Category Archives: Jayant Narlikar


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.

U-Turn of the Black Hole Research of Thanu Padmanabhan: Herd Behavior & Sociology of Modern Physics

As discussed in a previous blog, the noted scholar and researcher on gravitation, Prof. Thanu Padmanabhan, noted way back in 1988

 that the concept of Black Holes (BH) not only  leads to various inconsistencies but is unphysical too in certain ways. The  following paper coauthored by him

1. “The Schwarzschild Solution:Some Conceptual Difficulties’‘ by J.V. Narlikar & T. Padmanabhan published in Foundations of Physics, Vol. 18, pp.659-668 (1988)

mentioned that

“Nevertheless there are several conceptual difficulties associated with this simple and elegant solution that are usually ignored because of its manifest usefulness. Our purpose in this article is to highlight these problems since we feel that their eventual resolution will advance our understanding of the complex basic interaction of gravitation.”

Even before this the ABSTRACT of the paper may be recalled:

It is shown that inconsistencies arise when we look upon the Schwarzschild solution as the space-time arising from a localized point singularity. The notion of black holes is critically examined, and it is argued that, since black hole formation never takes place within the past light cone of a typical external observer, the discussion of physical behavior of black holes, classical or quantum, is only of academic interest. 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.”

About continued gravitational collapse to a point, these authors wrote that

“Thus we have arrived at an inconsistency at R = 0. It could be argued that a point source at R= 0 is unrealistic and that the Schwarzschild solution works for a distributed source only. This way out is unfortunately ruled out by the phenomenon of gravitational collapse that inevitably results in all the matter converging to R=0 in finite
comoving time.”

Accordingly, this paper repeatedly stressed so-called “Black Hole Candidates”  could be just  Quasi Black Holes rather than true BHs. Also in principle, a true BH cannot be observed directly. ”

Later in the following paper:

2.Phase volume occupied by a test particle around an incipient black hole

T. Padmanabhan, Physics Letters A, Volume 136,  p. 203(1989)

he wrote that

“The volume of phase space g(E) available for a system with a definite energy E plays an important role in statistical mechanics. We compute g(E) for a test particle in Schwarzschild geometry and show that it diverges as the source evolves to form a black hole.”

The consequence of the DIVERGENCE of g(E) is that Entropy of the ideal gas too would diverge in case an Event Horizon (EH) would form. Note that when ever unphysical or singular properties of the EH become manifest BH proponents shout “Oh! This must be a coordinate singularity”. But Padmanabhan correctly emphasized that blowing up of g(E) or entropy was not all any coordinate effect, on the other hand such a divergence shows true singular property of the EH. In fact one can easily trace the reason for the divergence of Entropy to the singular properties of the metric coefficients of the Schwarzschild Metric at the EH:

So as per the research carried out by Padmanabhan during the 1980s, atleast as far as General Relativity (GR) is concerned,

(i) The singularity of the Schwarzschild Metric at the EH can lead to genuine physical singularities, i.e., EH does not correspond to a mere “Coordinate Singularity”, and

(ii) The concept of BH is unphysical, and the so-called BH Candidates should be “Quasi- BHs” which to a far away observer look almost as compact and BLACK as theoretical BHs.

Following this ,a logical and honest future course of his Black Hole research could have been to explore the subtleties and nuances of the complex topic of General Relativistic Gravitational Collapse to see how continued collapse must prevent formation of Theoretical BHs, and on the other hand lead to only Quasi-BHs.

But This Was Not To Be

However, Padmanabhan did not pursue any such uncertain, un-trodden and tortuous path of research. Instead, after a hibernation in BH research;  he went on writing papers after papers which assumed that (iii) Gravitational Collapse Must Produce Black Holes and Event Horizons  and (iv) Black Holes and Event Horizons are one of the most important concepts in not only gravitation but physics as a whole IN DIRECT CONTRADICTION to his own previous ideas (i) and (ii).

 In particular, the idea of  Event Horizons became the basis of his future research in Gravitation which won him all accolades and awards. Recall his later papers:

3. “Quantum Structure of Spacetime and Entropy of Schwarschild Black Holes” : Physical Review Letters, Volume 81,  pp.4297 (1988)

4. “Event horizon of a Schwarzschild black hole: Magnifying glass for Planck length physics”: Physical Review D, Volume 59,  id. 124012 (1999)

5. “Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes”: Classical and Quantum Gravity, Volume 19, pp. 5387 (2002)

6. “Thermodynamics of Horizons”: Modern Physics Letters A, Volume 17, pp. 923-942 (2002)

7. “Why Gravity Has No Choice: Bulk Spacetime Dynamics Is Dictated by Information Entanglement Across Horizons”: General Relativity and Gravitation, v. 35, p. 2097-2103 (2003)

8. “Entropy of Horizons, Complex Paths and Quantum Tunnelling”: Modern Physics Letters A, Volume 19, pp. 2637, (2004)

9. “Gravity and the thermodynamics of horizons”: Physics Reports, Volume 406, p. 49-125 (2005)

10. “Entropy of null surfaces and dynamics of spacetime”: Physical Review D, vol. 75,  id. 064004 (2007) (Note: Null Surface is a surface similar to an “Event Horizon”).

11. “Einstein’s equations as a thermodynamic identity: The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons”: Physics Letters B, Volume 652, p. 338-342 (2007)

12. “Gravity as AN Emergent Phenomenon”: International Journal of Modern Physics D, Volume 17,  pp. 591-596 (2008)

13. “Ideal gas in a strong gravitational field: Area dependence of entropy”: Physical Review D, vol. 83, Issue 6, id. 064034 (2011)

14. “Structure of the gravitational action and its relation with horizon thermodynamics and emergent gravity paradigm”: Physical Review D, vol. 87, id. 124011 (2013)

Let us not first debate “Which of the two Self-contradictory phases of research of Padmanabhan is scientifically correct?”

Note that even if one would consider Quantum Gravity BHs, the latter must yield GR results for large quantum numbers or for macroscopic cases. Thus even QG BH research must be organically related to classical GR results. And let the latter phase, Papers 3-14 be correct.  Note there is no harm if an honest researcher would change his course of research as new arguments, new facts would emerge . In fact, up to 1996, I was a strong believer in the Black Hole paradigm, but I took a U-turn by 1998, as my research showed that my earlier belief was incorrect.

If Padmanabhan were an honest researcher and committed to scientific truths alone, he should have offered due explanation as to how his initial research which showed that GR should not allow BHs and EHs was incorrect. But he made no such attempts. In fact in a very mysterious manner, he never cited his own papers 1 & 2 in almost 100 papers, articles, books authored by him which involved the concept of “Black Holes” and “Horizons” or “Null Surfaces”!!!!!!

Now let us quickly adjudge which phase of his research is the physically and scientifically correct. In a series of peer reviewed papers by the present author, it has been shown that Black Holes and Event Horizons Indeed Correspond to Genuine Physical Singularities and Hence They Are Unphysical & Must Not Form. To cut a long story short, it has been shown that

  • A Radially Falling Material Test Particle Would Behave Like a Photon If It would Reach The Event Horizon Irrespective of the Coordinate System Used; and this is not allowed by GR.
  • The Coordinate Independent Scalar Acceleration Felt By An Observer At the EH is INFINITE indicating EH is a PHYSICAL SINGULARITY. In fact this is the reason that “nothing not even light can escape the Event Horizon”.
  • There are indeed scalars made out of Rimmanian Tensor which become singular at the Event Horizon and a Free Falling Observer Can Very Well Detect this singular surface contrary to the basic argument of BH paradigm: REF:

16. “A note on a local effect at the Schwarzschild sphere”

Karlhede, A.; Lindstrom, U.; Aman, J. E.: General Relativity and Gravitation, vol. 14, June 1982, p. 569-571 (1982)

17. “On the Local Detectability of the Passage Through the Schwarzschild Horizon”: Tammelo, Risto; Kask, Üllar, General Relativity and Gravitation, Volume 29, p.997-1009  (1997)

18.  “Detecting Event Horizons and Stationary Surfaces”: Gass, Richard G. et al. (arXiv:gr-qc/9808055)

Eventually, these questions were uniquely settled by this blogger by showing that  Black Holes Have a Unique Gravitational Mass M=0, and they represent only asymptotic final states of continued gravitational collapse which could be formed only after radiating away entire mass-energy and angular momentum of the collapsing body. REF: 

19. Comments on “The Euclidean gravitational action as black hole entropy, singularities, and space-time voids” [J. Math. Phys. 49, 042501 (2008)]: A. Mitra, Journal of Mathematical Physics, Volume 50,  pp. 042502 (2009);  (arXiv:0904.4754)

20. “Why the astrophysical Black Hole Candidates may not be black holes at all”: A. Mitra 2004, (arXiv:astro-ph/0409049)

It has also been shown that continued collapse should naturally give rise to Eternally Collapsing Objects (ECOs):

An Eternally Collapsing Object (ECO) is a compact star that resembles a ball of fire; and it is so hot that its radiation helps it stay put despite its intense pull of gravity. Being extremely compact, ECOs mimic mathematical “Black Holes” in many ways, but there are observational reasons to believe that the so-called astrophysical “Black Holes” are really ECOs: The eruptions and jet formations from the black hole candidates are better understood if they are indeed hot balls of fire rather than a cold piece of vacuum with an imaginary surface from which “nothing, not even light can escape”.
ECOs however asymptotically shrink towards the mathematical Black Hole state of infinite compactness.

21.“ Radiation pressure supported stars in Einstein gravity: eternally collapsing objects”: A. Mitra, Monthly Notices of the Royal Astronomical Society, Volume 369, pp. 492-496 (2006)

22. “Sources of stellar energy, Einstein Eddington timescale of gravitational contraction and eternally collapsing objects”: A. Mitra, New Astronomy, Volume 12, p. 146-160 (2006)

23. “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, Volume 404, pp. L50-L54 (2010)

24. “Kruskal Coordinates and Mass of Schwarzschild Black Holes: No Finite Mass Black Hole at All”, A. Mitra, International Journal of Astronomy and Astrophysics, vol. 2, issue 04, pp. 236-248 (2012), 10.4236/ijaa.2012.24031

25. “The Mass of the Oppenheimer-Snyder Hole: Only Finite Mass Quasi-Black Holes”, A. Mitra & K.K. Singh, International Journal of Modern Physics D, Volume 22,  id. 1350054 (2013)

26. “Does Pressure Accentuate General Relativistic Gravitational Collapse and Formation of Trapped Surfaces?”, A. Mitra,  International Journal of Modern Physics D, Volume 22,  id. 1350021 (2013)


What Could Explain The U-Turn of Thanu Padmanabhan?

Many greatest physicists including Eddington, Einstein and Dirac did not believe in Black Holes. However they could not offer any cogent reason which would resolve this paradox characterized by beautiful & exact GR solutions. The problem became even much more complex after it was found that indeed there are indeed valid Black Hole Candidates in X-Ray Binaries, Quasars and Many Galactic Centers. These massive compact objects certainly cannot be Neutron Stars or various other Alternatives like Star Clusters or Fictitious Super Massive Stars of Hoyle & Fowler.

And definitely Padmanabhan, though a great scholar and prolific researcher, was in no position to face this challenge. In fact his two initial papers 1 & 2, though profound and EXACT, are rarely cited. On the other hand, his latter papers affirming  his faith in the Black Hole Paradigm like a Neo-Convert got thousands of citations. Thanu is a bright researcher of course aspired to be not only a mainstream author but a sort of world leader, a Big Boss, in Gravitation Research. And this is possible only for a convert and not for someone who could be dubbed as a Crank or Heretic for being idiotic enough to challenge the holy scripture. A logical course of the initial phase of the Padamanabhan research could have been an attempt to show that the crucial assumption of “Trapped Surfaces” of Hawking & Penrose were incorrect. But  such a research would immediately make him cross swords with Penrose, Hawking and all “leaders” one can conceive of. It would have also led him to challenge the idea of a “Black Hole  Thermodynamics” and put on conflict course with Bekenstein, t’Hooft and who not?

Then let alone generous citation of his papers, they might not have been accepted in the first place, alteast in journals like PRD, PRL, CQG etc. May be despite his enormous talent, he would have been considered as a pariah with no entry to Blue Blood clubs. May be, no student would be ready to get associated with such iconoclastic research in order to protect their academic futures. May be, he would not have become “Distinguished Professor” and  the would be Director of IUCAA. Such possibilities are actually endless…

Incidentally, the fact that, continued GR collapse does not allow “Trapped Surfaces” in order that matter like trajectories of the elements of a collapsing star do not become lightlike was proved by the present author:

27. “Quantum information paradox: Real or fictitious?”: A. Mitra, Pramana, vol. 73,  pp. 615-622 (2009); (arXiv:0911.3518)

His books taught  everywhere are silent on the conceptual difficulties of black holes, his discussions on Gravitational Collapse do not go beyond the idealized pressure-less collapse model of  1939 by Oppenheimer & Snyder about which my take is the following:

28. “The fallacy of Oppenheimer Snyder collapse: no general relativistic collapse at all, no black hole, no physical singularity”: A. Mitra, Astrophysics and Space Science, Volume 332, pp.43-48  (2011)

He is leading researcher in Gravity; but his real uniqueness lies elsewhere: He is the only researcher in the world who hides in own research, appears to be mortally afraid of citing his own papers 1 & 2 which challenge the BH paradigm.

This is so when these two papers are scientifically correct because (i) the blowing up of g(E) at the EH is an EXACT & CORRECT result, (ii) EHs are indeed physical singularities and correspond to the central singularity because M=0 for BHs. Can QG qualitatively change these conclusions? It should not because Quantum Green’s Function blows up at the Event Horizon, which again can be traced to metric singularity of the Schwarzschild metric there.

28. “Quantum Field Theory in Schwarzschild And Rindler Spaces”:  D. Boulware, Phy. Rev. D11, 1404 (1975).

However, one may try to hide such singular behaviors by fudging the physics, by using the jargon of “near horizon” behavior, and by avoiding the EH. Even then most of the contributions of the relevant integrals naturally come from this “near horizon”. And then one can pretend that physics of the 3-D BH interior is encoded in the 2-D EH… then holography and justification of the most speculative hypothesis of BH Theromodynamics, Hawking Radiation can also follow. Of course, once on this glorious track, acceptance of the manuscript is guaranteed in any leading journal.

Incidentally, my prediction that the so-called BH candidates are ultramagnetized ECOs (MECOs) rather than true BHs have inspired several astrophysicists and here is a 2006 Center for Astrophysics, Harvard, press release to this effect:

One may partly appreciate this key to success of Padmanabhan research by recalling that post modern theoretical research, particularly, in the topics involving may not be about physical realities at all:

In their book A Different Approach to Cosmology, Fred Hoyle, Geoffrey Burbidge and Jayant Narlikar used the photo given below to illustrate the conformist approach to standard big-bang cosmology. ‘We have resisted the temptation to name some of the leading geese,’ they say. May be the same photo help explain the enormous success of  the research career of Thanu Padmanabhan:


Jayant Narlikar and Thanu Padmanabhan Correctly Challenged the Idea of Black Holes and Instead Argued for “Quasi Black Holes”

Prof. Jayant  Narlikar is a highly respected and noted cosmologist. He was closely associated with one of the greatest astrophysicists, namely, Prof. Sir Fred Hoyle who many believe should have got the Nobel Prize in physics. Prof. Narlikar is the founder Director of Inter University Centre for Astronomy & Astrophysics, a world renowned institute. And Prof. Narlikar is best known for Quasi Steady State cosmology. Similarly Prof. Thanu Padmabhan of IUCAA is a world renowned scholar in the area of gravitation & cosmology. And here I shall highlight the paper

“The Schwarzschild Solution:Some Conceptual Difficulties’‘ by J.V. Narlikar & T. Padmanabhan published in Foundations of Physics, Vol. 18, pp.659-668 (1988)…18..659N



It is shown that inconsistencies arise when we look upon the Schwarzschild solution as the space-time arising from a localized point singularity. The notion of black holes is critically examined, and it is argued that, since black hole formation never takes place within the past light cone of a typical external observer, the discussion of physical behavior of black holes, classical or quantum, is only of academic interest. 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.”


This paper was submitted on April 27, 1987, and got published in June 1988. And this delay indicates that, despite the academic stature of Prof. Narlikar, the referee may not have promptly accepted this manuscript. In those days, the editor of Foundations of Phys. was Prof. Alwyn van Merwe who was a very fair and liberal  editor quite unlike any editor of any mainstream journal of these days. Thanks to the wisdom of Prof. Merwe that he allowed publication of this very interesting and important paper.

Having made this introduction, I shall just paste excerpts from this paper which would convince anybody that Narlikar & Padmanabhan (NP) thought, and correctly argued that, finite mass black hole solution is unphysical and black hole formation should preferably be avoided in continued gravitational collapse:

“Nevertheless there are several conceptual difficulties associated with this simple and elegant solution that are usually ignored because of its manifest usefulness. Our purpose in this article is to highlight these problems since we feel that their eventual resolution will advance our understanding of the complex basic interaction of gravitation.”

As is well known, the Schwarzschild black hole solution begins with a Point Particle at r=0. And,  by studying this solution in the finite point mass limit, in pp. 663, NP comment:

“Thus we have arrived at an inconsistency at r = 0. It could be argued that a point source at r = 0 is unrealistic and that the Schwarzschild solution works for a distributed source only. This way out is unfortunately ruled out by the phenomenon of gravitational collapse that inevitably results in all the matter converging to r=0 in finite
comoving time.”

The latter comment by them that gravitational collapse must lead to a point singularity in a finite comoving time is actually based on the dust solutions which ignore all pressure, radiation, heat flow etc. 

The SECTION 3 of their paper is titled as


And then it elaborates,
“Black holes are generally believed to have formed by gravitational collapse of massive bodies. In a strict sense what is a black hole (BH)? It is an object surrounded by an event horizon. By contrast we may call an object a quasi-black hole (QBH) if it is highly collapsed and very dim but still outside its event horizon. It is well known that as an object collapses toward its event horizon the intensity of its radiation as received by an external Schwarzschild observer (with trajectory r=const) rapidly falls.
Thus a QBH will be invisible for all practical purposes although still outside the event horizon. Most astrophysical scenarios using black holes are concerned with QBH’s only. However, the laws of black hole physics apply to BHs, as does the notion of an evaporating BH. Our question is: are the BH phenomena really relevant to physics?

It is agreed that for a scientific hypothesis to be taken seriously, it must be testable–if not in practice (owing to limitations of technology available) at least in principle. So far as a BH is concerned, it is supposed to come into existence only when its outer surface enters the event horizon…..

For the detection of any object by whatever means, it must come within the observer’s past light cone. This does not ever happen for a BH. So none of the laws describing the behavior of BHs (as opposed to the QBHs) are in principle detectable or testable by the class of observers who stay outside their event horizons. Since most observers (including those on the Earth) are of this type, to them the BH’s are not relevant as physical objects.”

They correctly emphasize that the so-called “Black Hole Candidates”  could be just Quasi Black Holes rather than true BHs. Also in principle, a true BH cannot be observed directly. In SECTION 4, they concluded:

“In other words, no observers in the Schwarzschild metric (whether they stay outside  or inside the Event Horizon choose to fall inside) will ever be able to observe either the formation or the physical effects of a singularity at r = 0. We leave it to the reader to decide whether a singularity that can never be observed and that can never affect any physical process “exists” in any sense of the word.”

“To interpret the Schwarzschild solution as the metric produced by the point particle, one has to satisfy two conditions: (i) Einstein’s equations, with a point mass as the source, located at the origin, have to be satisfied by the metric and (ii) the source particle should follow a timelike trajectory. In Section 2 we pointed out that condition (i) is violated in the Schwarzschild metric. It is now clear that condition (ii) is also violated in any collapse that leads to a point source.”

Clearly, therefore, realistic gravitational collapse should not lead to the formation of a point singularity. However,  being misled by the Oppenheimer Snyder dust collapse picture, NP, at the same time thought, normal gravitational collapse should lead to a point singularity/black hole. It may be relevant here to mention that, in 2011, I categorically showed that the example of “Dust Collapse” is misleading, it is an illusion. In reality a dust has zero density (i.e., it does not exist physically), and dust collapse does not lead to any singularity/black hole

The fallacy of Oppenheimer Snyder collapse: no general relativistic collapse at all, no black hole, no physical singularity

A. Mitra, Astrophysics and Space Science, Volume 332,  pp.43-48 (2011)…43M

However NP were not aware of this, and their intellectual dilemma was more acute: To put the physical as well as mathematical inconsistency in the BH paradigm, defenders argue that, inside the Event Horizon, distance becomes time & time becomes distance. By analyzing such proposals, NP concluded:

“We therefore find that considering observers inside the event horizon makes the problems of interpretation even more difficult, and we wonder whether nature allows gravitational collapse to continue inside the event horizon at all.”

In the concluding section, NP demanded

“Those who believe that black holes (and not just the quasi-black holes) have physical relevance should produce a thought experiment to demonstrate to an

external observer that a black hole (and not a QBH) has formed in a given region.”

“We have no solution to offer for this difficulty nor do we believe that one exists within the conventional classical framework. These problems can be avoided by
introducing negative energy or stresses to reverse the gravitational collapse before the event horizon is formed.”

But how can matter with positive pressure suddenly develop negative pressure or negative energy? 

And clearly, here, they  forgot the famous quote by Sir Arthur Stanley Eddington:

“The star has to go on radiating and radiating and contracting and contracting until, I suppose, it gets down to a few km radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. … I think there should be a law of Nature to prevent a star from behaving in this absurd way!”

And in 2006-10, I along with Norman Glendenning showed that Eddington’s intuition was was correct: As the contracting object would plunge within its photon sphere, gravity becomes strong enough to hold in the radiation,  and sooner or later, the outward radiation pressure would arrest the catastrophic collapse of the star:

1. “A generic relation between baryonic and radiative energy densities of stars”

A. Mitra, Monthly Notices of the Royal Astronomical Society: Letters, Volume 367, Issue 1, pp. L66-L68 (2006);

2.  “Radiation pressure supported stars in Einstein gravity: Eternally Collapsing Objects”

A. Mitra, Monthly Notices of the Royal Astronomical Society, Volume 369, Issue 1, pp. 492-496 (2006);

3.  “Likely formation of general relativistic radiation pressure supported stars or `eternally collapsing objects”’

A. Mitra, Monthly Notices of the Royal Astronomical Society: Letters, Volume 404, Issue 1, pp. L50-L54 (2010);


And this was not enough. The popular idea that continued collapse must give rise to singularity is based on the “Singularity Theorems” of Hawking, Penrose & others. This theorem presumes that during continued collapse, a surface of no-return, “Trapped Surfaces” should form. In turn, this presumption is based on the idea of  pressureless “Dust Collapse”. And in full generality, I showed that this presumption is not realized:

“Quantum Information Paradox: Real or Fictitious?’  A. Mitra, Pramana, 73:615, (2009)

And this proof demands that any point singularity that may be assumed to be formed (in a mathematical sense) must have zero gravitational mass: M=0.

This means that while an extended object like the Earth, Sun, or galaxy must indeed have finite gravitational mass, their mass must shrink to zero if they would be assumed to shrink to a point.

And this consequence was also independently proved by me in the same year:

“Comments on “The Euclidean gravitational action as black hole entropy, singularities, and space-time voids”

A. Mitra, Journal of Mathematical Physics, Volume 50, Issue 4, pp. 042502-042502-3 (2009).….50d2502M

Thus the physical and mathematical conundrum experienced by Einstein, Dirac, Eddington, Hoyle, Rosen as well as by Narlikar and Padmanabhan (any many more) got completely resolved by work.

Finally Narlikar & Padmabhan were correct: General Relativity does not allow true Black Holes; on the other hand, it may permit  only “Quasi Black Holes” whose practical form could be the `Eternally Collapsing Objects”.