V.5 No 2

1

Supplement 2. The answers to Sergey Khartikov’s questions

Supplement 2. The answers to Sergey Khartikov’s questions

After the publication of our paper on black holes and in parallel with that of Supplement 1, we discussed this subject with Mr Sergey Khartikov and answered in the Supplement 1 to a part of his questions. He expressed a doubt and asked new questions, though, as it will be seen, much transformed. Since at all events these questions concern the basic aspects of our paper and crucial points of understanding the black hole conception wrong, we decided to write the Supplement 2 as Mr Khartikov’s questions and our answers.

To make this material more visual, we denoted our previous answers to Mr Krartikov on which he bases in olive colour, his new questions in bluish-green, citations from literature - in usual for our journal green colour, and the questions will be numbered in red.

1. Citation: “As to the reecord of metric transformation, here you are multiply incorrect”.

Sergey, the formula (18) of your work means the calculation of Jacobian of coordinate transformation in the plane Euclidean space. In the Schwarzschild problem, the space is initially Riemann’s, so your formula (18) wittingly never will match the Schwarzschild’s Jacobian. You derived your (18) for the equation (16). Calculate the same for the initial equation (14) and you will see, the results will differ strongly by the multiplier r² sin   being just the Jacobian calculated by Schwarzschild, i.e. things fully match the coordinate transformation in the Riemann space. Although Schwarzschild has yielded his Jacobian in the immediate calculation, I was not lazy to calculate it in your way – and it coincided with the Schwarzschild’s result.

I would repeat this result, to avoid any vagueness: the root of determinant for the metric (14) multiplied by the Jacobian r² sin   coincides with the root of the determinant for the metric (16). Will you use the 3-D either 4-D variant, you will yield the same. I have calculated by three techniques - yours, immediate calculation and after Schwarzschild - they all gave same results. This means, there is no mistake in Schwarzschild’s work.

Let us start from your point as if we determined the Jacobian in the plane Euclidean space and Schwarzschild – in that Riemann.

In transition from the metric in rectilinear coordinates ((14) of our paper)

to the metric in spherical coordinates ((16) of our paper)

Schwarzschild used the standard transformation formulas namely for the plane Euclidean space ((15) of our paper)

which you certainly could not omit, emphasising the previous and following formulas. In the Riemann space, the transformation (3) generally is not true. In the previous post to you, besides the cited above, we wrote also much, in particular to this point: “In your meaning it would be correct to write

taking separately the metric tensor and coefficient F_ik , though in this case it appears also necessary to account G and H. ;-)”. After this, we substantiated the necessity to account these coefficients in transformation. In this light, it would be interesting to recall our claim to (1) in (2) of our paper. “In accordance with the conventional mathematical formalism of tensor analysis, in this case of orthogonal metric

[1, p. 6]. In (5) we see the Jacobian in most general form, which is used in transformations in curvilinear coordinates. Just on the basis of (5) we concluded incorrect the following Schwarzschild’s opinion: “With it the element of volume in spherical coordinates is equal to

so that the Jacobian of transformation from old coordinates to those new equal to r² sin   differs from the unity” [2, p. 201]. We also pointed in the paper that “at the next his step, introducing the transformation

Schwarzschild already could not satisfy the equality

as he had to satisfy another equality,

[1, p. 6]. So your claims to our paper have no grounds in this case. Not we but Schwarzschild transformed in the metric of Euclidean space.

The same, the description of calculation of which you are writing does not fit the mathematical operations that have to take place really in passing from (1) to (2) with the following passing to (9). The point here is not that “the root of determinant of the metric (1) multiplied by the Jacobian r² sin coincides with the root of determinant for the metric (2) (should you use 3-D either 4-D variant, you will yield the same)”. The point is the value of Jacobian itself – and it has to be determined namely by (5), which we gave in the paper, not by that which Schwarzschild gave and on which you insist. See, by some reason you have multiplied the root of determinant of metric by the ‘Jacobian’ in your meaning. While in the previous post we showed you, in this case, in accordance with the fundamentals of analytical geometry: “Schwarzschild’s metric will be already not a metric but some curve, surface or volume in 3-D rectilinear or spherical space”. With it the invariant of this surface sought by Schwarzschild will be

To this invariant there have to be involved just the coefficients H and G whose presence in our computations you claimed illegal. Well, no one argued either is arguing that “results will differ exactly by the multiplier r² sin ”. But considering the transition between coordinate systems, we have to speak just of the transition between coordinate systems. Again, the issue was and is that with the multiplier r² sin   we have generally to account the coefficients of the very metric. Most interesting is that you did just this in your calculations, while, when we say it, you do not understand and even think erroneous.