On the breakdown of space-time in general relativity

TL;DR

This paper investigates the fundamental loss of space-time covariance in canonical quantum gravity, revealing it as an intrinsic property of general relativity that occurs in strong fields, contrasting with gauge theories where covariance is preserved.

Contribution

It demonstrates the breakdown of space-time covariance as a fundamental aspect of GR in the canonical quantization framework, extending the understanding of singularities and effective field theory limitations.

Findings

Loss of covariance occurs in first-order Einstein-Hilbert action due to second class constraints.

Covariance is restored in weak field perturbative regimes, enabling standard quantization.

Breakdown of space-time is a non-perturbative feature in strong gravitational fields, linked to singularity theorems.

Abstract

Based on the canonical quantization of $d>2$ dimensional general relativity (GR) via the Dirac constraint formalism (also termed as `constraint quantization'), we propose the loss of covariance as a fundamental property of the theory. This breakdown occurs for the first-order Einstein Hilbert action, whereby loss of diffeomorphism invariance, besides first class constraints, second class constriants also exist leading to non-standard ghost fields which render the path integral non-covariant. We also attempt, for the first time, the canonical quantization via calculation of the path integral for the equivalent Hamiltonian formulation of GR for which only first class constraints exist. However, loss of covariance still occurs in this action due to loss of diffeomorphism invariance and structures arising from non-covariant constraints in the path integral. In contrast, we find that covariance as a symmetry is restored and quantization with perturbative calculations is possible in the weak limit of the gravitational field of these actions. Hence, we firstly establish, for the first time, that the breakdown in space-time is a property of GR itself as its limitation, indicating it to be an Effective Field Theory (EFT). We further propose that the breakdown of space-time occurs as a non-perturbative feature of GR in the strong field limit of the theory. Besides GR, we also note that covariance is preserved when constraint quantization is conducted for non-Abelian gauge theories, such as the Yang-Mills theory. These findings are novel from a canonical gravity formalism and EFT approach, and are consistent with GR singularity theorems, yet are general and extend them, as the singularity theorems indicate breakdown at a strong field limit of GR in black holes.