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Foundations of Physics Letters

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

NON-OCCURRENCE OF TRAPPED SURFACES AND BLACK HOLES IN SPHERICAL GRAVITATIONAL COLLAPSE

  • Abhas Mitra
Article

Abstract

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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REFERENCES

  1. 1.
    J. Michell, Phil. Trans. 74, 35 (1784).CrossRefGoogle Scholar
  2. 2.
    P. S. Laplace (1799), see English translation in Ref. 3.zbMATHGoogle Scholar
  3. 3.
    S. W. Hawking and G.R.F. Ellis, The Large Scale Structure of Space-Time (University Press, Cambridge, 1973).CrossRefzbMATHGoogle Scholar
  4. 4.
    J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).ADSCrossRefGoogle Scholar
  5. 5.
    B. Waugh and K. Lake, Phys. Rev. D 38, 1315 (1988).ADSCrossRefGoogle Scholar
  6. 6.
    D. Christodoulou, Commun. Math. Phys. 93, 171 (1984).ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    D. M. Eardley and L. Smarr, Phys. Rev. D 19, 2239 (1979).ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    F. I., Cooperstock, S. Jhingan, P. S. Joshi, and T. P. Singh, Class. Quant. Grav. 14, 2195 (1997).ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Penrose, Nouvo Cimento 1, 252 (1969).ADSGoogle Scholar
  10. 10.
    E. Gourgoulhon and P. Haensel, Astron. Astrophys. 271, 209 (1993).Google Scholar
  11. 11.
    T. W. Baumgarte et al., Astrophys. J. 468, 823 (1996).ADSCrossRefGoogle Scholar
  12. 12.
    P. C. Vaidya, Proc. Indian Acad. Sci A33, 264 (1951)ADSMathSciNetGoogle Scholar
  13. 13.
    P. C. Vaidya, Current Sci. 21, 96 (1952).MathSciNetGoogle Scholar
  14. 14.
    P. C. Vaidya, Nature 171, 260 (1953).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th edn. (Pergamon, Oxford, 1985).zbMATHGoogle Scholar
  16. 16.
    S. Weinberg, Gravitation and Cosmology: Principles and Applications of General Theory of Relativity (Wiley, New York, 1972).Google Scholar
  17. 17.
    S. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983).CrossRefGoogle Scholar
  18. 18.
    C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Fransisco, 1973).Google Scholar
  19. 19.
    K. Schwarzschild, Sitzungsberichte Preuss. Akad. Wiss., 424 (1916).Google Scholar
  20. 20.
    C. W. Misner and D. H. Sharp, Phys. Rev. 1362B, 571 (1964).ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    M. May and R. White, Phys. Rev. 141, 1233 (1965).MathSciNetGoogle Scholar
  22. 22.
    C. W. Misner, in Astrophysics and General Relativity, Vol. 1 (Brandeis University Summer School Lectures, 1968), M. Chretien, S. Deser, and J. Goldstein, eds.; see pp. 180-181.Google Scholar
  23. 23.
    C. W. Misner, in Relativity Theory and Astrophysics, Vol. 3, J. Ehlers, ed. (1967); see pp. 120-121.Google Scholar
  24. 24.
    C. W. Misner, Phys. Rev. 137 B, 1360 (1965).ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    W. C. Hernandez and C. W. Misner, Astrophys. J. 143, 452 (1966).ADSCrossRefGoogle Scholar
  26. 26.
    P. C. Vaidya, Astrophys. J. 144, 943 (1965).ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    R. W. Lindquist, R. A. Schwartz, and C. W. Misner, Phys. Rev. B 137, 1364 (1965).ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    R. W. Lindquist, Ann. Phys. 37, 487 (1966).ADSCrossRefGoogle Scholar
  29. 29.
    J. M. M. Senovilla, Class. Q. Grav. 30, 701 (1998).MathSciNetGoogle Scholar
  30. 30.
    R. Penrose, Phys. Rev. Lett. 14, 57 (1965).ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    R. Schoen and S. Y. Yau, Commun. Math. Phys. 65, 45 (1979).ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    B. K. Harrison, K. S. Thorne, M. Wakano, and J. A. Wheeler, Gravitation Theory and Gravitational Collapse (University Press, Chicago, 1965), p. 75.Google Scholar
  33. 33.
    Ya. B. Zeldovich and I. D. Novikov, Relativistic Astrophysics, Vol. 1, p. 297 (University Press, Chicago, 1971).Google Scholar
  34. 34.
    S. Chandrasekhar, An Introduction To The Study of Stellar Structure (Dover, New York, 1967).zbMATHGoogle Scholar
  35. 35.
    R. F. Tooper, Astrophys. J. 140, 393 (1964).MathSciNetGoogle Scholar
  36. 36.
    K. Lake, Phys. Rev. Lett. 68, 3192 (1992).MathSciNetCrossRefGoogle Scholar
  37. 37.
    J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 375 (1939).ADSCrossRefGoogle Scholar
  38. 38.
    A. K. Raychowdhury, private communication, 1998.Google Scholar
  39. 39.
    M. D. Kruskal, Phys. Rev. 119, 1743 (1960).ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    A. Mitra, astro-ph/9803014 (1998).Google Scholar
  41. 41.
    A. Einstein, Ann. Math. 40, 922 (1939).ADSCrossRefGoogle Scholar
  42. 42.
    L. D. Landau, Phys. Z. Sowjetunion 1, 285 (1932).Google Scholar
  43. 43.
    N. Rosen, in Relativity, M. Carmeli, S.I. Fickler, and L. Witten, eds. (Plenum, New York, 1970).Google Scholar
  44. 44.
    M. W. Choptuik, Phys. Rev. Lett., 70, 9 (1993).ADSCrossRefGoogle Scholar
  45. 45.
    G. W. Gibbons, in Gravitation and Relativity: At the turn of the Millenium, N. Dadhich and J. Narlikar, eds. (IUCAA, Pune, 1998).Google Scholar
  46. 46.
    J. C. Miller, T. Shahbaz and L. A. Nolan, Mon. Not. R. Astron. Soc. 294, L25 (1998).ADSCrossRefGoogle Scholar
  47. 47.
    F. Wilczek, Physics Today 51, 11 (1998).ADSGoogle Scholar
  48. 48.
    S. R. Kulkarni et al., Nature 398, 390 (1999).ADSCrossRefGoogle Scholar
  49. 49.
    A. S. Frutcher et al., Astrophys. J. 511, 852 (1999).CrossRefGoogle Scholar
  50. 50.
    A. Mitra, Astron. Astrophys. 340, 447, (1998).ADSGoogle Scholar
  51. 51.
    Y. F. Huang, Z. G. Dai, and T. Lu, Astron. Astrophys 355, L43 (2000).ADSGoogle Scholar
  52. 52.
    Z. G. Dai and T. Lu, Astrophys. J. Lett. 519, L155 (1999); astro-ph/9904025.ADSCrossRefGoogle Scholar
  53. 53.
    A. Mitra, Astron. Astrophys. 359, 413 (2000)ADSGoogle Scholar
  54. 54.
    J. E. Rhoads, Astrophys. J. 525, 737 (1999); astro-ph/9903399.ADSCrossRefGoogle Scholar
  55. 55.
    Y. F. Huang, Z. G. Dai, D. M. Wei, and T. Lu, Mon. Not. Astron. Soc. 298, 459 (1998).ADSCrossRefGoogle Scholar

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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