand: in the Supplement 1 we gave a proof of absence of any out-of-centre singularity in the Schwarzschilds metric and after this any points of Schwarzschilds derivation become senseless. But we will answer.
We have to consider this question in several planes. On one hand, as we admitted in the previous Supplement, we actually did not account the condition x1 = . Yes, you can call it a mistake, as we had not an intention to show Schwarzschilds derivation in all its depth. We thought, his mistake in phenomenology description and mistake in passing to the x-metric are enough to stop any further consideration of the following derivation. We thought, the shown mistakes in the beginning of Schwarzschilds derivation are sufficient, the whole derivation to be recognised mistaken without reference which you caught. Well, in draft we had the whole analysis and showed it after you rose this question. Wishing or not, you have agreed with most our proofs. Thus, basically we answered your question and corroborated our conclusion: the Schwarzschilds derivation is wrong and it is illegal to model, basing on the Schwarzschilds metric and BH conception with the singularity out of a point gravitating body. In this sense your above statement, The singularity for the point source appears in the centre and at the Schwarzschilds radius, is erroneous.
But there exists another side of the medal. In the Supplement 1 we showed, the relativistic statement that locally linear metric of SR remains has not to be limited to phrases to which it usually was reduced. With it we showed a quite obvious criterion of consistence of the metric with this condition. According to this criterion based on the metric conservation in the local sense in Lorentz transformations, the initial Schwarzschilds metric is transformed in full accordance with conditions that we wrote for an infinitely distant point, which makes senseless any Schwarzschilds calculations. This is objective, as the local linearity of the metric just premises, there are true the conditions similar to those at the infinity where, as you said, it only in the limit, at the infinity, tends to that linear. We can only pity that this was not checked at due time by relativists who more than 80 years did not care to interrelate their statements with the formalism, confining themselves to the references to locality without a proper substantiation of this condition.
Furthermore, if speaking of Schwarzschilds mistakes, you have omitted one more basic point of our previous post to you, which touched the equatorial location of the point x2 = 0 at which Schwarzschilds sought the solution. This also is a mistake. Or rather, even not a mistake but a silent substitution with whose help he could, at all, yield whatever solution. We suggested you to show us the solution out of this ersatz-equatorial point which, after Schwarzschilds opinion, had to be identical to that he yielded. Have you a problem? Really? But you cannot deny, if the solution generally existed and was explicit, with full symmetry of the problem is has to remain explicit out of the ersatz-equatorial point. Not in vain Schwarzschild, before he published his derivation of solution, wrote so: Due to the symmetry about the turns, it is sufficient to write the field equations only in the equatorial plane (x2 = 0) [2, p. 203]. This means, we can yield this solution with, e.g., the equal-to-unity Jacobian and independence of the x-metric in Schwarzschilds derivation for the coordinate system turns. Your silence says, you understand: it is impossible to yield this solution, as the point is not equatorial and there is no symmetry of which Schwarzschild said. This also is the basic mistake and even, we can say, intentional adjusting of the result, although in the light of absence of event horizon as such, this all loses any sense. Nothing to discuss.
5. Citation: As to the collapse of dust sphere. I selected you exceptionally because in your first posts you seemed to be different from them. (Sergey, what for do you take this tone?)
I already wrote, in GR, for a distant observer a body will never go under the event horizon. What do you dislike, disagree? Or do you know the source of GR where something other was written?
As to my tone, these are emotions. I wrote you of it in the beginning, and you did not take my tone as improper. Something has changed? Let us discuss without emotions and to the point. Of course I know the difference in the intrinsic RF for which the calculations as if are made, and time interval for a distant observer. At the same time, if in BH for an extraneous observer the body never (i.e., during any finite time) goes behind the event horizon, the event horizon never can form, as the substance will fall infinitely long time onto the event horizon, doing not increasing the mass under Schwarzschilds sphere. But if we consider the process of so-called free fall of substance from the very beginning, when the radius of surface was much more than Schwarzschilds radius, we easily come to the fact that the main part of collapsing substance is unable to penetrate under the Schwarzschilds sphere. This means, the initial gravitating centre will not have the attractive force to provide the collapse. And we should not forget, in the Schwarzschilds derivation the whole substance of gravitating body was located in the physical point. This means, it already was under the Schwarzschilds sphere, i.e. in some way it already appeared there during some finite time before we start observing the collapse. Thus, the relativistic conclusion of infinite time, during which the outer observer as if will see the surface going under the Schwarzschilds sphere, will only show, how much mistaken was the derivation on whose basis the conclusion was made, as the very idea of BH will be simply absent and nothing to discuss. All consequences, in that number impossibility, this object to emit the light, can be revealed only if between the substance and outer part of the body there is a singularity in the form of event horizon. Thus, when you ask, Or do you know the source of GR where something other was written? - you and me, we both know. On this source as on the basis, the whole BH conception has been built. This is the Schwarzschilds work that we just are discussing. See, as we said above, Schwarzschild considered his model of the point gravitating body and yielded the event horizon not in the centre. Hence, the whole body was under this horizon. ;-) And if speaking of infinite time during which the surface goes under the Schwarzschilds sphere, there is nothing except usual relativistic fantasies. Actually, where from the condition of infinite time is derived? In reality, the very so much accustomed derivation on the stretch to infinity of time of approximation to SSh is yielded from the fact that the world line of the ray escaped the bodys surface however close to SSh goes however long (in time of any system!) near the world line of the point SSh [6, p. 401]. But this is studied with the formed event horizon, i.e. after the substance already came under the Schwarzschilds sphere. This discrepancy also originates from that Schwarzschilds paper. Specifically, from the expression (17) of Schwarzschilds paper:
(giving the time scale) [2, p. 206] and from (16) of the same paper [2, p. 206]:
In particular, from this second expression it follows that the light speed is constant and the derivative of with respect to s has to vary in inverse proportion to the squared distance R. But the light speed in the gravity field is inconstant, it grows with the growing gravity potential after the Einsteins law
Doing not touching yet, whether (23) is true (there will be your related question below), we can conclude that the light speed has to grow in proportion to the potential. Not in vain we in page 6 of our Supplement 1 raised the question, is it legal to represent the time parameter as the product of time by the light speed. With it, the time parameter of 4-D metric depends on spatial coordinates and the differentiation with respect to this parameter much complicates and becomes irrelevant to the Schwarzschilds operations. This also is the Schwarzschilds mistake. Gross mathematical mistake, due to which he got into the discrepancy with Einsteins formula, in accordance with which the light speed grows with the growing potential. Thus, when you are trying to find mistakes in our studies, we would be grateful if you first caught the mistakes in the conception which you defend, doing not accusing the opponent that he has to do it instead you and at the same time taking offence at him.
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