Carrollian limit of quadratic gravity

TL;DR

This paper investigates the Carrollian limit of quadratic gravity in four dimensions, classifying four distinct theories based on their parameters' dependence on the speed of light and analyzing their relation to general relativity and $R+R^2$ theories.

Contribution

It introduces a classification of Carrollian theories derived from quadratic gravity, detailing how parameters must depend on the speed of light and analyzing their equivalence to known gravity theories.

Findings

Four non-equivalent Carrollian theories identified.

Two theories are equivalent to general relativity at leading order.

Two theories are equivalent to $R+R^2$ at leading order.

Abstract

We study the Carrollian limit of the (general) quadratic gravity in four dimensions. We find that in order for the Carrollian theory to be a modification of the Carrollian limit of general relativity, the parameters in the action must depend on the speed of light in a specific way. By focusing on the leading and the next-to-leading orders in the Carrollian expansion, we show that there are four such non equivalent Carrollian theories. Imposing conditions to remove tachyons (from the linearized theory), we end up with a classification of Carrollian theories according to the leading-order and next-to-leading-order actions. All modify the Carrollian limit of general relativity with quartic terms of the extrinsic curvature. To the leading order, we show that two theories are equivalent to general relativity, one to $R+R^2$ theory, and one to the general quadratic gravity. To the next-to-leading order, two are equivalent to $R+R^2$ while the other two to the general quadratic gravity. We study the two theories that are equivalent to $R+R^2$ to the leading order and write their magnetic limit actions.