Faddeev-Popov ghosts and (1 + 1)-dimensional black-hole evaporation

Recently Callan, Giddings, Harvey, and the author derived a set of one-loop semiclassical equations describing black-hole formation and/or evaporation in two-dimensional dilaton gravity conformally coupled to $N$ scalar fields. These equations were subsequently used to show that an incoming matter wave develops a black-hole-type singularity at a critical value ${\phi }_{\mathrm{cr}}$ of the dilaton field. In this paper a modification to these equations arising from the Faddeev-Popov determinant is considered and shown to have dramatic effects for $N<\mathrm{}24\mathrm{}$, in which case ${\phi }_{\mathrm{cr}}$ becomes complex. The $N<\mathrm{}24\mathrm{}$ equations are solved along the leading edge of an incoming matter shock wave and found to be nonsingular. The shock wave arrives at future null infinity in a zero-energy state, gravitationally cloaked by negative-energy Hawking radiation. Static black-hole solutions supported by a radiation bath are also studied. The interior of the event horizon is found to be nonsingular and asymptotic to de Sitter space for $N<\mathrm{}24\mathrm{}$, at least for sufficiently small mass. It is noted that the one-loop approximation is not justified by a small parameter for small $N$. However an alternate theory (with different matter content) is found for which the same equations arise to leading order in an adjustable small parameter.