Topological variations in General Relativity: a rigorous perspective

TL;DR

This paper rigorously analyzes topological variations in General Relativity, revealing dimensional obstructions to the Einstein-Hilbert action's critical points in four dimensions and exploring effects of higher curvature terms.

Contribution

It introduces a rigorous framework for topological variations in gravity, including new types of variations and their impact on the Einstein-Hilbert action.

Findings

Dimensional obstructions prevent critical points in 4D

Higher order curvature terms influence critical dimension

Extended variational framework reveals new topological effects

Abstract

Motivated by recent developments in the theory of gravitation, we revisit the idea of topological variations, originally introduced by Wheeler and Hawking, from a rigorous perspective. Starting from a localized version of the Einstein-Hilbert variational principle, we encode the key aspects of the variational procedure in the form of a topology on a suitable space of variational configurations with low Sobolev regularity. This structure is the final topology with respect to the admissible variational maps and naturally lends itself to generalizations. We rigorously introduce two distinct types of topological variations, corresponding to the infinitesimal addition of disconnected components and to infinitesimal surgeries, both motivated by related physical concepts. Using tools from the theory of Sobolev spaces and precise asymptotics, we establish dimensional obstructions for the continuity and differentiability of the Einstein-Hilbert action with respect to these variations, and show that in the extended variational framework the action does not admit critical points in dimension $n=4$, while higher dimensions are free of this problem. Finally, we demonstrate the non-trivial effect of higher order curvature terms on the critical dimension.