In 1939, there were two important developments for the topic of General Relativistic gravitational collapse. Robert Oppenheimer and his student Hartland Snyder published a paper entitles “On Continued Gavitational Contraction’’ in Physical Review which, it is believed, that showed for the first time that continued GR collapse should lead to the formation of Black Holes (BHs):

Following this Einstein published a paper entitled “On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses” in Annals of Mathematics:

Here he concluded that

The essential result of this investigation is a clear understanding as to why the “Schwarzschild singularities” do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that mote general cases will have analogous results. The “Schwarzschild singularity” does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.’’

Einstein’s paper was intuitive, instead of studying actual non-stationary collapse process, he assumed that eventually the collapsing object would comprise rotating & counter-rotating mass points. But I have heard opinions that Einstein’s paper was incorrect

though I have rarely found papers which could pin point the incorrectness of this paper, e.g. one can find the citations of this classic paper here:…56..455O&refs=CITATIONS&db_key=AST

In science only scientific truth is sacrosanct and there is nothing wrong in criticizing even the stalwarts including Einstein, if appropriate.  However, I have a feeling that many pass adverse comment on it as a matter of hearsay without ever carefully going through it (I too have only browsed it) particularly because it attempts to refute the pet idea of “Black Holes’’. And here I would like to highlight a particular case of such adverse criticism.

Once internet became popular by late 1990s, the web got two pioneers of blogging on Theoretical Physics, particularly, GR. First it was John Baez, professor of mathematics in UCAL, Riverside and, second, his erstwhile close friend Chris Hillman (Ph.D. mathematics, University of Washington). Baez may be a good researcher in mathematics and some aspects of mathematical physics while Hillman has hardly any worthwhile research track record. This duo had good knowledge of Differential Geometry and many mathematical aspects of GR. And of course they were pioneers of use of internet. Unfortunately they mistook these two qualities as super expertise in GR itself: they virtually declared themselves as the custodian of GR & relativistic cosmology; and they decided to be ultimate judge for any research which they perceived to be “against the mainstream’’. Not that they cared to disprove such “anti-mainstream research papers’’ in a professional & painstaking manner. On the other hand, this duo would heap scorn & ridicule not only such research but sometimes on the authors i.e. almost personal attack. Sometime they would mention some mathematical/differential geometry mumbo-jumbo which might actually be irrelevant. In any case, this duo developed huge fan following on net and the fans would be impressed by mathematics, jargons, rants, and even ridicules.  They would compensate for the lack of substance in their critiques by liberal use of adjectives “CRANKY’’ & “CRACKPOT’’. Most of such attacks would be launched from the following site of Baez (now deleted by Baez himself)

In particular, this site contained a Chapter entitled “HALL OF SHAME’’. I recall, here they would ridicule and castigate claims like (i) Gravitational Waves May Not Carry Any Energy Momentum (view held by Einstein, Rosen, now Fred Cooperstock ..), (ii) Steady State Theory of Cosmology (Gold, Hoyle, Burbridge, Narlikar). But the No. 1. Target of scorn was the claim that “There cannot be true Black Holes’’. And it is in this context that Baez & Hillman slammed Einstein:

 In 1939, Einstein publishes a paper which presents a rather desperate (and entirely incorrect) argument that nobody could collapse past its Schwarzschild radius. The nature of the conceptual errors in this paper show that Einstein still did not understand either the distinction between a coordinate singularity (the boundary of a coordinate chart) and a geometric singularity, nor the distinction between local and global structure. (Indeed, there is no evidence that Einstein ever understood correctly the geometry of all exact solutions to his field equations).”


As already mentioned, Einstein’s assumption that the eventual state the collapsing object may be represented stationary circular orbits of mass points cannot be exactly correct, and thus his conclusion about non-formation of BHs is not convincing at all. But otherwise, is his paper “entirely incorrect’’?

As far as I am aware there only one research paper which complained of mathematical inaccuracy in his treatment:

whose abstract is “It has been found that the equations of the gravitational field of a large number of gravitating particles of equal masses with spherical symmetry not were given correctly by Einstein. The correct form of these equations has been obtained here. However, the conclusions, arrived at by Einstein, remain unaltered.’’

But there was another paper which showed that even Einstein’s mathematics was correct

“ It is shown in the present paper that, although Einstein’s paper is extremely confusing and contains some mistakes, Misra’s criticisms are completely unfounded. A general and clear derivation of Einstein’s results is given in this paper.’’


This paper was the maiden one which considered the importance of tangential pressure in the context of gravitational collapse.  And such a configuration of rotating & counter rotating point masses (DUST) is known as “Einstein Cluster’’ and hundreds of papers have been written on it.  Some examples:

  1.  “Critical Collapse of Einstein Cluster’’, A. Mahajan et al. Theor.Phys.118:865-878, 2007, which finds that “ We show analytically that the collapse evolution ends either in formation of a black hole or in dispersal..’’, i.e., indeed BHs may not form.
  2. “Gravitational Collapse with Tangential Pressure’’, Malafarina, Daniele; Joshi, Pankaj S, International Journal of Modern Physics D, 20, 463-495 (2011)

Here the authors claim that little addition of Tangential Pressure may inhibit BH formation. They however conclude that a “Naked Singularity’’ would form because they overlook the fact that an ideal p=0 fluid must have density=0 too.


Their paper too contained some minor algebric mistakes as has been pointed out in series of my peer reviewed papers. But we may ignore such minor mistakes and recognize it as a real pioneering paper. Yet this paper is physically vacuous & completely misleading. Why?

Primary reason is that no physical fluid can be completely pressure-less unlike what was assumed by Oppenheimer & Snyder (OS). This p=0 condition tacitly imply that the density of the fluid too is zero. This has been explicitly shown by me in several papers, but here, let me cite only two:

  •  “The fallacy of Oppenheimer Snyder collapse: no general relativistic collapse at all, no black hole, no physical singularity’’, A. Mitra, Astrophysics and Space Science, 332, Lett. 43-48 (2011); (arXiv:1101.0601)
  •  “The Mass of the Oppenheimer-Snyder Hole: Only Finite Mass Quasi-Black Holes’’, A. Mitra & K.K. Singh, International Journal of Modern Physics D 22, id. 1350054 (2013); DOI: 1142/S0218271813500545

The mathematical BH generated by OS collapse thus has a gravitational mass M=0 which would require infinite proper time to form, i.e., it would never form or at the most be an ETERNAL process. The fastest collapse results where p=0, and if such a case is eternal, then any other collapse involving resisting effects like pressure gradient, heat flow and radiation must too be an ETERNAL process. Hence continued GR collapse should result in “Eternally Collapsing Object’’ (ECO) rather than any finite mass BH or so-called “Naked Singularity’’.

In the extremely ultra-relativistic regime, the heat and radiation generated within the collapsing object get trapped by the self-gravity and the resultant luminosity must attain its Eddington value by which inward pull of gravity gets balanced by the outward push of radiation:

  • Radiation pressure supported stars in Einstein gravity: eternally collapsing objects, A. Mitra, Monthly Notices of the Royal Astronomical Society, 369, pp. 492-496 (2006)
  • Likely formation of general relativistic radiation pressure supported stars or `eternally collapsing objects’, A. Mitra & N.K. Glendenning, Monthly Notices of the Royal Astronomical Society: Letters, 404, pp. L50-L54 (2010)

Thus, Einstein’s physical intuition about non-existence of (finite mass) BHs was correct though he could not see (zero mass) BHs as the asymptotical solutions of physical continued gravitational collapse of a chargeless fluid. However, with regard, to a point particle possessing a charge, Einstein & Rosen, Physical Review,  48,  pp. 73-77 (1935) had written that

It also turns out that for the removal of the singularity it is not necessary to take the ponderable mass m positive. In fact, as we shall show immediately, there exists a solution free from singularities for which the mass constant m vanishes. Because we believe that these massless solutions are the physically important ones we will consider here the case m = 0”.

Most of the present day BH/Singularity “experts” and many of the GR experts having, in some cases, more mathematical/numerical skill than Einstein 30 were actually experts on either Differential Geometry, or Applied Mathematics relevant for GR studies or Numerical Computations riding on GR and not necessarily on the intricate and subtle physics lying at the throbbing heart of GR. And as to baseless critique by two upstart flamboyant experts, Baez & Hillman, we may recall Einstein’s famous quote:

Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.”

Despite Eddington’s unjustified public denouncement of Chandrasekhar’s correct result on upper limit of cold self-gravitating objects, Eddington’s physical intuition and insight were far superior to that of Chandrasekhar; he was the first to correctly visualize the unphysical Nature of (finite mass) BHs and

insisted hat

I think there should be a law of nature to prevent a star from behaving in this absurd way”

And as emphasized by my research, this “law of Nature” is nothing but the bending of radiation due to strong self-gravity and consequent attainment of a critical Eddington luminosity. Of course, at that time, Eddington too failed to recognize that the gravitational contraction process must be radiative and a BH (with M = 0) should indeed be the asymptotical solution of the continued collapse process. It would be recognized much later that Chandrasekhar’s result about upper limit of cold objects was almost universally misinterpreted, most notably by Chandrasekhar himself, as an upper limit on mass of all compact objects, hot or cold. Thus it would be recognized that Chandrasekhar’s discovery had a profound retrograde effect on the development of the physical theory of continued gravitational collapse and relativistic astrophysics in general. Probably this mis-interpretation  along with the misinterpretation that the OS collapse was physical and suggested formation of finite mass BHs (when in reality, there is no collapse without finite pressure and heat flux, or, mathematically, M = 0 in such a case), put the clock back by 60 years as far as the question of the final state of continued collapse is concerned.

Interestingly, this result is in agreement with the intuition of Oppenheimer & Snyder  too:

“Physically such a singularity would mean that the expressions used for the energy-momentum tensor does not take into account some essential physical fact which would really smooth the singularity out. Further, a star in its early stages of development would not possess a singular density or pressure, it is impossible for a singularity to develop in a finite time.”

Thus though OS could find “singularity’’ for their unrealistic toy model of collapse, they probably knew that realistic physical gravitational collapse must not generate any singularity. Alas this warning was overlooked my by most of the GR and mathematical relativists. And Einstein’s intuition that there cannot be true black holes has been proven correct.

Abhas Mitra, January 31, 2014!

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  • Yurij Baryshev  On January 7, 2015 at 3:24 pm

    This is very good explanation why even in GR there is no physical BH, i.e. the event horizons. Also the concept of BH is self-contradicting as recently was emphasized by Hawking (arXiv:1401.5761 + Nature, 24 Jan 2014 and see also paper by Chowdhury & Krawuss arXiv:1409.0187).

  • Yurij Baryshev  On January 11, 2015 at 6:14 pm

    There are additional very simple and hence very strong arguments which demonstrate why existence of BH contradicts basic concepts of modern physics.
    For example, all real physical objects have finite binding energy, which equals to the work needed for disperses the body to infinity. However for BH “the gravity force so strong, that even light cannot escape”, and this means that the binding energy of BH is equals to infinity. So formation of BH from a star would transform a finite energy to infinite energy which is absurd.

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