Eternally Collapsing Objects

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.

Suppose a massive spherical star started collapsing due to its own weight. Then by Newtonian physics, sooner or later,  in a FINITE time, the entire star material should collapse to a geometrical point forming a SINGULARITY  where density of matter and strength of / are infinite. It  must be so because in Newtonian physical space -time has an absolute meaning and the (initial)  actual/locally measured  radius of the star, say R0,  is Fixed & Finite:

In contrast, in Einstein’s “General Relativity” (GR),  in the presence of gravity, the actual interior  or  PROPER radius (L)  is always larger than the externally perceived radius (R); i.e., L =  a R > R, because a>1.  Furthermore, the value of   “a”  will always increase if  the strength of self-gravity will increase; i.e., if the star would get more compact.  In fact, in strong gravity, similar stretching  (dilation)  happens for  time intervals too.  In this sense, in GR,  space-time is like a rubber membrane which gets stretched with the increase of gravity:

Therefore, there is no a-priori certainty that a Newtonian type singularity must occur for GR collapse. Despite this , even after the introduction of GR, most general relativists believe  in essentially the same Newtonian picture of gravitational collapse  by which collapsing matter must converge in a point like singularity in a finite ( proper/ physically measured) time.  This is unfortunately the case though not only  Einstein himself,  but  also many great physicists like

Sir Arthur Eddington, the best theoretical astrophysicist ever, Sir Paul Dirac (Nobel Laureate), one of the finest theoretical physicists ever,

Nathan Rosen and many others argued that this picture must be incorrect and the star must not collapse beyond what is known as its “Schwarzschild Radius”:

Here G is the gravitational constant and c is the speed of the light. For a star like the Sun, the Rs ~3 km and for star ten times more massive, Rs ~30 km.

In particular,  Dirac’s comment made in Proc. Royal Soc. (London) A 270, 354 (1962),  was:

“The mathematics can go beyond the Schwarzschild Radius and get inside, but I would maintain that this inside region is not physical space, because to send a signal inside and get it out again would take an infinite time, so I feel that the space inside the  Schwarzschild radius must belong to a different universe and should not be taken into account in any physical theory”.

Such objections however got ignored because of two misleading mathematical developments, and effectively mathematics beat physics:

1.  In 1939, two American physicists J.R. Oppenheimer & H. Snyder APPARENTLY showed that for general relativistic collapse of a DUST (a fictitious form of matter having no pressure at all!), not only does the DUST collapse to a singularity but a Black Hole (BH) is formed whose imaginary surface lies at Rs  and nothing can escape from within this surface, now called, an Event Horizon [1].

2.  Further, in 1965, while studying generic gravitational collapse, Sir Roger Penrose, a highly regarded mathematical physicist, ASSUMED that for continued gravitational collapse, sooner or later, a one way trap – door will form, and once the collapsing matter would enter this trap-door it will be doomed to hurtle inwards  no matter how much powerful rocket it would employ  to escape [2]. He essentially ASSUMED that at the advanced stages of collapse,  any section of the star having mass M and radius R, must plunge within its running Schwarzschild radius:

R(t)  < 2 G M(t)/c2  (1)

Obviously, this idea was a sort of generalization of the concept of the  formation of   an “Event Horizon” and got known as  a “TRAPPED SURFACE”. Once this ASSUMPTION was made, then under reasonable physical conditions, the collapsing matter must again end up in a singularity, as intended. Ideally, Penrose should have proved the inevitability of  the formation of  a trapped surface. But instead, he just ASSUMED the crucial Eq.(1)  which he was expected to prove or at least  attempt to prove! Nevertheless,  post 1965, most of  the mathematical relativists  adopted  the same ASSUMPTION. If a singularity will indeed form, the space-time membrane, after the initial dip, would suddenly be pinched off and stellar matter would  hit the  BOTTOM of the pit:

It appears that, only one  humble mathematical physicist Kriele  offered a proof that at least for a homogeneous sphere, there cannot be any trapped surface [3] http://seminariomatematico.dm.unito.it/rendiconti/cartaceo/50-1/147.pdf . Unfortunately, Kriele could publish his result only in an  obscure journal and maybe  sensing trouble, he himself NEVER cited his important result!

Given this backdrop, in 1998,  completely unaware of Kriele’s work,  I showed that spherical GR collapse cannot form “trapped surfaces”  irrespective of whether the sphere is homogeneous or not.  I  published my proof  in peer reviewed journal first in 2000 [4] and in a broader context later [5,6].    Note, if a trapped surface did not exist, then in principle, collapsing matter could form
(i) Quasi-static Objects (Chandrasekhar mass limit becomes irrelevant for very hot stars),
(ii) Rebound due to heat , radiation or pressure, or
(iii) Simply keep on contracting indefinitely!

As already mentioned, while the last option is absurd in Newtonian physics, it is not so in GR.  As gravity increases, the space-time membrane elongates/dips and forms a pit; a particle undergoing collapse has to slide down this membrane into the pit. And if indeed the strength of gravity would grow indefinitely  (a–> Infinity) during continued collapse, then the

GRAVITATIONAL PIT TOO COULD BECOME ENDLESS & BOTTOMLESS

If so, continued gravitational collapse should indeed become ETERNAL without the formation of a true singularity.  Consequently, objects undergoing continued general relativistic collapse should end up as “Eternally Collapsing/Contracting Objects” rather than as Black Holes or “Naked Singularities” (Singularities not covered by any Event Horizon).

Later as  I interacted  with several American astrophysicists like Darryl Leiter , Stanley Robertson [7] and Norman Glendenning, it became clear that as the contracting object would become hotter and hotter under gravitational compression, the Radiation Pressure would become so strong that it would almost counter balance the inward pull of gravity. And it is in this way that  ECOs  conceived in 1998 could be realized  in real life [8, 9, 10, 11, 12]. Thus ECOs are

Extremely Hot and Relativistic Radiation Pressure Supported Stars

whose radius  hovers  just above the  INSTANTANEOUS “Schwarzschild Radius” as conceived by Einstein, Eddington, Dirac, Rosen and many other great physicists.

These quasi-static ECOs are however always radiating (unlike true black holes) and losing mass energy by  the E= Mc2 formula. Suppose  as a given ECO radiates  and its mass decreases, it contracts by 1cm. But as it contracts, its gravity becomes bit stronger and dips the rubber membrane of internal space may be by 1.1 cm. Thus it becomes a hopeless  unending chase for singularity.

Recall that the Acceleration Due to Gravity  as measured by any static observer on the Event Horizon of a black hole is g=Infinity (limitless).

(And this is the reason,  why nothing can escape  from within the EH)

Similarly, gravitational red-shift of the EH, as seen by a distant observer too is infinite: z=Infinity (unbounded, limitless)

This means that, even if an extremely energetic gamma-ray photon would start its journey near the EH, its energy would reduce by a factor of (1+z) as it would reach the distant astronomer. And since z=∞, no energy would reach the astronomer, and he would see the EH as black.

In contrast, on the surface of an ECO, ” g’” and  “z” are extremely large, may be thousands or even million times larger than the corresponding values for a Pulsar/Neutron Star which is supposed to be an extremely compact star (object). Accordingly, to a distant observer an ECO appears almost as a “black hole” though it has a physical surface made of hot plasma (fire) unlike the fictitious mathematical surface of a black hole (Event Horizon).  As mentioned, it is the relentless heating due to extreme gravitational compression which makes the ECO essentially a “Ball of Fire”.

  • ABSOLUTE GROUND STATE OF COLLAPSE

In classical physics, an absolute ground  state is characterized by E=0 . So, if a star would really become “dead” and the “singularity” would be its “dead body”  or the absolute ground state, then the singularity too should have

          E=Mc2 = 0; i.e.,  M=0::  Expected Mass of Singularity

Now recall that, the interior of an ASSUMED   (static) BH is swept clear of all matter/radiation because nothing  can stay put inside its  ”trapped region”, and the “singularity” is the only source of  matter/energy. Thus it is expected that a (neutral) BH may have unique mass M=0!  And indeed it was found that though the mathematical “black hole”  solution is a correct one, BHs nevertheless have unique gravitational mass M=0 [13, 14].

Note, in GR, an occurrence of M=0 need not mean absence of matter! This is so because gravitational mass is the sum total of all sources of energy including the NEGATIVE self-gravitational energy. Thus, an occurrence of M=0 may  imply that the (negative) self-gravitation has been so extreme that it has off-set the all other sources of energy like rest mass-energy of protons, neutrons constituting the star and all internal energies (heat, pressure etc).

And such a proof confirms that the so-called BLACK HOLE CANDIDATES having huge masses cannot be true black holes.

In effect the ECOs strive to become true mathematical black holes which have the unique mass M=0 [13]. However, an ECO never succeeds to attain this state. As an ECO continues to slide down self-created pit, the pit gets deeper and deeper, i.e. the depth increases eternally :

Actually, near the  singularity,” a” tends to blow up making interior radius L too blow up! Simultaneously, both “g” and “z” –> Infinity. Thus as if every sphere would internally  be stretched to Infinity if self-gravitation would push them towards gravitational singularity. Although such a scenario is weird, absurd from the point of view of Newtonian gravity,  it is plausible in GR;  as if all objects contain an Infinity within themselves!


Note, in principle, one should solve Einstein’s equations analytically to see whether singularities are indeed formed or not. Unfortunately, even the infinitely simplified Newtonian collapse equations cannot be solved analytically (or numerically, without many simplified assumptions) because one does not know the exact equation of state of matter and radiation transportation properties at extreme high density and temperature. Thus, so far no-body has analytically solved Einstein’s equations for truly realistic cases (without making various assumptions) to show that singularities are formed! And only way one can solve collapse equations analytically and without making convenient simplifications is by setting pressure=0! Of course, one must not hope that such an extreme unrealistic assumption would ever lead to a true physical picture.

In hindsight, Penrose overlooked that

(a)     Gravitational collapse must be radiative,  and the value of  M and Rs must keep on decreasing [15], and there is no fixed goal post which collapsing matter can target

(b)   Also, unlike what he ASSUMED, pressure always decreases effective gravity and  continues to oppose collapse in general relativity as it does so in Newtonian gravity [16].

Effectively,  at least for a spherical case,  Penrose assumed gravitational free fall of the star by ignoring resistive effects like pressure gradient and radiation/ heat flow etc.

In fact what Roger Penrose,  John Wheeler and many GR experts forgot was the intuitive comments by none other than Oppenheimer & Snyder THEMSELVES before they embarked on the idealistic, unrealistic computation about gravitational collapse:

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.

And may be  research works cited here  have indeed substantiated the above comment made in 1939 by the persons who are (incorrectly) credited with proving the inevitability of formations of “Black Holes’’ and “Singularities’’ in GR!

Incidentally, in 2011,  it was shown that even the apparent formation of “Trapped Surface” and “Black Hole” in Oppenheimer & Snyder Collapse was only a mathematical illusion and physically it does not correspond to any gravitational collapse at all as because when pressure is strictly zero, density too is zero, and there is no matter, no gravity [17]!

  • RADIATION PRESSURE SUPPORTED STARS (RPSS)

Most of the stars we know of, resist their gravity by means of gas pressure, i.e., the pressure due to electrons, ions and atoms. But, in the 1960s,   Sir Fred Hoyle, who many believe should have got the Nobel not once but twice,  and Nobel Laureate William Fowler  first pointed out that stars can be supported by radiation pressure alone if they would be sufficiently hot:

There is NO UPPER MASS LIMIT FOR RPSSs

They went on to suggest that QUASARS contain such hot  RPSSs  rather than true BHs. Similarly, ECOs too are hot RPSSs. But there are important differences:

The RPSSs conceived by Hoyle & Fowler  were  strictly static Newtonian stars (z <<1), and their power source was ascribed to central thermonuclear energy generation having an efficiency < 1%   a la Sun; and the possibility of power generation by slow gravitational contraction was ignored.  Thus the RPSSs of Hoyle & Fowler were perceived to run out of nuclear fuel.

In contrast, ECOs  get their luminosity simply by releasing part of the gravitational energy by means of infinitesimal  slow contraction:

As the star contracts and becomes more compact, its negative gravitational potential energy becomes even more negative. Then, in order to conserve total energy, the star creates additional positive energy. Part of this positive energy makes the star even hotter, and the rest of it is radiated away, a process explained independently by Lord Kelvin &  Hermann von Helmholtz approx. 150 years ago:

The net  energy which can be tapped in this way is 100%:

E0=M0 c

the initial mass-energy of the star.  In this case, there is no need for invoking any specific fuel! Furthermore ECOs are dynamic solutions and they are extremely relativistic with z>>1.

As the exterior radius of the contracting star R(t) shrinks, Rs = 2GM(t)/c2     too shrinks (very slowly)  because of loss of mass by radiation.  So as if there is a perennial chase R(t) –>Rs = 2GM(t) /c2  ,  and by the time R(t) would catch up with  Rs,  both would  attain nil values as  the final ground state must correspond to    E=Mc2 =0.  Yet, it is likely that L(t)= a(t) R(t)–>∞, because a(t) –>∞  infinitely fast!

A STAR CHASING ITS OWN “SCHWARZSCHILD SURFACE”? Or,

 AN AMBITIOUS ASTRONOMER “ZEROING IN” on the “EVENT HORIZON”?
  • REFERENCES:

1. “On Continued Gravitational Contraction’’

J.R. Snyder & H. Snyder, Physical Review, vol. 56, Issue 5, pp. 455-459 (1939)

2. “Gravitational Collapse and Space-Time Singularities’’

R. Penrose, Physical Review Letters, vol. 14, Issue 3, pp. 57-59 (1965)

3.  “A BOUND ON THE CONCENTRATION OF MATTER IN SPHERICALLY SYMMETRIC STARS & ITS APPLICATION FOR THE EXISTENCE OF BLACK-HOLES’’

M. Kriele, Rend. Sem. Mat. Univ. Poi. Torino, Vol. 50, 1 (1992)

See: http://seminariomatematico.dm.unito.it/rendiconti/cartaceo/50-1/147.pdf

4. “Non-occurrence of trapped surfaces and Black Holes in spherical gravitational collapse: An abridged version”

A. Mitra, Foundations of Physics Letters, Vol. 13 , pp. 543 (2000); eprint arXiv:astro-ph/9910408

5. “On the final state of spherical gravitational collapse”

A. Mitra, Foundations of Physics Letters, Volume 15, Issue 5, pp.439-471 (2000); eprint (arXiv:astro-ph/0207056)

6. “Quantum Information Paradox: Real or Fictitious?”

A. Mitra, Pramana, Vol. 73, pp. 615 (2009); eprint arXiv:0911.3518

7. “Does The Principle Of Equivalence Prevent Trapped Surfaces From Being Formed In The General Relativistic Collapse Process?”

D. Leiter & S.L. Robertson, Foundations of Physics Letters for February 2003 ;
arXiv:astro-ph/0111421v3

8.  “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); arXiv:gr-qc/0601025

9. “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); (arXiv:gr-qc/0603055)

See: http://adsabs.harvard.edu/abs/2006MNRAS.369..492M

10. “Sources of stellar energy, Einstein Eddington timescale of gravitational contraction and eternally collapsing objects”

A. Mitra, New Astronomy, Volume 12, Issue 2, p. 146-160 (2006); (arXiv:astro-ph/0608178)

See: http://adsabs.harvard.edu/abs/2006NewA…12..146M

11.“ 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); (arXiv:1003.3518)

See:  http://adsabs.harvard.edu/abs/2010MNRAS.404L..50M

12. “Black Holes or Eternally Collapsing Objects: A Review of 90 Years of Misconceptions’’

A. Mitra, in Focus on Black Hole Research, edited by Paul V. Kreitler. ISBN 1-59454-460-3. Published by Nova Science Publishers, Inc., NY, pp.1-94 (2006)

13. 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).; (arXiv:0904.4754)

14. “ On the non-occurrence of Type I X-ray bursts from the black hole candidates”

A. Mitra, Advances in Space Research, Volume 38, Issue 12, p. 2917-2919 (2006); (arXiv:astro-ph/0510162)

15. “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)

16. Does pressure increase or decrease active gravitational mass density

A. Mitra, Physics Letters B, Volume 685, Issue 1, p. 8-11 (2010)

17. “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, Issue 1, pp.43-48 (2011); (arXiv:1101.0601)

See:  http://adsabs.harvard.edu/abs/2011Ap%26SS.332…43M

TRIVIA:  From the Nobel Citation of  Louis de Broglie who was crazy enough to claim that electrons behave like waves as well:

“When quite young, you threw yourself into the controversy raging over the most profound problem in physics. You had the boldness to assert, without the support of any evidence whatsoever, that matter had not only a corpuscular nature but also a wave nature. Experiments came later and established the correctness of your view.”

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