The quantization of gravity: The quantization of the full Einstein equations

TL;DR

This paper presents a method for quantizing the full Einstein equations in a globally hyperbolic spacetime, deriving solutions expressed through spatial and temporal eigenfunctions, with special cases for certain dimensions and cosmological constants.

Contribution

It introduces a novel approach to quantize Einstein's equations globally, providing explicit solutions and spectral properties in specific dimensional and cosmological constant regimes.

Findings

Solutions expressed as products of spatial and temporal eigenfunctions

Eigenvalues are countable and eigenfunctions form an orthonormal basis in certain cases

Temporal eigenfunctions solve a second order ODE in b5

Abstract

We quantized the full Einstein equations in a globally hyperbolic spacetime $N=N^{n+1}$, $n\ge 3$, and found solutions of the resulting hyperbolic equation in a fiber bundle $E$ which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second order ordinary differential equation in $\mathbb{R}_+$. In case $n\ge 17$ and provided the cosmological constant $\Lambda$ is negative the temporal eigenfunctions are eigenfunctions of a self-adjoint operator $\hat H_0$ such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space.