Lorentzian Condition in Quantum Gravity

TL;DR

The paper addresses how to realize Lorentzian geometries in the quantum cosmology wavefunction by adopting a momentum representation in which the canonical momentum $\pi^{ij}$ is taken to be purely imaginary on the measurement surface $\Sigma$, linking the Lorentzian neighborhood to the Lorentzian phase of the wavefunction through a Laplace transform in $\Psi[\pi^{ij}]$. Complex saddlepoint methods are used to connect Euclidean and Lorentzian sectors, with the real part of the Euclidean action governing probabilities via $\Psi^*\Psi \sim e^{-2\Re(I)}$, and the approach is applied to de Sitter nucleation, the Nariai maximal black-hole case, and sub-maximal Schwarzschild–de Sitter nucleation. The results yield explicit creation rates, $\Gamma_{\rm SdS}=e^{-\pi bc}$ and $\Gamma_N=e^{-\pi/\Lambda}$ (in the Nariai limit), and show how conical singularities on $\Sigma$ can be accommodated within a consistent semiclassical framework that also relates to geometric entropy, providing a practical tool for Lorentzian quantum cosmology.

Abstract

The wave function of the universe is usually taken to be a functional of the three-metric on a spacelike section, Sigma, which is measured. It is sometimes better, however, to work in the conjugate representation, where the wave function depends on a quantity related to the second fundamental form of Sigma. This makes it possible to ensure that Sigma is part of a Lorentzian universe by requiring that the argument of the wave function be purely imaginary. We demonstrate the advantages of this formalism first in the well-known examples of the nucleation of a de Sitter or a Nariai universe. We then use it to calculate the pair creation rate for sub-maximal black holes in de Sitter space, which had been thought to vanish semi-classically.

Figures (2)

• Figure 1: The creation of a de Sitter universe (left) can be visualized as half of a Euclidean four-sphere joined to a Lorentzian four-hyperboloid. The picture on the right shows the corresponding nucleation process for a de Sitter universe containing a pair of black holes. In this case the spacelike slices have non-trivial topology.
• Figure 2: Carter-Penrose diagram of the Schwarzschild-de Sitter spacetime. The point $C$ is the location of the conical singularity in the Euclidean sector. The curved lines indicate a family of spacelike slices which all pass through the conical singularity. This is necessary since one must specify the metric there in order to ensure that the Euclidean solution is a saddlepoint. Regions I and II lie between the black hole and the cosmological horizon. Region III corresponds to an asymptotic de Sitter region, and region IV to the black hole interior.