Consistent and non-consistent deformations of gravitational theories

TL;DR

This paper investigates how gravitational theories can be consistently deformed when abelianized, revealing which models can be quantized perturbatively and potentially advancing quantum gravity research.

Contribution

It identifies which gravitational actions remain consistent under abelianization deformations in 2+1 and 3+1 dimensions, highlighting models suitable for quantization.

Findings

Cartan-Palatini and Holst actions are inconsistent deformations.

Husain-Kuchař and Euclidean self-dual actions are consistent deformations.

Potential for perturbative quantization of Euclidean general relativity.

Abstract

We study the internally abelianized version of a range of gravitational theories, written in connection tetrad form, and study the possible interaction terms that can be added to them in a consistent way. We do this for 2+1 dimensional and 3+1 dimensional models. In the latter case we show that the Cartan-Palatini and Holst actions are not consistent deformations of their abelianized versions. We also show that the Husain-Kucha\v{r} and Euclidean self-dual actions are consistent deformations of their abelianized counterparts. This suggests that if the latter can be quantized, it could be possible to devise a perturbative scheme leading to the quantization of Euclidean general relativity along the lines put forward by Smolin in the early nineties.