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WRONG "IDEES FIXES" IN GR

ANGELO LOINGER

Abstract. Six wrong idees fixes concerning the black holes and the

gravitational waves.

^^Idee fixe, n. Idea that

dominates the mind, monomania."

From "The Concise Oxford Dictionary" .

Idee fixe N°l

A widespread "Vulgate" of general relativity (GR) claims that if Schwarz-

schild problem - i.e. the problem of the Einsteinian gravitational field gene-

rated by a point mass M at rest - is solved using co-ordinate-free methods

(such as orthonormal bases, etc.), the result is necessarily the standard form

of solution, erroneously named "by Schwarzschild" , in reality due to Hilbert,

Droste, Weyl.

This is a wrong idee fixe: as a matter of fact, the procedure of the "Vul-

gate" is impaired by a vulgar logical fallacy, cf. e.g. R.M. Wald, General Rel-

ativity (The University of Chicago Press, Chicago; 1984), p. 121, eq.(6.15):

only by writing r^(di?^ -|- sin^??d(/3^) for the angular part, this author has

already chosen the standard HDW-form of solution. Remark that Birkhoff

theorem is not in question: indeed, Schwarzschild's original form of

solution 12 is diffeomorphic to the exterior part (r > 2m; m = GM/c?;

G is the gravitational constant) of the standard form. The interior part

(0 < r < 2m) of this form presents various mathematical and physical de-

fects 0; in particular, in the complete manifold of the HDW-form, i.e. for

< r < oo, it is impossible to assign the time arrow for every time geodesic,

according to rational criteria (private communication by S.Antoci). On the

contrary, this difficulty does not exist for the original Schwarzschild's form

0, which is regular in the entire spacetime, with the only exception of the

origin r = of the space co-ordinates, seat of the point mass M.

Idee fixe N°2

The undue privilege granted to standard HDW-form of solution of Schwarz-

schild problem and some physically senseless reflections on its interior part

(0 < r < 2m) have generated: i) the fictive notion of black hole (BH); ii)

the idee fixe according to which the continued gravitational collapse of a

sufficiently massive celestial body must give origin to a BH.

Thus, Chandra Digest of February 18th, 2004 tells us that "Thanks to

two orbiting X-ray observatories, astronomers have the first strong evidence

2

ANGELO LOINGER

of a supermassive black hole ripping apart a star and consuming a portion of

it. The event, captured by NASA's Chandra and ESA's XMM-Newton X-

ray Observatories, had long been predicted by theory, but never confirmed

. . . until now."

This is science fiction! In reality, the observations reveal only the action of

the very strong tidal effects produced by an enormous mass concentrated

in a "punctual" volume. The original Schwarzschild's ds^ [2j, for instance,

interprets perfectly the observational data.

The continued gravitational collapse can be described very generally in

a simple and concise way. Remark, first of all, that the necessary and

sufficient condition that a Riemann-Einstein spacetime admit the group of

spatial rotations is that its ds^ be reducible to the following form, where

r, 1?, </? are polar spherical co-ordinates [1]:

(1) ds^ = Ai{r, t)c^df - A2{r, t)dr'^ - A^{r, t)du:'^ ,

with

(2)

dw^ = di?2 + sin^ Mif^

Let us consider a collapsing spherical body of mass M . As it is well known,

the Einsteinian gravitational field outside the body is time independent, and

can be described consequently by the following ds^ |5j:

(3)

2m

W)

2m

W)

W{r)f - [f{r)f (d??2 + sin^ M^'^)

where /(r) is any regular function of the radial co-ordinate. If we put

r^ + (2m)'

1/3

we

fir) = r, we have the standard HDW-form; for r =

obtain the original Schwarzschild's form [2]; etc.

The physical results are independent of the choice of the function /(r) -

and this is true, in particular, for the above communication of the Chandra

Digest.

Idee fixe N°3

It consists of the conviction that in the motions of gravitationally inter-

acting masses an essential role is played by the gravitational waves. Now,

it can be proved with the well known method of Einstein-Infeld-Hoffmann

(EIH) that in a first approximation the above motions are Newtonian, then

the successive approximations yield some (relatively) small corrections. But

no gravitational wave is present ^ . In recent years I have given several exact

(i.e. non-approximate) proofs of the physical non-existence of gravitational

waves in the motions of the masses [7|.

WRONG "IDEES FIXES" IN GR

3

Idee fixe N°4

It is obvious that the notion of gravitational wave, as an object endowed

with a physical reahty, requires the existence of a class of physically privi-

leged reference frames.

Fock tried to prove quite generally that also in GR we have a class of this

kind: the set of the so-called harmonic systems But Fock's arguments are

not rigorous - and on the other hand GR, if properly understood, does not

allow the existence of privileged co-ordinate systems. Another vain attempt

is due to Bondi et alii J^, who tried to prove the existence of a class of

privileged frames insofar as the gravitational waves solely are concerned.

Idee fixe N°5

It regards the conceptual adequacy, for the treatment of the gravitational

waves, of the linearized version of GR. Now, in a beautiful article of 1944

Hermann Weyl proved the mathematical and physical inadequacy of the

linear approximation of GR, just under the above respect JHI- It is a pity

that the astrophysical community ignore Weyl's demonstration.

It follows, in particular, from Weyl's paper that the various computations

regarding the behaviour of the famous radiopulsar PSR1913-I-16 are fully

destitute of a physical value.

Idee fixe N°6

It consists of the settled belief that the gravitational wave generates a

certain gravitational field. Since this hypothetical wave does not possess

a true energy-momentum, but only a pseudo (false) energy-momentum (a

'mathematical fiction" , according to Eddington) the above idees fixe is pure

nonsense.

Final remarks

Many idees fixes regarding the BH's and the GW's are contained, respec-

tively, in the review articles by Celotti et alii and by Schutz jl2j .

In the golden booklet by Albert Einstein The Meaning of Relativity (1955)

there is no mention of BH's and GW's.

References

[1] G. Birkhoff, Relativity and Modem Physics, (Harward University Press, Cambridge,

Mass.) 1923, p. 253.

[2] K. Schwarzschild, Berl. Ber., (1916) 189; for an English translation see

arXiv:physics/9905030, May 12th, 1999. This translation has been published also

on Gen. Rel. Grav., 35 (2003) 951.

[3] Cf. e.g. A. Loinger, arXw:physics/0402088 vl, February 18th, 2004 - and the biblio-

graphy quoted here.

[4] See: J. Eiesland, Trans. Amer. Math. Soc., 27 (1925) 213; L.P. Eisenhart, Gontinuous

Groups of Transformations (Dover Publ., New York) 1961, sect. 58.

[5] Cf. A.S. Eddington, The Mathematical Theory of Relativity, Second Edition (Cam-

bridge University Press, Cambridge) 1960, p. 94.

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ANGELO LOINGER

[6] See e.g. L. Infeld and J. Plebanski, Motion and relativity (Pergamon Press, Oxford,

etc.) 1960, in particular Chapt. VI.

[7] Cf. e.g. A. Loingcr, arXiv:physics/0312149 v3, February 11th, 2004 - and the biblio-

graphy quoted here.

[8] V. Fock, The Theory of Space, Time and Gravitation, Second Revised Edition (Per-

gamon Press, Oxford, etc.) 1964, passim.

[9] H. Bondi, F.A.E. Pirani and L. Robinson Proc. Roy. Soc, A 251 (1959) 219.

[10] H. Wcyl, Amer. J. Math., 66 (1944) 591.

[11] A. Celotti, J.C. Miller and D.W. Sciama, Class. Quantum Grav., 16 (1999) A3.

[12] B.F. Schutz, Class. Quantum Grav., 16 (1999) A131.

DiPARTIMENTO DI FiSICA, UNIVERSITA DI MILANO, VIA CeLORIA, 16 - 20133 MiLANO

(Italy)

E-mail address: angelo.loinger@mi.infn.it

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