# Treating the Einstein-Hilbert action as a higher derivative Lagrangian: revealing the missing information about conformal non-invariance

## TL;DR

This paper shows that treating the Einstein-Hilbert action as a higher derivative Lagrangian reveals missing conformal invariance information, which is hidden in the standard formulation.

## Contribution

It demonstrates that the conformal properties of Einstein-Hilbert gravity become apparent when viewed as a higher derivative theory with independent extrinsic curvature.

## Key findings

- Conformal invariance is linked to specific first class constraints.
- Adding terms depending on the metric determinant or extrinsic curvature breaks conformal invariance.
- Treating the theory as higher derivative reveals missing conformal constraints.

## Abstract

The Hamiltonian formulation of conformally invariant Weyl-squared higher derivative theory teaches us that conformal symmetry is expressed through particular first class constraints related to the absence of the three-metric determinant and the trace of the extrinsic curvature from the theory. Any term depending on them which is added to this theory breaks conformal invariance and turns these constraints into second class ones. Such second class constraints are missing in the standard canonical formulation of the conformally non-invariant Einstein-Hilbert theory. It is demonstrated that such constraints do appear if the theory is treated as a higher derivative one --- if the extrinsic curvature is promoted to an independent variable, the apparently missing information about conformal behavior is revealed.