Ram Brustein, A. J. M. Medved, Hagar Meir

Classical general relativity predicts that a contracting, spherically symmetric matter system with a large-enough mass will result in the formation of a trapped region whose outer boundary is an apparent horizon where the gravitational redshift diverges. The incompleteness theorems then lead to the conclusion that the outcome of the collapse is the singular geometry of a Schwarzschild black hole. Both analyses rely on solving Einstein's equations, a set of partial differential equations, valid in the limit that the Schwarzschild radius is finite but the Planck length is set to zero, so that quantum fluctuations of the geometry are completely absent. Here, we keep both parameters finite, allowing the geometry to fluctuate quantum mechanically, and take the limit of vanishing Planck length only at the end. Expressing the geometry of a spherically symmetric, collapsing, thin shell of matter in terms of an effective quantum field theory in 1+1 dimensions, we show, using the standard techniques of quantum field theory in curved spacetime, that the production of particles as the shell approaches its would-be horizon is finite in the limit of vanishing Planck length. The total number of produced quanta of the gravitational field scales as the Bekenstein-Hawking entropy, while their total energy scales as the mass of the shell. Importantly, the quantum expectation value of the product of the scalar expansion parameters for the associated null vectors is never vanishing. The conclusion is that an apparent horizon is not formed even when the shell has reached its gravitational radius. As the collapse continues, the classical Schwarzschild geometry can no longer be used to describe the shell's exterior geometry. This provides the sought-after loophole that is needed to explain how astrophysical black holes could be compact objects that are regular and horizonless.