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.

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