Monthly Archives: December 2014


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!


An “Eternally Collapsing Object’’ (ECO) is a quasi-static ultra-compact ball of plasma (fire) which is so hot and whose self-gravity is so strong that even the light/radiation trapped within it can HARDLY move outward. In contrast the idea of a “Black Hole’’ is that it is infinitely compact and no light/radiation can move outward from it is a STRICT sense.  This essential difference may be quantified by introducing the parameter “Surface Gravitational Redshift’’, “z’’. While for an ECO, z>>1 (but finite), for the Event Horizon of a BH, one has z=∞. This notion of an ECO depends on the notion of the concept of EDDINGTON LUMINOSITY (L­­ed). When an object is radiating with a luminosity L=Led, for an in-falling plasma:


Although this crucial physics has been mentioned by A. Mitra, D. Leiter and S. Robertson several times in the past, formally this was highlighted in the following two papers

1. Likely formation of general relativistic radiation pressure supported stars or `eternally collapsing objects’, A. Mitra & N.K. Glendenning, Mon. Not. R. Astr. Soc. 04, pp. L50-L54 (2010); (arXiv:1003.3518)  DOI: 10.1111/j.1745-3933.2010.00833.x

2. Radiation pressure supported stars in Einstein gravity: eternally collapsing objects, A. Mitra, Mon. Not. R. Astr. Soc. 369,  492-496 (2006), (arXiv:gr-qc/0603055);  DOI: 10.1111/j.13652966.2006.10332.x

These papers showed that if continued gravitational collapse would indeed tend to form a BH having z=∞, then for certain z>>1, the trapped radiation within the contracting object would be so intense that Là Led, and the body is bound to be become an ECO. Yet, since z=Finite (and NOT Infinite), the object would continue to radiate & contract practically at an infinitesimal rate. Ideally, it would asymptotically (infinite proper time) become a true BH having z=∞. During this asymptotic infinite journey, the body is bound to radiate out its entire initial mass-energy so that the gravitational mass of the mathematical would be BH is M=0.

NOTE: This result never even invokes the notion of any `(i) `speed’’ of the stellar material or the (ii) speed of a test particle around a mathematical BH.

Therefore the claims by John Baez, Chris Hillman, Paulo Crawford, Ismael Tereno as well as some anonymous wiki editors of the MECO article are completely incorrect.

 However when I first claimed in 1998-99, that General Relativity (GR) does not allow either (i) Existence or (ii) Formation of finite mass BHs, I did not have a clear notion of ECOs. On the other hand my claims were based on the proof that in case a finite mass BH would be there, the fundamental GR principle that Motion of any material particle must be TIMELIKE rather than LIGHTLIKE. And in a strict sense, even this proof does not depend on the definition of any “Speed’’ contrary to the propaganda by Baez, Hillman; and claims by Crawford & Tereno. Let me expound on this in the next section.


Consider the motion of a test particle along X-axis. In Galilean physics, the infinitesimal distance, dX, traversed by it appear same (INVARIANT) to all observers. Similarly, the time elapsed during this motion,  dT too is same for all observers or all clocks. However, this is not so in Special Theory of Relativity (STR) where all observers are supposed to be moving at uniform speeds w.r.t. each other. As is well known, in STR, it is the spacetime interval (square)

ds2 = c2 dT2 – dX2 = INVARIANT (Appear same to all observers)   … (1)

ds2  also defines the nature of the underlying spacetime and is called “spacetime metric’’.  Further, by STR, for a particle having finite mass, the metric must be TIMELIKE, i.e., ds2 >0.

In contrast, the metric of a massless particle like photon must be LIGHTLIKE: i.e., ds2 =0.

On the other hand, GR involves accelerated motion and all kinds of non-inertial frames as a result of gravity.  Accordingly though the structure of a `metric’’ could be infinitely complicated in comparison to Eq.(1), one may broadly express

ds2 = c22 – dl2                                                                                      ..(2)

where  dτ is an appropriately defined PROPER TIME interval and dl is an appropriately defined proper distance.  dτ and dl are appropriate generalizations of  ordinary clock interval dT and spatial coordinate interval dX after taking into account spacetime curvature due to presence of gravity.

 Ref: R. J. Cook (2004) Physical time and physical space in general relativity, Am. J. Phys. 72:214–219

And I showed that if a material test particle will approach the EH of a Schwarzschild BH, its motion would tend to LIGHTLIKE, i.e., ds2 à 0 instead of ds2 >0 (different from 0 by a finite amount).  Since I need not precisely land on the EH in order to obtain this result, it cannot be hand waived by the convenient alibi that “There is a coordinate singularity at the EH’’. In any case, since ds2, is an INVARIANT, this result must be independent of the coordinates used, and I indeed used various other coordinates to obtain the same result. From this (and from many other considerations) I concluded that there cannot be any BH, any EH. And the mathematical EH must be a physical singularity. But the physical singularity lies at R=0. Therefore one must have Rg = 2 GMBH/c2 =0, from which it follows that mathematical BHs must have M=0. Later I offered an explicit proof to this effect

3. Comments on “The Euclidean gravitational action as black hole entropy, singularities, and space-time voids”, Mitra, J. Math Phys, 50,  042502 (2009); (arXiv:0904.4754); DOI: 10.1063/1.3118910

Similarly, I showed that in case a collapsing object would tend to form a Trapped Surface, the motion of the fluid would tend to be LIGHTLIKE, and hence there must not be any trapped surface. Though there are several publications toward this, let me cite the latest one:

  1. Quantum information paradox: Real or fictitious? A. Mitra, Pramana 73, 615 (2009) :


Look, Eq. (1) can be interpreted as ds2 = c2 dT2 (1-v2/c2) >0, or v = dX/dT <c. In other words a material particle must move with a 3-speed, v <c. However suppose an accelerator would accelerate the test particle for infinite duration, then, hypothetically, one can attain

v= c, and ds2   –>0

Similarly, in GR, for a test particle, one can define a physical 3 speed v= dl/dτ, and express Eq.(2):

ds2 =c22 (1- v2/c2)  >0, or v <c.                                                 (3)

And since the occurrence of a trapped surface or an EH implies ds2 à 0, a trapped surface or an EH must correspond to physical singularity and must not occur the same way an accelerator of infinite strength/duration cannot occur. And it is only for such an interpretation physical speed would be linked to my proofs.



In 1999, Tereno (astro-ph/9905144) claimed that, physical speed of a test particle vEH, calculated by Kruskal coordinates by me was incorrect. And in my response (astro-ph/9905175):

Comment on “Velocity at the Schwarzschild Horizon Revisited” by I. Tereno,

I wrote that “It appears that Tereno’s conclusion is driven by his inability to conceive proper limiting value of fractions. Similarly, his idea that, the velocity addition formula of Sp. Theory of Relativity breaks down when both the velocities approach unity is due his same inability. “

Here Tereno goofed up evaluating a 0/0 limit. Tereno implicitly admitted mistake in his preprint, and came up with a new one where he tried to find vEH in a numerical was for a special case, and then claimed that the geodesic of a test particle does not become LIGHTLIKE at the EH. Look, he did not contest my conclusion that ds2 (EH) =0 in other coordinates. Clearly before making his assertion that in Kruskal coordinates, nonetheless, ds2 (EH) >0, he forgot the fact that ds2 is an invariant and must be same in all coordinates. I again responded to his misplaced assertion

Final Comments on “Another view on the velocity at the Schwarzschild horizon” by Tereno (astro-ph/9905329). One may also look into

  1. “Kruskal Coordinates and Mass of Schwarzschild Black Holes: No Finite Mass Black Hole AT All’’
  2. Mitra , IJAA, 4, 26225 (2012):

Later, Crawford & Tereno came up with a journal publication on this topic:

P. Crawford and I. Tereno, “Generalized Observers and Velocity Measurements in General Relativity,” General Relativity and Gravitation, 34, 2075 (2002); (arXiv:gr-qc/0111073)

Here they mentioned that “In their very well known textbook [ p.342], Shapiro and Teukolsky also produce a similar statement: “. . . the particle is observed by a local static observer at r to approach the event horizon along a radial geodesic at the speed of light . . . ”

“And if one accepts that a particle has the speed of light with respect to a static observer (at r = 2m), using locally the velocity composition law from special relativity, he (or she) concludes that the particle has the same speed of light with respect to all observers. This is certainly something that conflicts with the physical observation that, in a vacuum, no material particle travels as fast as light.’’

“ Indeed, even if we use a coordinate system that has no difficulties at r = 2m, like the advanced Eddington-Finkelstein coordinates, we would still end up with the same result v → 1 as r → 2m.’’

Hence, they implicitly admitted that even in Kruskal coordinates, one would indeed have vEH à c in accordance with my calculations and in contravention of two earlier claims by Tereno.

Almost hopelessly,  they concluded that “Thus the real issue here is the choice of frame not the choice of coordinates’’ effectively claiming that the entire GR ansatz for finding “velocities’’ was incorrect! And in order to stick to their prejudice that one must have v(EH) <c (so that BH paradigm must survive), they came up with the diktat that the speed of one free falling particle must be measured by another free falling particle.  Assuming  this two free falling particles have energy/unit rest mass as E1 and E2, they showed that Vrel(EH) <1 (presuming E1≠E2). Let us assume for a moment that this result is meaningful. Yet such a result has no relevance for Eq.(3) which implies the notion of a physical velocity in arbitrary coordinate system because their NEW definition of “relative velocity’’ is different & does not satisfy Eq.(3). Accordingly their result does not affect the coordinate independent ds2 (EH) = 0.

Note, here Crawford & Tereno forgot the fact in a curved spacetime, the observer & the test particle must be at the same spatial position at the instant of measurement. But by using Galileo’s old Tower of Pisa experiment

It becomes clear that, in order that two free falling particles must be at the same spatial position, they must have the same initial conditions, i.e., one must have E1=E2. In such a case, the relative speed defined by Crawford & Tereno must always be not just Vrel <c, but Vrel=0!

I pointed out this in Ref. 5. Thus the awkward prescription of velocity measurement by Crawford & Tereno in no way invalidate my proof that ds2 (EH) à0, more importantly, this prescription is a physical nonsense.

In a subsequent paper

“Interior of a Schwarzschild Black Hole Revisited,” Foundations of Physics,38, 160 (2008), (gr-qc/0609042), R. Doran, F. S. N. Lobo and P. Crawford, wrote:

In this work, we have addressed some conceptual difficulties related to the notion of black holes. The solutions that do away with the interior singularity and the event horizon [13, 14, 15, 16], although interesting in themselves, sweep the inherent conceptual difficulties of black holes under the rug. In concluding, we note that the interior structure of realistic black holes have not been satisfactorily determined, and are still open to considerable debate.’’

 Similar conclusion was obtained by Jayant Narlikar & Thanu Padmanabhan way back in 1988

“The Schwarzschild Solution: Some Conceptual Difficulties’’

“It is suggested that problems related to the source could be avoided if the event horizon did not form and that the universe only contained quasi-black holes.’’

though later Padmanabhan converted himself  into an ardent BELEIVER of “Event Horizon’’ to atone for his past sins, and Narlikar chose to remain ambiguous on this issue, come what may.

 Well, after 100 years and after may be 1 lac articles & books, there is no resolution of self-contradictions & confusions associated with the notion of BHs. And the only resolution is that the mathematical solution of BHs correspond to M=0 (which arises from the relevant Integration Constant), and the so-called BHs are only quasi-BHs, and as suggested by my research, they are likely to be ECOs. Why it must be so has been explained in the following Plenary Talk:

Einsteinian Revolution’s Misinterpretation: No True Black Holes, No Information Paradox: Just Quasi-static Balls of Quark Gluon Plasma Mitra, World Scientific Publishing, 2014. ISBN #9789814578745, pp. 153-163

as well as this short talk:


In collusion with John Baez, in 2001, Hillman tried to publicly humiliate me through  sci.physics.research ,moderated by Baez, without pointing out any specific error in my papers:

“This preprint is by Abhas Mitra, a nuclear physicist by training, who apparently likes to claim expertise in astrophysics.  But, unfortunately,when it comes to gtr there is no simply polite way of describing his level of understanding, which literally does not rise to what is expected of -undergraduate students- taking a first course in gtr..’’

“However, some years ago I read in some detail an earlier draft of and it was quite frankly so riddled with -elementary- misstatements and misconceptions concerning gtr as to be not only worthless but frankly embarrassing.  Mitra is terribly confused about gtr, and so it is no surprise that it is usually impossible to be sure exactly what he is trying to say at any given place in this preprint, because a person who is thoroughly confused in his own mind cannot possibly express himself clearly!’’

“It is very regrettable that this particular preprint (astro-ph/9910408) was (shame! shame!) actually -published-: Found. Phys. Lett. 13 (2000) 543’’

About a related preprint, he could only comment:

“I don’t have the stength to go through this paper, but it sounds absurd to me: Of course it is absurd!’’

And John Baez, the supposed internet hero and a great promoter of GR shamelessly refused to publish my rejoinder to this character assassination:!msg/sci.physics/LIhGF9_VoIE/k_1jIza5pHkJ

Hillman & Baez together also heaped scorn & contempt on me and my research through a section called “HALL of SHAME’’ (Now Deleted! Why?)

In 2004, Stephen Hawking took an U-turn and declared that

The view seems to be forming in his mind that there isn’t a black hole in the absolute sense, there’s just a region where things take a very long time to escape,” says Gibbons. This suggests that black holes do not actually narrow to a singularity at all.’’

Hawking may have some vague & unspecified Quantum Gravity  ideas for arriving at such a conclusion.

But eventually, this is what was precisely shown (not just conjectured) by me. Also most dishonestly Hawking claimed that “I have solved the black hole reality’’, when it was me who had already solved this riddle in a precise manner:

Following this my research came into international focus again. In one internet forum, one user asked Baez “Can you kindly elaborate on the alleged “mistakes” and “blunders” of Mitra?!msg/sci.physics.relativity/pfBMXYZRgBU/qpCuQ_Lar0oJ

To this, Baez retorted “Mitra’s argument involve a combination of GR and mistakes. GR is a mathematically rigorous theory, so black holes either exist in this theory or not.  Mitra claims they don’t; most people know they do’’ (Note: actually he could never point out any precise “mistake’’).

As the user continued to press for “mistakes’’, all that Baez could blurt out is

“The errors lie with Mitra, not everyone else in the world.

Here’s some old discussion of Mitra’s mistakes. I also wrote my own analysis of his mistakes back when I had to reject some of his articles on sci.physics.research, but it’d take a bunch of work to find this now.’’

and having failed to point out any “mistake’’, he fell back on the character assassination post by his friend Hillman whose rejoinder he had refused to publish:

Hillman & Baez also claimed that Tereno had already punctured my papers which claimed there can be no finite mass BHs. But, here it was shown that even assuming Vrel(EH) <c, it has no relevance for the result ds2 (EH) = dτ2 (1- v2/c2) =0, where v≠Vrel. Thus even if Vrel <c, clearly và c at the EH. Therefore it is a Goebbelsian  lie propagated by Baez & Hillman that Crawford & Tereno punctured my proofs.

In another post whose original link has now been removed,

From: John Baez
Newsgroups: sci.physics.relativity
Subject: Re: Indian physicist vindicated in black hole controversy
Date: Thu, 19 Aug 2004 17:54:37 +0000 (UTC)

Baez wrote “The funny part is that he’s trying to do this using only general relativity! Starting from the solution which describes a black hole of mass m, he attempts by a calculation to show that m = 0. It’s a bit like taking an arbitrary prime number and proving that it must equal 37.

In an earlier version of his “proof”, Mitra’s mistake was simple to spot, since he was using the familiar Schwarzschild coordinates, and the mistake involved dividing by zero.’’

Here John, a noted mathematician, decided to behave like an idiot: When we solve differential equations, we get integration constants whose values, in principle could not only be zero, even be negative. For the Schwarzschild case, the integration constant is I = 2GM/c2. And the numerical value of I must change as the situation would evolve, e.g., if a contracting star would lose mass energy, the value of I must continue to decrease. Let for a point particle I =I0 = 2GM0/c2. In principle, I0 (and M0) could be –ve: I0 <0. In such a case, the solution for the point particle would indicate a NAKED SINGULARITY. On the other hand, the assumption I0 >0 gives rise to the BH paradigm. And if I0 can be even –ve why it cannot be zero?

And my paper A. Mitra, J. Math Phys, 50,  042502 (2009); (arXiv:0904.4754) showed that I0=0 so that M0=0.

 Note that it was Tereno’s preprint which made a mistake in finding the limit of 0/0 form and which was pointed out by me: (astro-ph/9905175). But see the mischief monger Baez: In his frustration to find any error in my paper over 5 years (1999-2004), he ascribed Tereno’s mistake to me!

Now let me say for once: SHAME: John Baez: University of California, Riverside would be ashamed if it would know your academic misdeeds! Also shame on some past wiki editors for MECO who chose to pick up the internet trash left by Baez & Hillman by ignoring papers published in Phys Rev, Mon. Not. R. Astron. Soc., J. Math Phys, Astr. Sp. Sc., New Astronomy etc!

December 25, 2014, from Abhas Mitra

This blog is dedicated to my friend Dr Sabbir Rahman whom I am yet to meet, but long to do so.


Even before Einstein proclaimed that General Relativistic (GR) gravitational collapse cannot form Black Holes (BH) in 1939 “On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses’’ Annals of Mathematics, Second Series, Vol. 40, p. 922 (1939) many other founding fathers of GR shared similar opinions. The most notable of which was Eddington’s intuitive comments “I think there should be a law of Nature to prevent a star from behaving in this absurd way!” However, neither Eddington, nor Einstein researched on GR collapse and showed how this could be possible. On the other hand GR collapse equations were solved by Oppenheimer & Snyder (OS) for the case of a uniform density fluid assumed to have no pressure at all. Under such patently unrealizable assumptions, it appeared that BHs may form in GR collapse. Then in the 1960s Penrose, Hawking and others constructed “Singularity Theorems’’ by which it appeared that all continued GR collapse must give rise to formation of appropriate singularities provided one would assume beforehand that following OS example, continued collapse would apriori generate “Trapped Surfaces’’ (2GM/R c2 >1)  from which nothing not even light can escape. Thus the assumption of prior formation of “Trapped Surfaces’’ is like “assuming what one intended to prove’’. Unfortunately not only the astrophysicists but even many GR researchers are not even aware of this vicious cyclic structure of the “Singularity Theorems’’. Simultaneously, by arguing that the unphysical nature of Schwarzschild BHs for the region R < Rg= 2GM/c2, were only due to malfunctioning of coordinates was only due to “Coordinate Singularity at R=Rg =Event Horizon (EH), the concept of BHs became one of the most influential paradigms of Physics as well as Astrophysics by 1970. And my paper: “Non-occurrence of trapped surfaces and Black Holes in spherical gravitational collapse: An abridged version’’ (arXiv:astro-ph/9910408) was the maiden one which seriously attacked this paradigm from multiple directions:

  • It purported to show that contrary to the crucial assumption behind revered singularity theorems, Trapped Surfaces DO Not Form in GR collapse, i.e., one will always have 2GM/R c2 <1, or
  • R >2GM/c2.
  • Like in Special Theory of Relativity, in GR too, the motion of material particles is “Time like’’; technically, ds2 <0. In contrast motion of light is always “Light like’’; technically for photons, ds2=0. My proof was based on the fact that in case a trapped surface would form the motion of the stellar material would violate this basic GR principle.
  • I also showed that if a test particle would ever reach the Event Horizon (R=Rg) of an assumed BH, its motion would cease to be “Timelike’’ and instead it would behave “Lightlike’’ in violation of GR principles.
  • From such dual considerations, I concluded that GR must not allow formation/existence of true BHs having gravitational mass M>0. On the other hand, my research concluded that though the “Black Hole’’ solutions are formally correct, they pertain only to a limiting case of a BH with M=0, whose formation would take infinite proper time even by a commoving observer, i.e. collapse would continue eternally in the form of Eternally Collapsing Objects. Accordingly, in 2000, I concluded that the so-called “Black Hole Candidates’’ cannot be true BHs and instead likely to be ECOs.
  •  Simultaneously, I pointed out that the concept of upper mass limits of compact objects like “Chandrasekhar Mass Limit’’ and “TOV Limit’’ were based on the existence of COLD quantum degeneracy pressures alone. And in the presence of various other effects like intense Magnetic Field or Intense Radiation Pressure, such limits may get completely invalidated.

My paper got accepted for publication only after two anonymous referees got convinced: 1. Mitra, Phys.Lett. 13 , 543 (2000) However much before this, my preprint got noticed by the entire world and my request for open critique of it did not evince any GR collapse researcher. Nonetheless, my research as well as I as a person came into vicious attack from two scholars:

  1. Christ Hillman, Ph.D. Mathematics from Washington University &
  2. John Baez, a Mathematical physicist & professor in UCAL, Riverside.

To give credit to them, they were the pioneers of Internet blogging & they ran several internet groups, they knew Differential Geometry & many aspects of mathematical GR, and in particular, Baez may indeed be a very good and insightful mathematician.  But none of them had ever worked on classical GR collapse problem, and in particular, Hillman may not have a single GR research paper to his credit.  In contrast I was a naïve & had no idea of such matters, and got personal email ID only in 1998-99. And I received an email from one Dr Sabbir Rahman (Ph.D. in particle physics from Cornell University) that Hillman & Baez had been viciously attacking my preprint through their widely read google groups & elsewhere!topic/sci.physics.research/22VAhEOyals%5B1-25-false%5D In particular one of the leading sites which would criticize any GR research perceived to be “against the mainstream’’ was maintained by Baez:

Baez’s site, in those days, contained a long section entitled “HALL OF SHAME’’ where Hillman & Baez would ridicule all GR claims which they would perceive as “against the mainstream’’. They would also indiscriminately put labels like “Cranky’’ and “Crackpot’’ to the persons associated with researches. I told Sabbir that in my preprint posting I had already invited readers to criticize my work if they would choose to do so, and I was not bothered with such below the belt attacks which would be considered in any journal as a repudiation of my research. But Sabbir told me that readers and researchers actually would be misguided by the Hillman & Baez’s internet posts which were read world over. Following his prodding, I sent an email to Hillman asking him to send me his critiques directly so that I would be able to respond. I also told him that he was free to criticize my research in a proper manner, i.e. by posting a preprint or by a research paper. But Hillman pooh poohed my suggestion and gave a derogatory reply. And I did not even know how to register and give apt response to Hillman’s baseless criticisms in such web groups. To see the text of Hillman-Baez’s criticism, please refer to APPENDIX 1. Following the publication of my paper in FPL, in the context of another related preprint, Hillman really indulged in hitting me below the belt (see APPENDIX 2). He started off his attack with comments like this: “This preprint is by Abhas Mitra, a nuclear physicist by training, who apparently likes to claim expertise in astrophysics.  But, unfortunately, when it comes to gtr there is no simply polite way of describing his level of understanding, which literally does not rise to what is expected of-undergraduate students- taking a first course in gtr (c.f. the well known textbook by Schutz). -Of course- Mitra is flat out wrong.’’ In reality I was already an award winning astrophysicist, one of the pioneers of theoretical High Energy Astrophysics research in India. I happened to be a reviewer for The Astrophysical Journal, my paper cited in Nature News & Views, my papers discussed in Sky & Telescope magazine, I was the 1st speaker for the prestigious Huntsville Conf on Cosmic Gamma Ray Burst from India; I was one of the Invited Speakers for the IAU colloquium at Montana State University …… On the other hand, I never did any research on Nuclear Physics! In short, Baez & Hillman never refereed to any equation in my paper and pointed out any precise error anywhere. There was however a subtle error which they missed as they had never worked on GR collapse. Though this subtle error did not change the ultimate result, I corrected it later arXiv:astro-ph/0408323 as well as in On the other hand Baez & Hillman went on beating about the bush by using Differential Geometry jargons and vague insinuations. Essentially, their point was that

  •  If BHs were discussed as ultimate truth in renowned text books, my conclusion about their non-existence must have resulted from my idiocy.
  • Since Penrose, Hawking and others claim that singularity is inevitable, my proof on non-occurrence of trapped surfaces & singularities is bound to be incorrect.

Here one may recall that the entire subject of physical gravitational GR collapse arose from the discovery of “Vaidya Metric’’ by P.C. Vaidya. And he found that my preprints were on correct paths, both physically & mathematically: Willy-nilly, I forwarded my rejoinder to Sabbir who was supposed to post the same on my behalf (See APPENDIX 3); my rejoinder ended with the following comments:

I invite Hillman and all other readers to write a proper scientific critique pointing out specific errors in my equations and interpretations in case any of them think that my work is erroneous. Incidentally, through email, I have repeatedly requested Profs. S. Hawking. R. Penrose, K.Thorne, C.W. Misner and many others to send their critique, if any, to me. But none of them have acted so far.’’

But Sabbir could not post my rejoinder! Why? In 2001, the science.physics group was essentially managed by John Baez, Chris Hillman and Matt McIrvin with Baez & McIrvive as “moderators’’. Naturally these wicked moderators refused to post my rejoinder showing the hollowness of the critique of Baez & Hillman. Then Sabbir published my rejoinder in other web groups with an introduction (see APPENDIX 4):!msg/sci.physics/LIhGF9_VoIE/k_1jIza5pHkJ



From 1997, I was in the “Theoretical Astrophysics Division’’ of Bhabha Atomic Research Centre. In 1998, TPD got a new head named Dr D.C. Sahni.

During his entire career, he failed to any meaningful research in Theoretical Physics and all could do was churn out few mathematical papers on 1-D Neutron Transport. That time, Director of BARC too was a noted reactor cum nuclear engineer. And in 2000, the younger brother of DC, namely Dr V.C. Sahni    

became the director of BARC  physics group. VC too wanted to be leading Theoretical (condensed matter) physicist, but badly failed to be so. Then both these siblings became rabidly“ Anti- Theoretical Physics’’. They particularly targeted and two other bright award winning Theoretical Physicists (Arun Pati: and Sudhir Jain). DC & VC dictated that we must do either some Reactor Physics or Defence related “Strategic Research’’.  And what a coincidence: Most likely Baez or Hillman contacted DC & VC telling that my research on GR was completely erroneous & embarrassing for my institute. This was an ideal fodder for two real cranky and aggressive siblings. Eventually all three of us (me, Pati, Jain) had to leave “Theoretical Physics Division’’. It is also likely that DC & VC pasted the clandenstine baseless comments on my research in my ANNUAL CONFIDENTIAL REPORTS to put my career in BARC in jeopardy! My promotions were almost stopped apart from various other harassments & humiliations I faced. To this extent Baez & Hillman have been greatly successful in their attempt to stifle my research career. They may also have been successful in misguiding thousands of readers and researchers (also WIKI EDITORS) against the ECO paradigm BUT WERE BAEZ & HILLMAN EVENTUALLY SUCCESSFUL IN THEIR ANTI-SCIENCE MISSION? Note that even by 2014, neither Hillman nor Baez nor DC nor VC have been able to write a single paper/preprint criticizing my peer reviewed research. Further, series of my peer reviewed papers have shown that

  •  Even if a there would be no source of central nuclear energy generation, any contracting self-gravitating object would generate heat & radiation because of the attractive nature of the gravity:

2. “Sources of stellar energy, Einstein Eddington timescale of gravitational contraction and eternally collapsing objects” Mitra, New Astronomy, Volume 12, Issue 2, p. 146-160 (2006); (arXiv:astro-ph/0608178) 3. Why gravitational contraction must be accompanied by emission of radiation in both Newtonian and Einstein gravity, A. Mitra, Physical Review D, vol. 74, Issue 2, id. 024010 (2006); (arXiv:gr-qc/0605066)

  •  As the contracting object become extremely relativistic, sooner or later, the outward force due to the trapped radiation would counter the inward pull of gravity signaling the formation of ECOs:

4. A generic relation between baryonic and radiative energy densities of stars, A. Mitra, Monthly Notices of the Royal Astronomical Society: Letters 367, pp. L66-L68 (2006) 5. Radiation pressure supported stars in Einstein gravity: eternally collapsing objects, A. Mitra, Monthly Notices of the Royal Astronomical Society, 369, pp. 492-496 (2006) 6. 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) As to the Oppenheimer-Snyder paradigm of BH formation, it was shown that a p=0 collapse is a  mathematical illusion, and the BHs so produced have M=0: 7. The fallacy of Oppenheimer Snyder collapse: no general relativistic collapse at all, no black hole, no physical singularity, A. Mitra, Astrophysics and Space Science (L) 332, , pp.43-48 (2011) 8. 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) More significantly, it was shown that Schwarzschild BHs have unique gravitational mass: M=0; the integration constant appearing in the appropriate differential equations is actually zero. Note the BH paradigm arose by presuming this integration constant to be positive definite.  Way back in 2006, there were reasonable observational evidence that the so-called “Black Holes’’ in quasars were actually (Magnetized) ECOs (Center for Astrophysics, Harvard Press Release)  Chris Hillman boasted of being an alumnus of Washington University, but the latter deleted his E-id long ago. Baez himself deleted his site several years ago. He however continued to prowl the net HIDING UNDER THE PSEUDONYM “T. ESSEL” for several years! As to John Baez, he too erased his site several years ago. Nonetheless he is still quite active on web and pretends to be a great intellectual & physicist. I wish, UCL Riverside too removed him from his position for his irresponsible & unethical attacks against me.


APPENDIX 1: COMMENTS MADE BY HILLMAN IN 1999:!topic/sci.physics.research/22VAhEOyals%5B1-25-false%5D

Mitra ( has claimed that the singularity theorems are gtr are wrong (at least, a casual reading seems to suggest that this is his claim), that gravitational collapse to a black hole is -not- after all predicted by gtr (i.e. that Chandrasekhar, Oppenheimer and Synder, Wheeler, Penrose, Hawking, etc., are all wrong), that the Schwarzschild solution does not
contain a curvature singularity (similar comments), that the standard interpretation of the Kruskal coordinates in wrong and that timelike radial infalling geodesics become null at r = 2m, and so on and on.  To my knowledge, none of these papers have been accepted, and Tereno has written one preprint rebutting the claim about the alleged “unphysical” motion of infalling particles at r = 2m in the Schwarzschild solution. I haven’t read any of this stuff closely, but casual perusal of some of Mitra’s
preprints suggested to me that this work rests upon various serious misconceptions concerning semi-riemannian geometry.  Any comments?


  • Subject: Black Hole Preprints by Abhas Mitra [Was: charged black holes]
  • From: Chris Hillman <>
  • Date: 19 Jul 2001 16:31:20 GMT

On Sat, 14 Jul 2001, Squark wrote:

> On Sat, 14 Jul 2001 03:14:25 GMT, zirkus wrote (in

> <>):


> >Btw, I have not looked at the following paper but, according to its

> >abstract, GTR only admits the existence of extremal black holes and

> >the paper might discuss how this result is related to string theory:

> >

> >

This preprint is by Abhas Mitra, a nuclear physicist by training, who apparently likes to claim expertise in astrophysics.  But, unfortunately, when it comes to gtr there is no simply polite way of describing his level
of understanding, which literally does not rise to what is expected of-undergraduate students- taking a first course in gtr (c.f. the well known textbook by Schutz).

-Of course- Mitra is flat out wrong.  Specifically:

(1) Mitra claims that gtr only allows extremal Reissner-Nordstrom electrovacuums (q = m in relativistic units in which G = c = 1)!  In fact, any decent undergraduate student of gtr can easily check that the subextremal (q < m) RN hole is a perfectly legitimate exact electrovacuum solution to the EFE; that is, it solves the Maxwell source-free field equation on curved spacetime and also the Einstein tensor matches the EM stress-energy tensor, so the RN electrovacuum models a the exterior fields, both electric and gravitational, of a massive charged object. The maximal extension has a global conformal structure

which Mitra doesn’t like (or understand), but no matter how many preprints he posts to LANL claiming otherwise, he cannot change the fact that it -is- a perfectly legitimate exact solution to the Einstein-Maxwell field equations.  Indeed, a standard problem for beginning students of gtr is to -derive- the this electrovacuum.  See for example the discussion of the RN
electrovacuum in this review paper

or in the well-known monograph

Stephen W. Hawking and G. F. Ellis, The Large Scale Structure of Space-Time.
Cambridge University Press, 1975.  In print, ISBN 0-521-09906-4; list price $47.95 (paperback)

or in widely used gtr textbooks such as

Ray A. d’Inverno, Introducing Einstein’s Relativity
Oxford University Press, 1992 In print, ISBN 0-19-859686-3; list price $42.95 (paperback).

Charles W. Misner, Kip S. Thorne, and John A. Wheeler, Gravitation,
W. H. Freeman & Company, 1973. In print, ISBN 0-7167-0344-0; list price $63.95 (paperback).

as well in as these high quality on-line course notes:
Mitra has posted many other cranky preprints to the LANL server, including

in which, as anyone can easily verify:

(2) Mitra claims that the tangent vectors to a timelike geodesic in the Schwarzschild vacuum must become -null- at the event horizon r = 2m; this is of course completely incorrect!  One need only start with -any-timelike vector at r = 2m and evolve backward in time a timelike geodesic, parametrized by proper time, by using this initial data in the geodesic equations.  Mitra appears to be completely unaware of the Painleve chart

ds^2 = -dt^2 + (dr + sqrt(2m/r) dr)^2 + r^2 (du^2 + sin(u)^2 dv^2)

-infty < t < infty, 0 < r < infty, 0 < u < pi, -pi < v < pi

in which it is very easy to find explicitly the world lines of “LeMaitre
observers”, namely

r(t) = (9m/2)^(1/3) (t0-t)^(2/3),

-infty < t < t0

Here, r(t1) = 2m for t1 = t0-4m/3, and it is true that the -coordinate
slope- dr/dt equals -1 there, but if you draw the -light cone- there using
the LeMaitre ONB of vectors

e_1 = d/dt – sqrt(2m/r) d/dr

e_2 = d/dr

e_3 = 1/r d/du

e_4 = 1/(r sin(u)) d/dv

(to draw the light cones in the tr plane, use the null vectors e_1 + e_2, e_1 – e_2), or if you simply compute the squared magnitude of the tangent vector, you will see that of course this tangent vector is -timelike-, not null!  Part of Mitra’s confusion throughout his preprints rests upon persistent failure to distinguish clearly between coordinate slopes and physical velocity measured relative to some “very close” observer (e.g., defined by the unit timelike vector X = e_1 in an ONB, such as the LeMaitre ONB).

(3) Mitra claims that the Kruskal-Szekeres chart has a coordinate singularity at r = 2m; this absurd (and very incorrect) conclusion may also be very quickly debunked: it is easy to write the K-S chart in closed form, without the constraint (used in most textbooks) which confuses Mitra, by using the “Lambert W function”:

m^2 W[(R^2-T^2)/(2me)] [-dT^2 + dR^2]
ds^2 = ————————————-
(1 + W[(R^2-T^2)/(2me)]) (R^2-T^2)

+ 4m^2 (1 + W[(R^2-T^2)/(2me)])^2 (dU^2 + sin(U)^2 dV^2)

-1/e < (R^2-T^2)/(2me) < infty, 0 < U < pi, -pi < V < pi

Here, the Lambert W function is the holomorphic function defined by

z’ = z exp(z)  iff z = W(z’)

We choose the -principal branch- of the W function, which is real valued precisely where we need it to be, namely on the interval (-1/e,infty). At the horizon (the locus R^2=T^2 in the KS chart), it is easy to check
that the line element given just above reduces to

ds^2 = 8m/e [-dT^2 + dR^2] + 4m^2 [dU^2 + sin(U)^2 dV^2]

where the underlined part is the metric of an ordinary sphere of “radius” 2m.  Thus, the K-S chart has no “coordinate singularity” at the event horizon, contrary to Mitra’s claim.

(4) Mitra claims that the Oppenheimer-Snyder collapsing dust ball model can only yield a black hole with -zero mass-.  This is of course absurd; the OS model is carefully and correctly analyzed in many gtr textbooks, for example

Hans Stephani,  General Relativity: An Introduction to the Theory of the  Gravitational Field, 2nd ed. Cambridge University Press, 1990  In print, ISBN 0-521-37941-5, $39.95 (paperback).

One of Mitra’s persistent confusions arises from an apparent inability to understand the matching employed in building the OS model (in which we match across the world sheet of a collapsing spherical surface from a dust ball which is a region of the collapsing FRW dust with E^3 hyperslices, to a exterior vacuum region, which is a region of the Schwarzschild vacuum). Mitra also appears to be completely ignorant of the well-known Vaidya null dust in which a collapsing spherical shell of massless radiation
(Minkowski region vacuum inside the shell, incoherent massless radiation in the interior of the shell, Schwarzschild vacuum region outside the shell) collapses from scri^- to form a black hole; see

The point is that no matching is required to construct or analyze this exact solution. The Vaidya null dust is briefly discussed in this review paper

and is also extensively discussed here

and in the monograph

  1. D. Novikov and V. P. Frolov, Black Hole Physics: Basic Concepts and New Developments,
    Kluwer Academic Publishers, 1998, In print, ISBN 0-79-235145-2; list price $98.00 (paperback)

(5) Mitra claims that “the proper time for the formation of any black hole” (tellingly, he cannot clearly explain what he means by this claim) is “infinite”; this is also absurd, under any sensible interpretation
(e.g.  the proper time measured by an observer riding on the surface of the OS collapsing dust ball), as every competent student of elementary gtr knows.  This issue is discussed at length in most gtr textbooks; a particularly well illustrated discussion can be found in

Charles W. Misner, Kip S. Thorne, and John A. Wheeler, Gravitation, W. H. Freeman & Company, 1973. In print, ISBN 0-7167-0344-0; list price $63.95 (paperback).

(6) Mitra claims that the well-known “trapped surface” singularity theorem is -false-!  The (correct) statement and proof can be found in

Robert M. Wald, General Relativity, University of Chicago Press, 1984. In print, ISBN 0-226-87033-2; list price $34.00 (paperback).

Mitra has repeated the claims (1)-(6) in other preprints, and has made still more incorrect claims, and he has not accepted correction of his errors:

> I don’t have the stength to go through this paper, but it sounds absurd to me:

Of course it is absurd!

> certainly reasonable solutions of the GR equation exist which describe gravitation collapse formation of non-charged black-holes. They actually claim the later cannot form! Has anyone read this?

In fact, it is only necessary to skim the abstracts of Mitra’s preprints on gtr to see that they all make ludicrous claims which anyone who has worked through an elementary gtr textbook (e.g Schutz or Stephani) will know right away are dead wrong.

However, some years ago I read in some detail an earlier draft of and it was quite frankly so riddled with -elementary- misstatements and misconceptions concerning gtr as to be not only worthless but frankly embarrassing.  Mitra is terribly confused about gtr, and so it is no surprise that it is usually impossible to be sure exactly what he is trying to say at any given place in this preprint, because a person who is thoroughly confused in his own mind cannot
possibly express himself clearly!  Nonetheless, I believe that anyone who reads the abstracts of his preprints posted to LANL will see that he-does- make the claims I listed above.

It is very regrettable that this particular preprint (astro-ph/9910408) was (shame! shame!) actually -published-:

Found. Phys. Lett. 13 (2000) 543

The irate abstract to an earlier draft of this preprint, unfortunately no longer available on LANL (but I have a hard copy printout) stated that Mitra was having a great deal of trouble with the referees; I cannot understand why the editor apparently decided in the end to publish the paper with all the misstatements of the original left untouched.  The published version is -shorter- than the original draft but it is no less -erroneous-.

Chris Hillman

Home Page:


From: Abhas Mitra,
Newsgroup: sci.physics.research, sci.physics.relativity

Subject: Rejoinder to Hillman’s posting entitled “Black Hole Preprints by Abhas Mitra”

None of the various posting(s) made by Chris Hillman to this forum [can be said to be] a scientific critique of my preprints/reprints related to the non-occurrence of finite (gravitational) mass Schwarzschild Black Holes (BHs), for the following basic reasons:

(i) It seems that Hillman has not read my papers properly and has attributed or implied several aspects which are not contained in my papers. Neither has he made any sincere attempt to understand whatever portions he might have read. It is possible that he has carefully read only the abstract of these preprints
and subsequently given free rein to his prejudices and predetermined notions.

Some of the comments posted in this forum by others which were lapped up byHillman are also likely to have been based [solely] on reading of the abstract. [For example], as quoted by Hillman, someone [referred to as] “zirkus” wrote, “I have not looked at the following paper but, according to its abstract…”.

(ii) Most of my preprints are based on simple and exact analytical calculations (WITHOUT ever involving a single assumption or simplification) and associated equations. A meaningful critique of such works must point out the definitive errors/shortcomings in specific equations, quoting specific equation numbers.
If there is a conceptual error, a meaningful critique must point out which specific equation is based on those conceptual errors, or else provide an interpretation of which equations have been [derived or used] incorrectly.

But as one can see, Hillman has been unable to point out any such specific error. His comments are either vague or misplaced (by reason of not having read my preprints carefully). In general, his style of criticism is non-academic, non-professional, and non-collegial. On the other hand, his
critique, which is meant to be of a purely scientific nature, is actually filled with personal attack, calumny, derision and intolerance.  He has probably [resorted to] this style because he is apparently incapable of posting his critique to refereed journals, and he may actually be [resorting to] this style [out of necessity] to hide and make up for the absence of objective scientific content.

(iii) In the following, I shall show that whatever little analytical elements (non-numbered equations) there [happen to be] in the present critique by Hillman, actually CORROBORATE the results of my preprints!

  1. One of my previous preprints (gr-qc/9807197) tried to show that:

If we follow the radial geodesic of a test particle around a BH using any coordinate system including Lemaitre coordinates (r, t, t=comoving time), the geodesic, which must be TIMELIKE (ds^2 >0) at any non-singular region of spacetime, would become null (ds^2=0) at the Event Horizon (EH) at
R=R_g=2M. Hillman asserts that my derivation to this effect is incorrect because “Part of Mitra’s confusion throughout his preprints rest upon persistent failure to distinguish clearly between coordinate slopes and physical velocity…”.

While he makes this accusation, note that he has NOT pointed out which EQUATION is based on such “confusion” and in turn, which specific result is incorrect because of such “confusion”. To hide his inability to actually pinpoint the specific location of error or “confusion” he has unnecessarily introduced grandiose-sounding mathematical jargon, [i.e. the] “Painleve chart”, and claims, WITHOUT ACTUALLY SHOWING IT, how the “Painleve Chart”‘ disproves my result. However, his writing makes the pretention that he has actually disproved my result. For the benefit of the serious readers, I give below
the essence of my proof:

In Lemaitre coordinates, the radial geodesic (angular part=0), the metric of a test particle around a BH is

ds^2 = dt^2 – g_rr dr^2                      (1)


g_rr = [(3/2R_g) (r-t)]^{-2/3}               (2)

The invariant circumference coordinate R is related to r, t in the following way:

R = [(3/2R_g) (r-t)]^{2/3} R_g               (3)

Thus at R=R_g (2M),

[(3/2R_g)(r-t)]^{2/3} = 1                    (4)

Using Eq.(4) in Eq.(2), we find that,

g_rr = 1   at   R = R_g                      (5)

Using Eq.(5) in Eq.(1), we have

ds^2 = dt^2 – dr^2   at   R=R_g              (6)

Note that while he uses the symbol “r” for circumference coordinate, I am using “R” for the same; also while Hillman uses “m” for the gravitational mass of the BH, I use “M” for the same.

We would require here a standard result:

dR      -(1-2M/R)
— =  ———- [E^2 -(1-2M/R)]^{1/2}      (7)
dT          E

where E is the conserved energy per unit rest mass of the test particle. Since t is the comoving time, we have

dt = (1-2M/R)^{1/2} dT                       (8)

Now by using the above equations, it can be found that,

(dr/dt)^2 = 1   at   R = R_g                 (9)

Hillman also writes that “it is true that the coordinate slope equals to -1 here”; by ‘here’ he means at R=2M. To verify the correctness of Eq.(9), however, it would be better to see the Eq. 3.12.5, pp. 112 of
Zeldovich and Novikov, Rel. Astrophysics, Vol. 1, Univ of Chicago (1971):

(dr/dt) = +/- (R_g/R)^{1/2}                  (3.12.5) of ZN

Note that the tau of ZN is our t, r of ZN is our R and vice-versa, and recall that we have taken c=G=1.

By putting Eq.(9) in Eq.(6) one can find that INDEED

ds^2 = 0   at R=R_g=2M following the radial geodesic.  (10)

If the EH R=R_g were a mere coordinate singularity and actually a regular region of spacetime, GTR demands that the geodesic must remain timelike there and we should have had ds^2 > 0. Thus Eq.(10) implies that the R=2M is NOT a non-singular region of spacetime. [Rather], it corresponds to a true physical singularity. But, for a BH, we know that the true physical singularity is at R=0. Therefore we can reconcile Eq.(10) with this knowledge by recognizing that we must have

R = R_g = 2M = 0                             (11)

In other words, the mass of the BH must be

M = 0                                        (12)

Does this proof ever involve the accusations levelled by Hillman? Did we ever mention “physical velocity” in this proof? Or did we [suffer from] any “confusion” between “coordinate slope” and “physical velocity”? Thus all that Hillman writes in the guise of a critique is actually gibberish. But to hide the gibberish, he strews some unnecessary mathematical and geometrical jargon around in his postings about my work.

  1.  In Sec 3 of his posting, Hillman writes that “Mitra claims that the Kruskal-Szekers chart has a coordinate singularity at R = 2M; this absurd (and very incorrect) conclusion…”

Now the fact is that, I HAVE NEVER EVER MADE SUCH A CLAIM IN ANY OF MY PREPRINTS. This again clearly shows that Hillman has not even cared to read my papers carefully before ridiculing them by abusing the internet. On the other hand, I have pointed out, in several of my preprints, that
even if one uses the Kruskal coordinates u and v, one would find that the EH is a true physical singularity. In other words, I have derived Eq.(10) using [these] coordinates. In fact, I have done so in a most straightforward manner using Schwarzschild coordinates too. For the benefit of the serious readers, I shall, again, give below the essence of my proof that (du/dv)^2 =1 at R=2M in this regard:

In one of his earlier postings [webpages?] entitled “Hall of Shame”, Hillman had ridiculed my work purely on the basis of prejudice without even attempting to put up any scientific critique. All that he could say
was to mention a preprint, (astro-ph/9905144), by I. Tereno which had scientifically, albeit erroneously, criticised my preprint. Nevertheless, here Hillman had conveniently forgotten to mention any of my REBUTTALs to Tereno’s work:

  1. Mitra, astro-ph/9904163 and 9905329

Anyway let me proceed with my proof:

From Eq.(7), it follows that, dR/dT =0 at R=R_g=2M. And since dT is an infinitesimal quantity by definition (not to be confused with delta T, which could be finite or even infinite), we have

dR = 0   at R=R_g=2M   along a radial geodesic   (13)

The Kruskal coordinates obey, everywhere in the Kruskal diagram, the equation

u^2 – v^2 = (R/2M -1) exp (R/2M)              (14)

so that,

u^2 = v^2,   (v/u)^2 = 1    at R=2M           (15)

Now differetiating Eq.(14) w.r.t. R, and using Eq.(13) on the LHS, it follows that

u du – v dv = 0    at R=2M                    (16)


(du/dv)^2 = (v/u)^2    at R=2M                (17)

Invoking Eq.(15) in Eq.(17), we see that

du^2 = dv^2   at R=2M                         (18)

Now for a radial geodesic, it can be seen that the Kruskal metric at R=2M is:

ds^2 = (16M^2/e) (du^2 – dv^2)                (19)

Invoke Eq.(18) here and obtain

ds^2 = 0    at R=2M along a radial geodesic   (20)

Again note that this proof neither involved any mention of “physical velocity” or any associated “confusion”. Hillman unnecessarily and irrelevantly invokes the “Lambert W function” without showing how the W function or any other function would actually disprove my Eqs. 13-20. Here Hillman has used two variables “R” and “T” without even mentioning what they are (note that I use R and T for Sch. coordinates); presumably they are proportional to our u and v. For the radial part of
the metric, Hillman too finds, through a convoluted route using the “Lambert W function” that at the EH

ds^2 = (8m/e) (-dT^2 + d R^2) ,  R and T not defined by Hillman.

By comparing with Eq.(19), it seems that R= 2m u and T =2m v (at least at the horizon). Then, we would have (dR/dT)^2 =1 at the EH, and Hillman’s Eq. too would give ds^2=0 at the EH. As explained earlier,
this would mean that the mass of the BH, M=0.

  1. Since Hillman has been unable to point out any real error in any of my preprints, he goes on citing one standard book after another (without showing how those books actually negate my precise derivations). If at any given point of time, the existing scientific literature and interpretation of laws of Nature were the ultimate without leaving scope for new analysis and interpretations, there would not have been any
    scientific or intellectual progress, and, intellectually, we would have continued to be like the prehistoric cavemen.

My criticism of the Oppenheimer-Snyder paper is based on the fact Eq.(36) of it overlooked the fact that the ARGUMENT OF A LOG FUNCTION CANNOT BE NEGATIVE. This statement cannot be negated by citing books and pouring forth ridicules. And this statement demands that during the collapse of
the dust-ball, we must have

2M/R < = 1                                    (21)

This mathematical result means that TRAPPED SURFACES DO NOT FORM in the O-S dust collapse. As I have shown (astro-ph/9904163, 9910408), unless Eq.(21) is incorporated into the O-S analysis, the behaviour of the metric coefficients would be inconsistent and unphysical at R=0.

Here Hillman makes mention of the “Vaidya null-dust” without knowing that Prof. P.C. Vaidya himself has found my work to be completely correct.

  1. Hillman writes that “Mitra claims that the well-known ‘trapped surface’ singularity theorem is false! The correct statement and proof can be found in..”

Firstly, I never mentioned “trapped surfaces” as singularities. [Secondly,] it appears that Hillman is unaware of the fact that the well-known singularity theorems are based on several ASSUMPTIONS. The
most crucial assumption here is that there is a “trapped surface” in the spacetime. Now when, by definition, existence of “trapped surfaces” is an ASSUMPTION how can any book PROVE the existence of trapped surfaces?

On the other hand, I have actually proved, in a general manner, for spherical collapse of baryonic matter, that trapped surfaces do not form at all. In other words, I have shown that the crucial assumption behind
the singularity theorems is incorrect (for collapse of isolated bodies). And since the essence of my proof is so straightforward, for the serious readers, I present it below:

All spherical collapse involving baryonic matter and radiation obey (see ref. in my paper) a relation

Gamma^2 = 1 + U^2 – 2M/R                      (22)


Gamma = dR/dl                                 (23)

and dl is an element of proper length along the radial worldline of the collapsing fluid. Also,

U = dR/d tau                                  (24)

where dtau is an element of proper time following the fluid element. Clearly, Eqs.(23) and (24) are correlated as

U = Gamma V                                   (25)


V = dl/d tau                                  (26)

and let us treat V as a pure symbol.

By putting Eq.(25) in Eq.(22) and by transposing, we have

Gamma^2 (1- V^2) = 1 – 2M/R                   (27)

Now by using the result that the determinant involving the metric coeff. of any metric must be negative, I have shown that, if (1-V^2) is negative, then so must be Gamma^2, so that the LHS of Eq.(27) is always positive. Then it follows that

2M/R < = 1                                    (28)

a result obtained independently from the O-S work. Eq.(28) shows that, if the fluid [were to] collapse to a singularity at R=0, under positivity of mass, one must have

M–>0 as R–>0                                (29)

  1. The preprint astro-ph/9910408 was published in Foundation of Physics Letters and this has obviously greatly disappointed Hillman and others. If they are so sure that it is all wrong, I would suggest that they
    submit an academic critique of my paper to Foundations or any other standard refereed journal.

This paper was published in Foundations after several referees failed to point out any specific errors in my work, after some referees found that the work is “mathematically” correct, and eventually after two anonymous referees recommended its publication with some revisions. This is hardly
a “very regrettable” or “shame shame” procedure.

  1. Hillman’s posting started with criticism of my work hep-th/9905182. In this work I never claimed that non-extremal BHs with mass M>Q (charge) are not exact GTR solutions, as implied by Hillman. In fact they are as exact solutions as the Schwarzschild BHs. But if one starts with a M>Q case and slowly reduces Q to Q=0, one should recover the Sch. BH. In such a case, one would obtain a finite mass M>0 BH. But since I have already shown, by several independent modes, that the only allowed value of M is
    0, eventually, it is [only] the extremal BHs with M=Q solutions which are to be accepted because they lead to the correct result M=0 when Q=0.

I invite Hillman and all other readers to write a proper scientific critique pointing out specific errors in my equations and interpretations in case any of them think that my work is erroneous. Incidentally, through
email, I have repeatedly requested Profs. S. Hawking. R. Penrose, K.Thorne, C.W. Misner and many others to send their critique, if any, to me. But none of them have acted so far.


Dear Friend and Colleagues,

It is with some disappointment and reluctance that I am forwarding this message to the sci.physics and sci.physics.relativity newsgroups, which are unmoderated. I hope, nevertheless, that it will be given due
consideration by those readers of this newsgroup who are open-minded and sincere enough to seek scientific truths even if they seem to go against their mathematical or physical “intuition”.

On 19th July, 2001, a rather scathing attack was made by Chris Hillman on the work of Abhas Mitra, regarding the formation and existence of black holes (or rather, lack thereof). This is still available on the
sci.physics.research archives:

Note that the tone used in this email is quite intolerant and undignified, being full of personal attack and “flames” and is clearly quite contrary to the charter of the sci.physics.research newsgroup. Nevertheless, the moderators of the sci.physics.research newsgroup (John Baez and Matt McIrvin) accepted Hillman’s posting for distribution. Note that John Baez is currently hosting Hillman’s relativity website, so one can safely presume that the three of them know each other fairly well.

I made Abhas Mitra aware of Hillman’s posting, and he prepared a detailed rejoinder to it, including mathematical details, which I forwarded to the newsgroup. This rejoinder was rejected by the moderator, Matt McIrvin, on the basis that it contained “personal attacks” against Chris Hillman. This of course was surprising, as (i) Mitra’s rejoinder was a completely justified and direct response to Hillman’s abusive posting, and (ii) was much tamer in content besides, with logical mathematical arguments being presented to show the mathematical misconceptions in Hillman’s original posting.

Both Abhas Mitra and myself independently complained to Matt McIrvin, expressing our surprise at his rejection of the rejoinder. To this we have yet to have, nor do we any longer expect, a response. Matt
McIrvin did suggest that if we sent a version without the personal attacks, he would consider it for posting. On this basis I prepared a second version (appended below). This has neither appeared on the
sci.physics.research newsgroup, nor has McIrvin had the courtesy to give an explanation as to why.

The moderators of the sci.physics.research website are quite clearly guilty of double-standards here. Apparently it is fine to publish inflammatory articles so long as they are protecting established
physical preconceptions – and clearly it is fine to censor ideas which would seem to contradict these established preconceptions). Scientists, and indeed communities of scientists have often made
mistakes or had serious misconceptions in the past. This human attribute is of course not limited purely to science.

It would be reassuring if we were to see some humility and willingness to accept this human fallibility amongst our scientists – no matter how intelligent or how objective we may hold *ourselves* to be. Arrogance and prejudice are no replacement for honesty and sincerity in searching
for scientific truth, no matter where or what its source.

To the end of this introduction, I have attached the edited rejoinder, as well as Chris Hillman’s original, and the original rejoinder. I will let the readers judge for themselves on the matter.

Sincerely yours,

Sabbir Rahman.

Having made Prof Abhas Mitra aware of the postings to the newsgroup sci.physic.research regarding his work on the non-existence of black holes, he has kindly prepared the following rejoinder. He has asked me to edit the English and forward it to the newsgroup on his behalf. I would request that any questions be sent to Prof Mitra directly – as I understand it he does not have access to newsgroups and has certainly never posted to one before. I have cross-posted this message to sci.physics.relativity as the contents are obviously relevant to this newsgroup as well.

The only changes made to Prof Mitra’s original are fairly minor spelling and grammatical changes to clarify the meaning where this may have been unclear. Non-trivial adjustments requiring some element of
personal interpretation of the originally intended meaning have been placed in square brackets, though the resulting text has been kept as faithful to the original as possible. The equations and mathematical
arguments should, however, have remain unchanged.

For reference, the original posting by Chris Hillman to which this reply is addressed can be found at:

There seems to be some confusion regarding Prof Mitra’s academic background. So to clarify these on his behalf: Abhas Mitra is a theoretical high-energy astrophysicist by training. His PhD thesis was
entitled “A New Theory of Ultra High Energy Gamma Ray Production in Cygnus X-3” from the University of Mumbai, India. He was a full member of the American Astronomical Society from 1993-95, is a member of the International Astronomical Union and has been a life member of the Astronomical Society of India since 1983. He has published many papers in journals such as Astrophysical Journal, Astronomy and Astrophysics, and Physical Review Letters. He has been an invited speaker on various
topics of High Energy Astrophysics in many conferences, and has worked as a referee for the Astrophysical Journal amongst others. As far as research on the physics of the Central Engine of Gamma Ray Bursts is concerned, he happens to be the only individual having publications in
refereed journals.




I dedicate this note to my friend Dr Sabbir Rahman whom I am yet to meet.

Abhas Mitra, 27/12/2014