Foundations of Physics Letters

, Volume 13, Issue 6, pp 543–579 | Cite as


  • Abhas Mitra


By using the most general form of Einstein equations for General Relativistic (GTR) spherical collapse of an isolated fluid having arbitrary equation of state and radiation transport properties, we show that they obey a Global Constraint, 2GM(rt)/R(rt)c2≤1, where R is the “invariant circumference radius”, t is the comoving time, and M(rt) is the gravitational mass enclosed within a comoving shell r. This inequality specifically shows that, contrary to the traditional intuitive Newtonian idea, which equates the total gravitational mass (Mb) with the fixed baryonic mass (M0), the trapped surfaces are not allowed in general theory of relativity (GTR), and therefore, for continued collapse, the final gravitational mass Mf→0 as R→0. This result should be valid for all spherical collapse scenarios including that of collapse of a spherical homogeneous dust as enunciated by Oppenheimer and Snyder (OS). Since the argument of a logarithmic function cannot be negative, the Eq. (36) of the O–S paper (T∼In\(T \sim \ln \tfrac{{y_b + 1}}{{y_b - 1}}\)) categorically demands that yb=Rb/Rgb≥1, or 2GMb/Rbc2≤1, where Rb referes to the invariant radius at the outer boundary. Unfortunately, OS worked with an approximate form of Eq. (36) [Eq. 37], where this fundamentalconstraint got obfuscated. And although OS noted that for a finite value of M(rt) the spatial metric coefficient for an internal point fails to blow up even when the collapse is complete\(e^{\lambda (r < r_b )} \) ≠ ∞ for R→0, they, nevertheless, ignored it, and, failed to realize that such a problem was occurring because they were assuming a finite value ofMf, where Mf is the value of the finite gravitational mass, in violation of their Eq. (36).

Additionally, irrespective of the gravitational collapse problem, by analyzing the properties of the Kruskal transformations we show that in order that the actual radial geodesics remain timelike, finite mass Schwarzschild Black Holes can not exist at all.

Our work shows that as one attempts to arrive at the singularity, R→0, the proper radial lengthl=∫\(\sqrt { - g_{rr} } \)dr→∞ (even though r and R are finite), and the collapse process continues indefinitely. During this indefinite journey, naturally, the system radiates out all available energy, QMic2, because trapped surfaces are not formed. And this categorically shows that GTR is not only “the most beautiful physical theory,” but also, is the only, naturally, singularity free theory (atleast for isolated bodies), as intended by its founder, Einstein. However, this derivation need not rule out the initial singularity of “big bang” cosmology because the universe may not be treated as an “isolated body”.

There is a widespread misconception, that recent astrophysical observations have proved the existence of Black Holes. Actually, observations suggest existence of compact objects having masses greater than the upper limit of static Neutron Stars. The present work also allows to have such massive compact objects. It is also argued that there is evidence that part of the mass-energy accreting onto several stellar mass (binary) compact objects or massive Active Galactic Nuclei is getting “lost”, indicating the presence of an Event Horizon. Since, we are showing here that the collapse process continues indefinitely with local 3-speed Vc, accretion onto such Eternally Collapsing Objects (ECO) may generate little collisional energy out put. But, in the frame work of existence of static central compact objects, this small output of accretion energy would be misinterpreted as an “evidence” for Event Horizons. Thus the supposed BHs are actually massive compact ECOs.

black hole eternally collapsing object gravitational collapse gamma ray burst 


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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Abhas Mitra
    • 1
  1. 1.Theoretical Physics DivisionBhabha Atomic Research CenterMumbai-India

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