No-Go Theorem for Singularity Resolution

Abstract

We prove a No-Go theorem for singularity resolution in gravitational collapse: within any analytic gravitational theory, including general relativity and all theories with polynomial actions, quantum corrections introduced solely as effective matter sources are insufficient to halt singularities. This rules out singularity resolution via effective energy density in a broad class of quantum gravity approaches, including asymptotic safety and noncommutative geometry, which inevitably yield finite-time singularities or geodesic incompleteness. Resolution strictly requires either non-analytic modifications to the gravitational action, or a vanishing effective energy density at high densities (as realized in loop quantum gravity's Planck stars). The theorem is proved via an intrinsic $f(\mathbb{Q})$ gravity framework, extended universally to general relativity, $f(\mathbb{R})$, and $f(\mathbb{T})$ through the geometrical trinity--with regularity criteria and junction conditions grounded in non-metricity, free of standard GR tools.

Figures (1)

• Figure 1: $a-H$ phase diagram for collapse progress. The upper ($H>0$) and lower ($H<0$) branches represent the expansion and collapse processes respectively. (a) the combination of a polynomial $f(\mathbb{Q})$ ($f(\mathbb{Q})=\mathbb{Q}+\alpha \mathbb{Q}^2$, with $\alpha=0.01$) with asymptotic safety (orange); (b) the combination of a divergent $f(\mathbb{Q})$ at the origin ($f(\mathbb{Q})=\mathbb{Q}-\lambda/\mathbb{Q}$, with $\lambda=0.1$) with asymptotic safety (blue); (c) the combination of a polynomial $f(\mathbb{Q})$ ($f(\mathbb{Q})=\mathbb{Q}+\alpha \mathbb{Q}^2$, with $\alpha=0.01$) with LQG (red). In the first case, the system is strictly divided into upper and lower branches. In the latter two cases, the collapsing and expanding branches are connected.