Topology Change and Nonperturbative Instability of Black Holes in Quantum Gravity

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 S2 × R. On the other hand, nonclassical topologies Σg × R, where Σg 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 Plancksized 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. ⋆ e-mail address: [email protected] The topology of space or spacetime may play a fundamental role in any reasonable theory of quantum gravity [2-6]. The fascination with the possibility of topology change has led to several recent works [4,6,7,8,12-16]. It has recently been proposed that quantum gravity at ultra-short distances may be an example of a theory in which the interaction is introduced by a topological principle [13]. Indeed, splitting of Universes, or topology change can be thought of as a way to introduce interactions between extended objects (Universes). Second-quantized field theory of Universes will have topological perturbative expansion with topology change (cobordisms) playing the role of “Feynman diagrams” of an ordinary field theory. String theory (or 1+1 dimensional induced gravity) is the simplest example of such a model [13,15]. In analogy to an interacting string theory a model of an interacting second-quantized geometry was proposed [13]. The plethora of different topologies of low-dimensional manifolds for which the oriented cobordism groups are trivial (D=2,3) (D+1 dimensional Universes) seems to suggest that an interacting geometry model may indeed be very rich and may lead to many surprizes. In this essay I will address three issues of topology change, demonstrating that one may expect more surprises to come in the near future. First I will demonstrate how, in the 2+1 dimensional Chern-Simons gravity, an exact wave function of the Universe arises as an effect of a quantum anomaly. The Chern-Simons model is an example of the topological field theory in 2+1 dimensions introduced some time ago [14]. In this model the spacetime metric does not seem to play any role on a compact manifold without boundary. However, we will see that the metric reappears as a dynamical degree of freedom as an effect of a quantum gravitational anomaly induced by the presence of boundaries. This phenomenon has an attractive physical interpretation. Starting with a theory without a metric explicitly appearing in the action, we obtain an induced gravity due to the effect of a quantum anomaly. This anomaly is exactly calculable and as its effect one obtains the wave function of the 2-dimensional Universe