Quantum theory of k(phi )-FLRW-metrics its connection to Chern-Simons-models and the theta vacuum structure of quantum gravity

J Hristov

doi:10.1140/epjc/s10052-021-09315-1

J Hristov

Open Access

https://doi.org/10.1140/epjc/s10052-021-09315-1

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Abstract

We show how a foliated 4-dimensional FLRW-metric becomes a gravitational instanton, if the spatial metric minimizes a three-dimensional Einstein–Hilbert action with positive cosmological constant, which is equal to the demand, that the scale factor satisfies the Bogomolny-equation, where the curvature parameter varies over the one-parameter family of hyperslices and takes the role of a potential depending on the scale factor. Additionally, we draw the connection to SO(4)-Chern–Simons theory and show how the established interpolating solutions describe the gradient flow between the minima of the vacuums of the Einstein–Hilbert action, as well as how they can be used to calculate tunnelling-amplitudes of gravitons and trivialize the calculations of path integrals in quantum gravity. All the calculations are carried out particularly for k admitting a {mathbb {Z}}_{2}-symmetry.

Highlights

We would like to describe the evolution of spacetime as a trajectory of a three-dimensional pseudo-Riemannian manifold (, h) with positive, constant scalar curvature equipped with metric tensor hi j xk in a special kind of double-well potential
We associated the systems (1.3) and (1.9) to each other by turning k into a dynamical parameter through a foliation of a gravitational instanton and viewing it as the potential of a one-dimensional system, of which the scale factor of the four-metric is a topological solution to. k is proportional to the scalar curvature and the evolution of the scale factor inside the potential of the intrinsic curvature determines the structure of the gravitational instanton
The ideas of the previous section are supported by the identification of the “k(φ)-theory” with the Chern–Simons theory

Summary

Introduction

We would like to describe the evolution of spacetime as a trajectory of a three-dimensional pseudo-Riemannian manifold ( , h) with positive, constant scalar curvature equipped with metric tensor hi j xk in a special kind of double-well potential. 2 p is the hight of the false minima of k (φ), which is set to 1, so that the curvature reaches a maximum for t = 0 By construction this would mean, that every solution φ of (1.8), trivial or nontrivial, which minimizes the action functional (1.9), is the scale factor of a four-dimensional manifold M with metric gμν defined above, that satisfies the Einstein vacuum equations and is a gravitational instanton. Only if r ≤ l p, (φ) in (1.13a) admits a non-singular behaviour, r = l p is clearly favoured cosmologically speaking due to the absence of a zero at the origin, as it is present for r < l p In this case, even though φ (0) = 0, a singularity is avoided because curvature and energy density stay finite due to the finiteness of the false maximum of k (φ), which is.

Concluding

Gravitational instantons and the gradient flow of the Chern–Simons invariant

One-dimensional QFT of the scale factor