Quantum collapse of a self-gravitating shell: Equivalence to Coulomb scattering

A spherically symmetric thin shell of dust with a fixed rest mass $M$ is considered as a model for gravitational collapse in general relativity. For a special choice of the time variable, the dynamical equations of the shell have the same form as those of a charged relativistic particle moving radially in an external Coulomb potential. The critical charge of the Coulomb potential, $Z=\frac{137}{2}$, corresponds to the rest mass $M$ of the shell attaining the Planck mass value $M_P$. A boundary condition for wave functions at the singularity is determined by requiring that the Klein-Gordon product and the total energy be conserved. This leads uniquely to the spectrum of the relativistic "scalar hydrogen" obtained long ago by Sommerfeld, if $Z=137\left(\frac{M^2}{{2M}_P^2}\right)$ is substituted for the "central charge." All stationary wave functions are expressed by means of standard special functions. The scattering states are symmetric under time reversal for arbitrarily high energies. In particular, their asymptotic form shows that precisely the same amount of probability and energy comes out as was sent in. This is surprising, because energy and/or information losses down black holes are to be expected. The full solvability and the analogy to the charged particle does not, however, automatically remove some interpretational problems typical for quantum gravity.