Topology Change and Nonperturbative Instability of Black Holes in Quantum Gravity

Abstract

Topology change in quantum gravity is considered. An exact wave function of the Universe is calculated for topological Chern-Simons 2+1 dimensional gravity. This wave function occurs as the effect of a quantum anomaly which leads to the induced gravity. We find that the wave function depends universally on the topology of the two-dimensional space. Indeed, the property of the ground state wave function of Chern-Simons gravity which has an attractive physical interpretation is that it becomes large in the infrared (large distances) if the Universe has ``classical'' topology $S^2\times R$. On the other hand, nonclassical topologies $Σ_g\times R$, where $Σ_g$ is the Riemann surface of genus g, are driven by quantum effects into the Planckian regime (``space-time foam''). The similar behavior of the quantum gravitational measure on four-manifolds constructed recently is discussed as the next example. We discuss the new phenomenon of the nonperturbative instability of black holes discovered recently. One finds that the Planck- sized black holes are unstable due to topology change. The decay rate is estimated using the instanton approximation. A possible solution to the primordial black hole problem in quantum cosmology is suggested.