Topological variations in General Relativity: a rigorous perspective

TL;DR

The paper develops a rigorous framework for topological variations in general relativity by recasting the Einstein–Hilbert action as a localized functional on Sobolev metrics and equipping the configuration space with a final topology that accommodates both geometric and topological changes. It introduces disconnected and connected topological variations and proves continuity of the action under these variations, uncovering a critical dimension at $n=4$ where the action loses the ability to admit critical points in the disconnected case, with higher dimensions alleviating the issue. By refining the variation topology and employing gluing/surgery constructions, the authors define topological derivatives and show how higher-order curvature terms shift the critical behavior, including cases with quadratic actions and Gauss–Bonnet terms. The results suggest that topological fluctuations are mathematically tractable within a Sobolev–based framework and hint at possible resolutions (e.g., extra dimensions) for maintaining a well-posed variational principle in quantum gravity. Collectively, the work provides a rigorous foundation for analyzing topology-change in GR and its implications for classical and quantum consistency.

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.

Figures (3)

• Figure 1: Graphic impression of the topological variation. A domain of variation $\Omega$ is replaced with a new one $\tilde{\Omega}$ with new topology and metric, and matching boundary.
• Figure 2: Graphic impression of the disconnected topological variation. The topology of $\Omega$ remains unchanged, and a new disconnected component $M'$ is added. In the most general case, variation of the metric is permitted in $\Omega$, to reproduce the effects of classical geometric variations.
• Figure 3: Graphic impression of the connected topological variation. A ball of diameter $\sqrt{\epsilon}$ in $\Omega$ is replaced with another manifold with matching boundary, whose diameter also scales as $\sqrt{\epsilon}$. Since the metric scales by $\epsilon$, this is the topological equivalent of the geometric variation $g \rightarrow g+\epsilon h$.

Theorems & Definitions (51)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Definition 2.6
• Remark 2.7: Moduli space structure
• Remark 2.8: Associated diffeological structure
• Theorem 2.9