Dmitri Rabounski wrote:

Dear Prof Bruhn,

Thank you for your effort to pay attention to Crothers' papers on the black hole problem. I looked through your web-page you referred for. I however found, in brief, that you are still talking about different things than we are. We told you about the representation with the observable quantities which give really observed effect, while you told about the coordinate quantities. Of course, no one discuss that the black hole solution can be obtained with the coordinate quantities where the spatial section is free of fixation so that the space-time is not split into spatial sections pierced by time lines, but is still remaining a whole continuum. But all that had been said by me (and also by Larissa), fighting for Crothers' papers, is related to physical observable quantities which give really observable effect and no the black hole solution on Schwarzschild's metric.

You're waiting for my discussing incorrecteness of Crothers' claim or that by yours, but these are different approaches which give different results. I only try to give you understanding that there are two different mathematical approaches to the same problem. The metrics g and h are the same; the are different in the properties, and so on in the sequel. Meanwhile, Crothers' approach can be bound meeting close to observations, because the based on observable quantities, in contrast to the coordinate representation used before.

Thank you for your attention,

Dmitri

you wrote as essential in you last email: ''. . . these are different approaches which give different results. I only try to give you understanding that there are two different mathematical approaches to the same problem. ... ''

This statement of yours is incorrect. The advantage of mathematical methods is just that one can check the correctness of considerations by using different mathematical approaches.

If there are two authors which obtain contradicting results from approaching the same problem by two different mathematical methods then at least one of them is in error.

So who is in error in our case???

The problem is to describe the spacetime manifold that is given by the Schwarzschild metric.

(i) Standard methods of the theory of manifolds yield that the Schwarzschild metric has two disjoint validity regions S' and S'', while from your side the existence of one of both, namely S'' (0 < r < r_s), is repeatedly denied.

(ii) Standard methods of the theory of manifolds, namely the transformation of the Schwarzschild coordinates t,r to Painlevé-Gullstrand coordinates T,r yield that die validity regions S' and S'' are belonging to the same manifold S.

(iii) Standard methods of the theory of manifolds, namely the discussion of the spacetime manisfold S by means of the Painlevé-Gullstrand coordinates, yield the existence of Gravitation collapse which is denied by your side.

So please take note of the fact that there are essential contradictions of your claims to the results of standard methods of the theory of manifolds.

Since the means of getting the results of the theory of manifolds by transforming Schwarzschild to Painlevé-Gullstrand coordinates are completely elementary you are challenged now to explain what is wrong in the Standard methods.

Conversely I have repeatedly pointed to the errors in the considerations on your side. Especially your argument of "observability" is strictly connected to the Schwarzschild metric in S'. You did never answer to my question what happens to your observability concept in case of say Painlevé-Gullstrand coordinates.

Kind regards

Gerhard W. Bruhn

Dmitri Rabounski schrieb:

Dear Prof. Bruhn,

Different pictures can be obtained in different coordinates. That's as a matter. For instance, as you know all the effects of General Relativity can be obtained in the curvilinear coordinates of a Minkowski's space, etc. I therefore think that your words are not related to the considered problem.

No doubts that the same method should give the same result with no dependance of the number of the authors, or their personality. No one disputes you that the gravitational collapser solution can be get with the usual technics. On the other hand, another method gives another result. A condition true with one coordinate technics can be denied if represented with another coordinate technics. That's Crothers' case. No one disputes with you about the usual technics or results obtained with. That's normal. I therefore think that there is no reason to discuss anything.

Sincerely yours,
Dmitri

you wrote:

''Different pictures can be obtained in different coordinates. That's as a matter.''

However, geometric properties are those which are invariant under changes of coordinates. Other ''properties'' are only reflecting the properties of the coordinates (which are not of interest) This insight has led to the invention of tensor calculus where the transformation properties can be seen from the indices.

''No one disputes you that the gravitational collapser solution can be get with the usual technics.''

However, your author Crothers the articles by whom you have published is denying that.

''A condition true with one coordinate technics can be denied if represented with another coordinate technics. That's Crothers' case.''

NO!!! See above what I have told you about geometric properties. True in ONE coordinate sytem means NOTHING. A geometric property is true under ALL coordinate systems.

Our discussion has shown that there is a lot of math you have to think over. So let's close the discussion for this time.

Thanks

Gerhard W. Bruhn

PS I shall post this discussion with you as an appendix to
http://www2.mathematik.tu-darmstadt.de/~bruhn/toRabounski030508.html

Dmitri Rabounski schrieb:

Dear Prof Bruhn,

The properties of the coordinate systems we refer for w\are those standards we observe. These are really observed quantities, in contrast to the formally compoinents of the general covariant quantities. That's as a matter. If you are not interested in the result really expected in practice, this is only reflects your scientific interest. (Many people are working on abstract results, nothing insulted.) Crothers used another technics that the mathematical apparatus of general covariant quantities, so he gets another result. You try to analyse his result from the basics of general covariant technics, so you have that you get.

I only try to give you this idea, as a corner-stone of your problem with Crothers' papers on the gravitational collarger problem. Nothing more, just this one.

Thank you for your responsibility,
Auf wiedersehen!

Dmitri

Betreff: Re: Open letter to you, editor-in-chief of PP
Datum: Fri, 9 May 2008 09:21:58 -0700 (PDT)
Von: Dmitri Rabounski
An: GWB

Dear Prof Bruhn,

I pity that you dislike to study the coordinate methods in the General Theory of Relativity. These methods are described enough in Both The Classical Theory of Fields, Fock's book, Zelmanov's book, and many others. I appreciate your belief in the general covariant methods, but the professionally relativists, who are me Dr Borissova and many others know many other methods, the coordinate methods which are different from the general covariant methods you belief. Those are different methods which can give different results on the same phenomenon in consider. If you would take an interest, and many years of research in General Relativity, you would appreciate many methods, not only the general covariant method, I sure.

Thank you very much for your attention to my letter(s).

Auf wiedersehen,
Dmitri

I have already once replied to your opinion that . . . Those are different methods which can give different results on the same phenomenon in consider. . This statement of yours is WRONG:

Let me repeat: In mathematics different methods applied to the same math problem cannot give different results. Then there is only one conclusion: One of the opponents does not do his math correctly.

And let me add this: Your informations about your sources are remarkably imprecise. I'm prepared to clear up any alleged contradiction if you give me detailed information (author, book or paper, page and equ. no.) as is use in science. However, your citation method is completely insufficient. Then we shall immediately recognize WHO is going astray. As I have pointed out e.g. the inexact informations given by L. Borissova cannot be found in Landau-Lifshitz' ''Field Theory''.

I don't know from where you take your claim that I should dislike coordinate methods. The only thing I'm really disliking in science are preconceived ideas. Special coordinates are advantagous for the formulation of fundamental laws. But afterwards it is necessary to obtain a formulation that is invariant under changes of coordinates which are required by the symmetries of the problem under consideration, e.g. in case of spatial spherical symmetry, think of the Schwarzschild metric which can be transformed to several other coordinate systems of important physical meaning, e.g.

− the Painlevé-Gullstrand coordinates for from infinity radially infalling (or outgoing) test-particles,

− the tortoise coordinates (t,R), which have the property that radially infalling and outgoing light rays satisfy
                t + R = constant         or         t − R = constant         respectively.

− the Eddington-Finkelstein coordinates for light radially infalling (or outgoing) from (to) infinity ,

You mentioned V. Fock's work. Surely, V. Fock was an outstanding physicist as can be seen at
        http://people.bu.edu/gorelik/Fock_ES-93_text.htm
He introduced the concept of harmonic coordinates x^λ satisfying
                D^κD_κ x^λ = 0         (λ= 0, 1, 2, 3).
where D^κD_κ denotes the covariant d'Alembertian.

However, just these ''harmonic coordinates'', though surely being of some theoretical value, had less success in practical problems. For instance, all coordinates with spherical symmetry characterized by a metric of the form

                ds˛ = g_tt dt˛ + 2 g_tr dtdr + g_rr dr˛ + r˛ (dθ˛ + sin˛θ dφ˛) ,

where g_ttg_rr−g_tr˛ is independent of the angular coordinates, fail to be harmonic as can easily be shown by checking the well-known criterion

                ∂_κ (g^½g^κλ) = 0 .

Take λ = 2 ~ θ to obtain

                ∂_θ (g^½g^θθ) = (g_ttg_rr−g_tr˛)^½ ∂_θ sinθ ≠ 0 ,

which is especially fulfilled by the Schwarzschild coordinates.

Seeing forward to your response with precise information about your sources. Otherwise any further discussion will be useless.

Best regards

Gerhard W. Bruhn