Quantum Perfect-Fluid Kaluza-Klein Cosmology

TL;DR

This paper quantizes perfect-fluid cosmology in higher-dimensional Kaluza-Klein spacetimes, analyzing quantum effects on the evolution of extra dimensions and showing they can avoid singularities, with solutions for specific equations of state.

Contribution

It provides exact and approximate solutions to the Wheeler-DeWitt equation in higher-dimensional Kaluza-Klein cosmology with perfect fluids, revealing quantum effects that prevent singularities in extra dimensions.

Findings

Flat d-spaces expand forever.

Compact D-spaces avoid finite-time collapse.

Quantum effects prevent singularities in extra dimensions.

Abstract

The perfect fluid cosmology in the 1+d+D dimensional Kaluza-Klein spacetimes for an arbitrary barotropic equation of state $p= n \rho$ is quantized by using the Schutz's variational formalism. We make efforts in the mathematics to solve the problems in two cases. For the first case of the stiff fluid $n=1$ we exactly solve the Wheeler-DeWitt equation when the $d$ space is flat. After the superposition of the solutions we analyze the Bohmian trajectories of the final-stage wave-packet functions and show that the flat $d$ spaces and the compact $D$ spaces will eventually evolve into finite scale functions. For the second case of $n \approx 1$, we use the approximated wavefunction in the Wheeler-DeWitt equation to find the analytic forms of the final-stage wave-packet functions. After analyzing the Bohmian trajectories we show that the flat $d$ spaces will be expanding forever while the scale function of the contracting $D$ spaces would not become zero within finite time. Our investigations indicate that the quantum effect in the quantum perfect-fluid cosmology could prevent the extra compact $D$ spaces in the Kaluza-Klein theory from collapsing into a singularity or that the "crack-of-doom" singularity of the extra compact dimensions is made to occur at $t=\infty$.