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By using the most general form of Einstein equations for General Relativistic (GTR) spherical collapse of an isolatedfluid having arbitrary equation of state and radiation transport properties, we show that they obey a Global Constraint, 2GM(r, t)/R(r, t)c2≤1, where Ris the “invariant circumference radius”, tis the comoving time, and M(r, t) 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 surfacesare 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 Rbreferes 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(r, t) 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, ignoredit, and, failed to realize that such a problem was occurring because they were assuming a finite value ofMf, where Mfis 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 rand Rare finite), and the collapse process continues indefinitely. During this indefinite journey, naturally, the system radiates out all available energy, Q→Mic2, 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 staticNeutron 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 V→c, accretion onto such Eternally Collapsing Objects (ECO) may generate little collisional energy out put. But, in the frame work of existence of staticcentral 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.

In sequel to an earlier work (Mitra, 1984), I set out to find the appropriate equation of state of an isothermal (self-gravitating) cosmic gas sphere. It will be shown that because of self-gravitational interaction, a cosmic gas which is perfect otherwise, is destined to behave like an imperfect gas. It is found that for a cosmic gas in hydrostatic equilibrium the only mode of free expansion (or contraction) is by a homologous evolution. This theorem enables one to construct the equation of state in terms of the local thermodynamical parameters of the inhomogeneous medium in such a way that it is consistent with the boundary conditions at the centre and at the natural boundary of the system. This method can be extended even when there is a temperature gradient in the system. This analysis predicts that the density profile should be the same at least for all nondegenerate Newtonian stars having the same temperature profile and which can be considered to be in hydrostatic equilibrium. And the density profile will be that of a polytrope of indexn=3, modulated by a temperature profile.

Depending upon the various models and assumptions, the existing literature on gamma-ray bursts (GRBs) mentions that the gross theoretical value of the duration of the burst in the hydrodynamical models is ? ~ r?/2?2c, where r?is the radius at which the blast wave associated with the fireball (FB) becomes radiative and sufficiently strong. Here ? ? E/Mc2, cis the speed of light, Eis the initial laboratory frame energy of the FB, and Mis the baryonic mass of the same (Rees & Mészáros). However, within the same basic framework, some authors (e.g., Katz and Piran) have given ? ~ r?/2?c. We intend to remove this confusion by considering this problem at a deeper level than has been considered so far. Our analysis shows that none of the previously quoted expressions are exactly correct, and in case the FB is produced impulsively and the radiative processes responsible for the generation of the GRB are sufficiently fast, its expected duration would be ? ~ ar?/2?2c, where a~ O(10). We further discuss the probable change, if any, of this expression, in case the FB propagates in an anisotropic fashion. We also discuss some associated points in the context of the Mészáros & Rees scenario.

There has been a recent suggestion that Cyg X-3 contains a black hole (BH) of mass M1 ~ 17 M. This interpretation is closely linked to a previous claim that Cyg X-3 contains a Wolf-Rayet star of mass M2 ~ 10 M. The latter interpretation would imply that the X-ray source in this close binary (P = 4.8 hr) is enshrouded by a relatively cool superstrong wind with M ${&ucpmathaccent;{M}{"705F}}$ [?]10 [?]5 M yr-1. It can be shown that such a wind would be completely opaque to the low-energy X-rays observed from the source, and hence the Wolf-Rayet hypothesis cannot be a correct one. By pursuit of the same argument, it also follows that the compact object must be of modest mass, ruling out the existence of a massive BH in Cyg X-3. Note, however, that at present we cannot strictly rule out the probable existence of a stellar-mass BH in Cyg X-3 or any other X-ray binary that is believed to contain a neutron star as the compact object. Yet a more probable scenario for Cyg X-3 would be one where the compact object is a canonical neutron star and the companion is an extremely low mass dwarf, M2 ~ 10-2 M, much like PSR 1957+20.