## Open letter to Dr. D. Rabounski, Editor-in-Chief of the JournalPROGRESS IN PHYSICS (PP)

(NB.: PP should not be mixed up with the prestigious and much older IOP-journal ''Reports on Progress in Physics''.)

06.05.2008

### Dear Dr. Rabounski,

Thanks for your somewhat curious letter. Summing up, it seems that you are telling me that your co-editor's, Mr. Stephen J. Crothers', work is correct except for some minor errors. And then, you tell me that all great geniuses of the past have made their errors; so why not the genius Crothers.

As I have pointed out on my website the PP-articles by Stephen Crothers are substantially wrong.

### Mr. S. Crothers has not understood the concept of black holes and is fighting against his own misconceptions.

Let me sketch here for you the State of the Art of the Schwarzschild metric which is well-known now for several decades, together with some comments of mine:

### 1. The Schwarzschild Metric

(1.1)                 ds² = − (1− rs /r ) dt² + (1− rs /r )−1 dr² + r² (dθ² + sin²θ dφ²)         (rs > 0 fixed)

defines two regions of regularity:

The ''outer'' region S' given by

(1.2)                 rs < r <         with metric signature         (−,+,+,+)

and the ''inner'' region S'' where

(1.3)                 0 < r < rs .         with metric signature         (+,−,+,+) .

### 2. The ''Event Horizon''

The regions of regularity S' and S'' are separated by the ''event horizon'', defined by r = rs. The event horizon is not a ''point'' as asserted by your co-editors S. Crothers and L. Borissova, but a 2-sphere with radius rs. L. Borissova wrote to me as follows: (with \alpha = rs):
''Crothers is right saying: r_0 = alpha is a point, in the "proper" quantities. He uses d\sigma^2 = h_{11} dr^2, where d\sigma is the physically observed length along the radial coordinate r. If \alpha = r, we have d\sigma = 0, hence \alpha = r is a point.
The definition of the "proper" interval is given in one of the paragraphs of "The Theory of Fields" by Landau and Lifshitz.
''

In the Schwarzschild metric (1.1) the parameter r has the geometrical meaning of a curvature radius of the spherical shells (dr = 0, dt = 0). The parameter r can, of course, be replaced with the transformed parameter Rp(r) = ∫rsr (1− rs /r' )−½ dr'. But this parameter transform does not affect the induced metric on the event horizon which due to eq. (1.1) is

(2.1)                 ds² = rs² (dθ² + sin²θ dφ²)

and does not depend on Rp(r). Thus, the event horizon remains a 2-sphere of radius rs as before.

I recommend especially that Mrs L. Borissova should refresh her GRT knowledge by carefully studying the section ''Gravitational Collapse'' in the book Classical Field Theory by Landau-Lifshitz which book she herself is referencing. At the beginning of that section Landau-Lifshitz are stating:

''For the Schwarzschild metric the coefficient goo is vanishing at r = rs (i.e. on the Schwarzschild sphere) and g11 there becomes infinite. This could lead to the conclusion . . . that bodies with a given mass having a radius less than the gravitational radius rs cannot exist. However, in reality this conclusion is wrong. This is already obvious from the fact that the determinant g = − r4 sin²θ does not have any singularity at r = rs and the condition g < 0 is not violated. . . .

Who is right? The authors Landau & Lifshitz talking of the ''Schwarzschild sphere'' and ''the condition g < 0 is not violated'' in the interior, or Mrs Borissova, who denies the validity of the interior region S'' when writing to me as follows: (with \alpha = rs)
''Crothers is right, saying that the interval 0 < r < \alpha is indefinite by the Hilbert metric. By this condition, we have the component h_{11} of the physical observed three-dimensional metric tensor h_{ik} is
h_{11} = - ( 1 - \alpha/r ) <0.
This condition doesn't satisfy to the signature conditions (see "The Theory of Fields" by Landau and Lifshitz).
''

### 3. Equivalent metrics

Since 1922 it has been known that the Schwarzschild metric can be transformed to the Painlevé-Gullstrand metric with coordinates (TP,r) where r in addition has the meaning of the spatial distance to the center (measured at constant P.G.-time TP). Two years later, in 1924 the Eddington-Finkelstein (EF) metric was introduced, re-discovered in 1958 bei David Finkelstein. Both validity regions of the Schwarzschild metric can be unified by applying the EF-transform

(3.1)                 cT = ct + rs ln | r/rs−1|,         c dT = cdt + rs/r (1− rs/r)−1 dr

to the Schwarzschild metric (1.1) in the region S' and in the region S'' as well to obtain the EF-metric

(3.2)                 ds² = − (1−rs/r ) c²dT² + 2 cdT dr + r² (dθ² + sin²θ dφ²),         valid for 0 < r < .

This transformation is invertible.

### Thus, the two validity regions S' and S'' of the Schwarzschild metric belong to different parts of the same spacetime-manifold S.

Therefore the whole PP-editorship falls into error when claiming the opposite.

4. What about the existence of (Schwarzschild-) blackholes?

The Schwarzschild black hole does exist in the following sense:

(1) No light signal starting at some point inside the event horizon can leave the event horizon towards the outer world. All that light ''falls'' into the center. The same occurs to all test particles inside the event horizon. This effect is called ''gravitational collapse'', e.g. by Landau-Lifchitz who named the corresponding section of their book by that effect.

(2) An observer using the Painlevé-Gullstrand metric or the Eddington-Finkelstein metric does not remark anything while passing the event horizon at r = rs. But after a finite interval of his proper time his fate will be a crash at the center r = 0. (c.f. the section The Gravitational Collapse in Landau-Lifshitz' Field Theory.)

### Conclusion

The contributions published in PP on Schwarzschild black holes deny the scientific facts listed above and/or contradict them diametrically. Therefore we can only conclude that these publications are dubious and must be considered to be attempt of misleading the less informed readership.

Dear Dr. Rabounski, this is my response to your refusal to publish my objections on the black hole considerations in your Journal PROGRESS IN PHYSICS. Please consider the question as to whether your refusal to publish well-based scientific objections to articles in PP are compatible with your own high sounding words posted on the PP-website.
''All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media.''

Be assured that any future scientific arguments from your side will be published with commentary on my website in the sequel of my article on ''Crothers Views . . .''.

Kind regards

Gerhard W. Bruhn