Magnetospheric eternally collapsing object

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A Magnetospheric Eternally Collapsing Object or MECO is a proposed alternative to a black hole. In essence, this theory states that massive objects that suffer gravitational collapse never actually form black holes since the build up of gravitationally trapped radiation pressure slows the collapse to a very small rate when the object becomes sufficiently compact (see Eddington luminosity). The main differences between MECO's and black holes lies in the fact that a MECO has a magnetic field and emits a significant amount of radiation (black holes can't have magnetic fields and "emit" only weak Hawking radiation).

Contents

[edit] History

The term Eternally Collapsing Object (ECO) was first coined by astrophysicist Abhas Mitra in 1998[1]. He argued that the so-called observed Black Hole (BH) Candidates must actually be quasistatic ultra compact objects (UCO), called ECOs, though they are asymptotically approaching the true BH state. He mentioned that "Much more importantly, the ECOs may possess magnetic fields whose value could be modest (in extragalactic cases) or extremely high (in stellar mass ECOs). In contrast, the intrinsic magnetic field of supposed BHs is zero. And ECOs might be identified as objects different from BHs by virtue of the existence of such intrinsic magnetic fields". He also claimed that the beamed emission from the Gamma Ray Bursts could be better understood if they are associated with birth of highly magnetized ECOs rather than non-magnetized BHs "[2]. The essential idea was that the non-singular ECOs are something like the Relativistic version of Neutron Stars. In several other preprints[3] and papers[4], he claimed that the so-called black holes are ECOs with physical surfaces and intrinsic magnetic fields.

The observational and theoretical arguments which actually proved that the Galactic Black Hole Candidates (GBHC) are Eternally Collapsing Objects containing strong intrinsic equipartition magnetic field in lieu of an Event Horizon (EH) was first shown in an important paper published by two American astrophysicists, Robertson and Leiter[5]. In this context of this new development, it was natural to add the term "Magnetospheric" to ECO to make it MECO, as was done by Darryl Leiter, Abhas Mitra and Stanley Robertson in 2001[6].

However, Mitra later withdrew his name from a subsequent revised version of the preprint he had written with Leiter and Robertson. The reason that Mitra withdrew his name was that he wanted the MECO picture to be a generic extension of the more general ECO concept, which would have no crucial assumptions and is thus model independent. But Leiter & Robertson were keen to make a specific model of MECO with some assumptions in order to immediately apply it to astrophysical observations, and their paper bearing the same title was subsequently published[7].

Robertson and Leiter elaborated on this important aspect by using the idea of disk-magnetospheric interaction [8]. Further, by invoking the well known "Magnetic Propeller" mechanism, they offered a novel explanation for a hitherto unexplained universal correlation between observed radio and X-ray luminosities in black hole candidates[9]. In turn, such works strongly suggest, from an observational point of view, that BHCs are actually magnetized ECOs. Yet one might argue that the evidence that BHCs are actually magnetized ECOs is circumstantial, like much of the other "evidences" in Astrophysics. One might also argue that, in principle, there might be alternative explanations based on other models.

[edit] Quasar Q0957+561

The more or less direct evidence to this effect came in 2005 largely because of the pioneering and protracted efforts by another American astronomer Rudy Schild:[10].

"Schild monitored the quasar's brightness for 20 years, and led an international consortium of observers operating 14 telecopes to keep the object under steady around-the-clock watch at critical times."[11]

The quasar Q0957+561 revealed its structure with very high resolution because of microlensing by the stars of an intervening galaxy along its line of sight. And this structure has broad similarity with the one expected from a magnetized neutron star endowed with an accretion disc.

The accretion flow in the equatorial plane is halted by the magnetic pressure of the central compact object and the inner radius of the disk is determined by the Alfven Radius. The accretion plasma flow then gets guided by the central dipole magnetic field towards the poles of the central magnetized compact object. During this process, the joint effect of spin and magnetic field may fling part of the accretion flow in an outward relativistic jet. Probably the fastest known astrophysical jet has a bulk Lorentz Factor of > 10, and it is associated with a X-ray binary Cir X-1 which contains a magnetized neutron star rather than BHC. This shows that presence of strong central magnetic field may be necessary to launch strong jets. There may be a wind outflow from the accretion disk itself. The magnetic push due to the central dipole field may also lend the wind an oppositely moving twin cusp-like structure. This "cusp like" picture may be compared with the hour glass like picture of magnetic field structure recently observed in a collapsing magnetized Molecular cloud[12].

Thus, the observations of the accretion disc of this quasar made with the aid of a gravitational lens seem to indicate that Q0957+561 has a magnetic field, which a black hole cannot have. The researchers deduced the existence of a magnetic field from the fact that the accretion disk has a gap of 4000 AU around the central object. A small part of the disk just outside of the gap seems to be glowing, which is interpreted as being a sign that the material is heated by a strong magnetic field. Such a glow is also expected because most of the accretion power is released at the inner edge of the disk, tructated by the magnetic pressure of the central compact object. In contrast, for BH accretion, eventhough, the inner edge of the disk formally lies at 3 Rg, where Rg is the Schwarzschild Radius, the flow is actually never truncated and, on the other hand, proceeds first to the EH and then, all the way up to the central singularity. The flow might get quasispherical in between and a glow is expected from region lying between the EH and the inner edge of the disk.

[edit] Basic Generic Theory

The basic underlying concept of an "ECO" is a model independent generic result:

Strength of gravity around a compact object or any object may be expressed by the observed surface gravitational redshift parameter z. For the Sun, one has, z \approx 2 \cdot 10^{-6}. For a neutron star which is usually considered as one of the best examples of a relativistic compact object, one has z \approx 0.1 -0.2. But the Event Horizon of a Schwarzschild black hole has z= \infty. Thus compared to an EH, all other known objects practically have z \approx 0. Hence, during continued gravitational collapse, the collapsing object has to traverse an infinite distance in the z-space, starting from z \approx 0 to z = \infty, in order to become a Black Hole.

It is known that, in strong gravity, i.e., for large z, many important general relativistic effects must occur. For instance, it is known that, in strong gravity, photons and neutrinos (not to talk of massive particles) move in curved paths. The corresponding angle of deviation from a rectilinear path is given by \theta \approx 4z if z < < 1, and this result constituted the historical verification of GR by Eddington in 1919.

Further, it is also known that realistic collapse of gravitational collapse must heat up the collapsing object and it must radiate out part of the energy gain due to release of gravitational energy during collapse. In the context of GR, this assertion has been made only in 2006[13].

Thus the quanta of "heat" i.e., photons and neutrinos generated within the collapsing object must move in increasingly curved trajectories as the continued collapse would proceed towards the = z= \infty Black Hole stage. As a result they would increasingly fail to move out of the compact object due to its increasing self-gravity. In other words, the collapse generated radiation would tend to get trapped by self-gravity. This process of pure self-gravitational trapping during continued collapse has recently been elaborated.[14].

It has been shown here that the energy density of the gravity trapped radiation would increase dramatically as \rho_r \proptor ~(1+z)2 as the collapsing body would attain z >>1 during continued collapse. Since by definition, a BH EH is marked by z= \infty, the trapped radiation density would tend to increase unabated until collapse process would be halted. This halting would occur when the the luminosity of the trapped radiation would attain the corresponding Eddington Luminosity at an appropriately high value of z. It must be so because the very definition of an Eddington Luminosity implies

The outward PUSH due to radiation = Inward PULL due to gravity ..(1)

In other words, sooner or later, at a sufficiently high z, the catastrophic collapse must be halted by the outward trapped radiation pressure and the object must become a quasistatic Radiation Pressure Supported Star(RPSS):

And this is how ECOs are inevitably born in realistic continued gravitational collapse.[15] In general, it has been shown that when the luminous compact object becomes extremely relativistic, it must become HOT and Radiation Dominated like the very early Universe unlike the "matter dominated" universe we know of [16].

ρr / ρ0 ~ z >> 1; if z >> 1 ...(2)

Here,

ρr= Radiation Energy Density; ρ0 = Rest Mass Energy Density

It is most interesting to note that the local mean temperature of such an extremely relativistic RPSS, i.e., an ECO is uniquely determined, by virtue of Eq.(2), by its mass alone:

T = 600 MeV (M/Msolar)-1/2 ...(3)

where Msolar is the mass of the Sun.

[edit] Comparison With True Black Holes

A true BH has an EH with z=\infty. In contrast, a MECO, the "intermediate state", has a finite but large z>> 1 and as it becomes more and more compact during continued collapse, in principle, its z \to \infty. While a true BH has a radius R = Rg = 2GM / c2, the Schwarzschild radius, a MECO has practically the same radius because its z>>1.

By definition, a MECO is radiating at its Eddington rate, which is the maximal luminosity. However, because of joint effect of gravitational redshift (of emitted quanta) and gravitational time dilation (slower ticking of clocks in stronger gravity), the observed luminosity of an ECO is

L(observed) = 1.26. 1038 (1+z)-1 (M/Msolar) erg/s ....(4)

Suppose, z =108 during one of these intermediate states. Then we will have

L(observed) ˜ 1031 erg/s ..(5)

for a 10 Msolar MECO. Note that the corresponding Eddington luminosity of a Neutron Star would have been

L (observed, Newtonian) ˜ 1039 erg/s ....(6)

which would be larger than a factor of (1+z) ˜108.

Thus unless the ECO would be very closeby, it would be extremely difficult to detect this dim quiescent ECO luminosity. Therefore, observationally, it may be mistaken as a true Black Hole.

One can appreciate Eq.(4) by noting that a true Event Horizon with z=\inftywould correspond to

L (observed, EH) =0 ..(7)

This is another expression of the concept that "nothing can escape the EH". Physically, the occurrence of an EH would correspond to an ABSOLUTE trapping of light (z=\infty). In contrast, the ECO is an intermediate state with z>> 1 during the formation of this "absolute trapping".

The most important difference between a MECO and a true BH is that the former is a dense ball of radiation supported plasma having a "physical surface" while the latter is the ultimate asymptotic singular state having no physical surface. Consequently, a MECO must possess strong intrinsic magnetic field like all other compact astrophysical plasmas while a non-spinning non-charged BH has no intrinsic magnetic field.

[edit] Evolution of a MECO

The total mass energy of a MECO (or anything) is E = Mc2 and it is losing energy as per Eq.(4). Then it follows that, its observed time-scale at a given z=z is

t(observed) = E/L ˜ 4. 108 (1+z) yr ...(8)

This time scale has recently been termed as "Einstein -Eddington" time scale [17].

As the MECO evolves to approach z \to \infty, naturally,

t(observed) → \infty ......(9)

This explains the rationale behind the phrase ``Eternally Collapsing Object". During this infinite journey towards the true BH state, a MECO burns its entire mass into energy/radiation and hence the eventual BH has M=0. In fact, this profound result has been obtained in a most straight forward manner by using the basic fact of Differential geometry that the Proper 4-volume associated with a BH (or anything) is independent of the coordinates used to measure it[18]. Recall that the Schwarzschild solution corresponds to the spacetime structure for a "Point Mass" or a "Massenpunkt". And it was shown long ago that even if there would be "Bare Mass" Mb for a point particle, the "Clothed Mass" obtained after accounting for the negative self-gravitational interactions is zero for a "Neutral Point Particle"[19].

Thus it is natural that the gravitational mass of a neutral Schwarzschild BH, i.e., a neutral massenpunkt is just zero. Hence any object with finite mass, M >0, cannot be a Schwarzschild BH. In particular, the observed BHCs with masses much higher than the Chandrasekhar mass limit, Mch, are least likely to be true BHs.

In reality, it would take not only infinite observed time but also infinite comoving proper time to attain this ultimate state indicated by the vacuum Schwarzschild solution of a Massenpunkt. To appreciate this, recall that the comoving proper time of BH formation in the Oppenheimer - Snyder Dust collapse is

t ~ M-½ ...(10)

so that, t → \infty as M→ 0. Thus the aymptotic (true) BH state is never ever formed. And the MECOs with z>>1, in many ways, mimic this idealized asymptotic BH state.

Since the MECOs are radiation supported HOT objects, the Chanhrasekhar Mass Upper limit has no relevance for them. This limit is applicable only to COLD objects supported either by cold quantum degeneracy pressure or ordinary kinetic gas pressure.

[edit] Oppenheimer Snyder Collapse

In 1939, Oppenheimer and Snyder indeed exactly solved the GR spherical collapse equations. And it is this exact solution which consolidated the idea of (finite mass) BHs apparently suggested by the static vacuum Schwarzschild solution in 1916. But it has generally been ignored that the Oppenheimer and Snyder "exact solution" was obtained at an enormous physics cost:

They were bound to completely ignore pressure, p=0, for the fluid. Consequently, there is no internal energy (e=0) and heat flux (q=0) for the Oppenheimer & Snyder idealized fluid termed as "Dust".

In contrast, for physical gravitational collapse, the radiation density and heat flux must increase dramatically much before any Event Horizon would form (cf. Eq.2):


Thus not only the Oppenheimer and Snyder Dust collapse, but all adiabatic collapse, in reality means no collapse because there cannot be any collapse without radiation. On the other hand, if one would insist for a collapse on the strength of only mathematics, the corresponding density of the fluid is

ρ = 0 ! .... (11)

For a pressureless ``Dust", this result can also be obtained from simplest thermodynamical considerations too. The pressure of a fluid, including localized quantum degeneracy pressure, arises because of internal random motion of the fluid particles either on a macroscopic scale or on localized microscopic scale. And thus pressure can be absent, p=0, only when either

  1. There is no fluid : ρ = 0 or,
  2. The fluid has no motion, i.e., it is at zero temperature.

In realistic cases, a fluid at T=0 (a perfectly degenerate case) would be already at a lowest thermodynamical state and cannot contract further. Thus if ``collapse would be still be feigned, one must have ρ = 0. In such a case, the mass of the fluid ball would be

M= 0 ..(12)

Thus all the BHs, mathematically generated in the Oppenheimer Snyder collapse or Adiabatic Collapse or any collapse that ignores the fact that ρr/ρ0 ≈z >>1, corresponds to zero gravitational mass! To recall, the standrad GR collapse and the BH paradigm assumes that once the collapse would cross the Neutron Star stage with z ≈0.1, it would directly proceed to the infinitely far away z=\infty BH stage in a flash (free fall time scale)!

[edit] Relevant Radiative Processes

One may also wonder what generates the luminosity of an ECO? Is it nuclear fuel or something else? It is definitely not because of any nuclear fuel. Recall that cold Giant Molecular Clouds with an initial temperature of T˜10 K contract to make extremely hot pre-main sequence stars. This heating during the preceding gravitational collapse is certainly not due to burning of any nuclear fuel. The key to the understanding of this physical phenomenon is again virial theorem :

Virial Theorem dictates that the fluid must become hotter during collapse. Although the MECO is quasi-static, in the strictest sense, it is always undergoing secular contraction. And it is this secular contraction which is generating fresh internal energy and radiation by the virtue of E=Mc2 formula.

But how does the fluid actually become hot?

The fluid becomes hot because of various radiative processes like Bremsstrahlung, Compton Scattering, Synchrotron Process and other inherent Radiative processes. All that is required here is that the fluid must be ionized. In addition, in order to have the synchrotron process, there must be an internal magnetic field and which is always the case. In fact, in the ultimate microphysical analysis, all energy generation by either nuclear or chemical means can be seen to be mediated by such basic radiatve processes. At sufficiently high temperature and densities matter can directly burn mass to form "energy" because of increased rate of such microphysical processes and by virtue of E = Mc2 formula. In contrast, for a contracting cold molecular cloud, the energy release is mediated by interaction between various molecular energy levels which is a very inefficient process. Further since in strong gravitational field, charged particles moving in curved trajectories, they emit radiation due to Gravitational Bremsstrahlung[20] and Gravitational Synchrotron [21].

[edit] Discussions

The observed radiation from the MECO surface or interior would be extremely redshifted by a factor of (1+z) >>1. The observed magnetic field would be smaller approximately by the same factor. The corresponding redshift factor could also be (1+zs), where zs is the redshift of the MECO "photosphere". At present we do not have any basic theory to predict the value of either z or zs for a MECO with a given mass M. But despite such uncertainties, it is expected that the quiescent and extremely faint MECO thermal radiation could peak in the infrared, millimeter, microwave or radio band.

A very pertinent question here would be how can MECOs have z>> 1 when the Buchdahl inequality states that for a spherical object z is bounded: z <2. Recall that the Buchdadl inequality corresponds to absolutely cold objects, having no radiation, and whose external spacetime is determined by strictly static vacuum Schwarzschild Metric. In contrast, in a strict sense, an ECO/MECO is always contracting and radiating. The spacetime exterior to an ECO is thus given by the famous radiative and dynamic Vaidya Metric rather than by non-radiating and static Schwarzschild metric. In such a case, the occurrence of z>>1 is allowed.

The existence of ECOs/MECOs is certainly not widely accepted at present.

But this is not necessarily because of any theoretical inconsistency or of any lack of observational evidence for ECOs. On the other hand, it could be so simply because the concept of a static Black Hole looks simpler and exact; and, also, four generations of astrophysicists and physicists have got used to working within the BH paradigm. To appreciate this statement, note that a true observational signature of a BH would be the detection of its EH. But neither has any EH ever been detected nor is it possible in principle. Also one faces innumerable puzzles and contradictions in the BH paradigm due to presence of an infinte redshfit EH. Yet, at present, most of the astrophysicists use the BH paradigm because (i) of the existence of the unique and exact vacuum Schwarzschild solution, (ii) the existence of the unique and exact Oppenheimer - Snyder collapse solution. But we saw that such exact solutions actually correspond to a unique mass of the eventual BH : M=0. And a MECO is just the intermediate finite mass state preceding this unique and ultimate zero mass BH state. Hence the concept of a MECO is really not in conflict with the mathematical idea of a BH. Far from it, it is an integral part of true BH formation process because both the concepts involve the same physical ingredient of bending of light in strong gravity.

Formation of a black hole EH may be seen as an extreme result of trapping of radiation by gravitation (z= \infty). But before this ultimate gravitational trapping state would occur, there must be initially minor and then gradually severe stages of light trapping. Thus there is no denying that the collapse generated quanta would tend to move within the fluid in almost closed circular orbits until they collide with electrons, pairs, protons etc. Even after collisions, the quanta would tend to move in almost closed orbits because of extreme self-gravity (z>> 1).

In fact, by using a perturbation technique, three relativists found that collapse of Newtonian Supermassive Stars would produce an ECO rather than a BH.[22]

From this view point a MECO theory never denies the fundamental importance of the exact vacuum Schwarzschild solution. But it relies on the fact that the value of the INTEGRATION CONSTANT a= 2M happens to be zero for a MASSENPUNKT (and not for an object with finite radius) as is also indicated by the classic ADM paper (cf. Ref. 18, 19). Thus a MECO is really not an "alternative" to BHs; far from it, it is the natural and necessary quasistatic state prededing a true BH state. Without the MECO phase, i.e., without any radiation trapping, there would not be any BH at all. It is a different matter that the eventual BH state would have M=0 because the MECO would radiate away its entire mass energy during this infinite journey.

A MECO has properties of mass and spin, like a black hole, but unlike a black hole, it has other physical properties that make it more like a highly redshifted spinning magnetic dipole. Black holes do not have magnetic moments, so a MECO is expected to have a structure that can generate a magnetic field. MECOs do not have event horizons. Instead, approaching a MECO, the rotating magnetic field would accelerate infalling material away, and any which do not get flipped away is gradually slowed down and stopped at or near the highly redshifted surface. This phenomenon allows only a very weak interaction between those within the grasp of the MECO and the outside universe.

One may also wonder how a star with an initial finite "radius" R can continue to contract indefinitely howsoever slow the rate of contraction may be. In General Relativity, the actual radius of the star measured locally is larger than R. Imagine a horizontal rubber membrance fixed across a flexible circular ring. The externally perceived "radius" of the star is akin to the radius of the peripheral ring, R while the internal radius of the star is the radius of the membrane l measured along its surface.

When there is no weight on the membrane, it is "flat" and l = R. But now, suppose, a marble is placed on the membrane. Then the membrane will sag and its radius measured along its surface will be larger than the radius of the ring: l > R. At the same time, the ring will shrink a bit. The increasing grip of gravity during continued collapse may be simulated here by imagining that the weight of the marble keeps on increasing indefinitely. If so, while on one hand, l \to \infty, on the other hand, the ring would tend to shrink to a point, R \to 0. And since the membrane always sags with a finite speed (less than the speed of light) the process would continue eternally. Thus while externally perceived space may shrink towards a point, the gravity may stretch the internal space indefinitely! If the membrane represents the local space of a plasma, it will always have some intrinsic magnetic field. During the contraction, the value of this field, in a naive theory, would increase as B \propto R^{-2}. Thus the star or anything else undergoing continued collapse may eventually become a MECO.

[edit] Further reading

  • "Magnetospheric Eternally Collapsing Objects (MECOs): Likely New Class of Source of Cosmic Ray Particle Acceleration", A. Mitra, Proc. 29th Int. Cos. Ray Conf., Vol 3. pp.125-128 (2005), arXiv:physics/0506183
  • "The Magnetospheric Eternally Collapsing Object (MECO) Model of Galactic Black Hole Candidates and Active Galactic Nuclei", S.L. Robertson and D.J. Leiter, in "New Developments in Black Hole Research", ed. P.V. Kreitler (Nova Sc., NY, 2006), pp. 1-43, [ISBN 1-59454-641-X], arXiv:astro-ph/0602453
  • "Sources of Stellar Energy, Einstein - Eddington Time Scale of Gravitational Contraction and Eternally Collapsing Objects", A. Mitra, New Astronomy, Vol. 12(2), pp.146-160 (2006) arXiv:astro-ph/0608178
  • "Eternally Collapsing Objects or Black Holes: A Review of 90 Years of Misconceptions", A. Mitra, Invited Review Article in "Focus on Black Hole Research", ed. P.V. Kreitler (Nova Sc. NY, 2006), p. 1-94, [ISBN 1-59454-460-3]
  • "Radiation Pressure Supported Stars in Einstein Gravity: Eternally Collapsing Objects", A. Mitra, Mon. Not. Roy. Astron. Soc., Vol. 369, pp. 492-496 (2006) arXiv:astro-ph/0603055

[edit] References

  1. ^ "Final State of Spherical Gravitational Collapse and Likely Sources of Gamma Ray Bursts", A. Mitra, astro-ph/9803014 (1998)
  2. ^ "Non-occurrence of Trapped Surfaces and Black Holes in Spherical Gravitational Collapse: An Abridged Version", A. Mitra, Found. Phys. Lett., 13(6),543 (2000), astro-ph/9910408
  3. ^ "On the Question of Trapped Surfaces and Black Holes", A. Mitra (2001), astro-ph/0105532
  4. ^ "On the Nature of the Compact Condensations at the Centre of Galaxies", A. Mitra, Bull. Astron. Soc. India, 30, 173 (2002), astro-ph/0205261, "On the Final State of Spherical Gravitational Collapse", A. Mitra, Found. Phys. Lett., 15, 439, (2002), astro-ph/0207056
  5. ^ "Evidence for Intrinsic Magnetic Moments in Black Hole Candidates", S. Robertson and D. Leiter, Astrophysical J., 565, 447 (2002), astro-ph/0102381
  6. ^ "Does the Principle of Equivalence Prevent Trapped Surfaces From being Formed in the General Relativistic Collapse Process?", D. Leiter, A. Mitra and S. Robertson, (2001), astro-ph/0111421
  7. ^ "Does the Principle of Equivalence Prevent Trapped Surfaces From being Formed in the General Relativistic Collapse Process?", D. Leiter and S. Robertson, Foundations of Physics Lett., Vol. 16, pp.143 (2003)
  8. ^ "On Intrinsic Magnetic Moment in Black Hole Candidates", S. Robertson and D. Leiter, Astrophysical J. Lett., 596, L203 (2003), astro-ph/0310078
  9. ^ "On the Origin of the Radio/X-Ray Luminosity Correlation in Black Hole Candidates", S. Robertson and D. Leiter, Mon. Not. Roy. Astron. Soc., 350, 1391, 2004, astro-ph/0402445.
  10. ^ "Observations Supporting the Existence of an Intrinsic Magnetic Moment Inside the Central Compact Object Within the Quasar Q0957+561" R. Schild, D. Leiter and S. Robertson, Astronomical J., 132, 420 (2006), astro-ph/0505518.
  11. ^ Center for Astrophysics, Harvard, Press Release, July 25, 2006
  12. ^ Center for Astrophysics (CfA), Harvard, Press Release, Aug 10, 2006
  13. ^ "Why Gravitational Contraction Must be Accompanied by Emission of Radiation Both in Newtonian and Einstein Gravity?" A. Mitra, Phys. Rev., D74, 024010 (2006), gr-qc/0605066
  14. ^ "Black Holes or Eternally Collapsing Objects?" A. Mitra and N.K. Glendenning, (Preprint, 2006, Oral Presentation, 11th Marcel Grossmann Meeting, Berlin)
  15. ^ "Radiation Pressure Supported Stars in Einstein Gravity: Eternally Collsing Objects" A. Mitra, Mon. Not. Roy. Astr. Soc., 369, 492 (2006), gr-qc/0603055.
  16. ^ "A Generic Relation Between Baryonic & Rest Mass Densities of Stars", A. Mitra, Mon. Not. R. Astr. Soc. Lett., 367, L66 (2006), gr-qc/0601025
  17. ^ "Sources of Stellar Energy, Einstein -Eddington Timescale of Gravitational Contraction and Eternally Collapsing Objects", A. Mitra, New Astronomy, Vol. 12(2), pp. 146-160 (2006), astro-ph/0608178
  18. ^ "On the Non-occurrence of Type I X-ray Bursts from Black Hole Candidates", A. Mitra, Advances in Sp. Res. (2006, in Press), astro-ph/0510162
  19. ^ "The Dynamics of General Relativity", R. Arnowitt, S. Deser and C.W. Misner, in "GRAVITATION: An Introduction to Current Research" (ed. L. Witten, Wiley, NY, 1962), gr-qc/0405109
  20. ^ "Relativistic Gravitational Bremsstrahlung", P.C. Peters, Phys. Rev., D1(6), 1559 (1970)
  21. ^ "Gravitational Synchrotron Radiation in the Schwarzschild Geometry", C.W. Misner et al., Phys. Rev. Lett., 28, 998 (1972)
  22. ^ "Gravitationally Eternally Collapsing Objects" http://www.fma.if.usp.br/100years/pl6.html H.J. Cuesta, J.M. Salim and N.O. Santos, "Gravitationally Eternally Collapsing Objects", Paper presented in "100 years of Relativity", Sao Paula, Brazil, 2005

[edit] External links

[edit] See also

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