Metric dimensional reduction at singularities with implications to Quantum Gravity

TL;DR

This paper demonstrates that dimensional reduction at singularities arises naturally from General Relativity without modifications, potentially explaining the high-energy dimensional reduction needed in Quantum Gravity theories.

Contribution

It shows that dimensional reduction effects are due to classical General Relativity at singularities, avoiding ad-hoc assumptions and linking geometry to quantum gravity.

Findings

Dimensional reduction occurs at singularities due to degenerate metrics.

Regular extensions of singularities do not require modifying General Relativity.

Singularities induce effective lower-dimensional behavior in spacetime properties.

Abstract

A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimensional reduction would still be desirable. The main contribution of this article is to show that dimensional reduction effects are due to General Relativity at singularities, and do not need to be postulated ad-hoc. Recent advances in understanding the geometry of singularities do not require modification of General Relativity, being just non-singular extensions of its mathematics to the limit cases. They turn out to work fine for some known types of cosmological singularities (black holes and FLRW Big-Bang), allowing a choice of the fundamental geometric invariants and physical quantities which remain regular. The resulting equations are equivalent to the standard ones outside the singularities. One consequence of this mathematical approach to the singularities in General Relativity is a special, (geo)metric type of dimensional reduction: at singularities, the metric tensor becomes degenerate in certain spacetime directions, and some properties of the fields become independent of those directions. Effectively, it is like one or more dimensions of spacetime just vanish at singularities. This suggests that it is worth exploring the possibility that the geometry of singularities leads naturally to the spontaneous dimensional reduction needed by Quantum Gravity.