Non-occurrence of Trapped Surfaces and Black Holes in Spherical Gravitational Collapse: An Abridged Version

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(r, t)/R(r, t)c2 ≤ 1, where R is the “invariant circumference radius”, t is the comoving time, and M(r, t) is the gravitational mass enclosed within a comoving shell r. The central derivation to this effect is simple and exact. 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 must 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 can not be negative, the Eq. (36) of O-S paper (T ∼ ln yb+1 yb−1) categorically demands that yb = Rb/Rgb ≥ 1, or 2GMb/Rbc 2 ≤ 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 fundamental constraint 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,