Are there Einsteinian gravities involving covariant derivatives of the Riemann tensor?

TL;DR

This paper investigates whether higher derivative gravity theories involving covariant derivatives of the Riemann tensor can have Einsteinian spectra, concluding that such theories likely contain ghosts and hinting at non-locality in quantum gravity.

Contribution

It provides a no-go result showing the difficulty of constructing Einsteinian gravities with covariant derivatives of the Riemann tensor, suggesting inherent ghost issues.

Findings

Finite order Lagrangians with covariant derivatives of Riemann likely have ghosts.

Generalized quasi-topological gravities without derivatives have Einsteinian spectra.

Evidence points to non-locality in consistent UV completions of gravity.

Abstract

We study the particle content of higher derivative theories of gravity built with contractions of the Riemann tensor and its covariant derivatives. In the absence of the latter, there is a family of theories exhibiting an Einsteinian spectrum known as generalized quasi-topological gravities. In turn, we present a no-go result for the construction of Einsteinian gravities involving covariant derivatives of the Riemann tensor. We find evidences suggesting that (truncated series) finite order Lagrangians with covariant derivatives of the Riemann tensor generically present ghosts in their spectrum. This might be interpreted as a hint of non-locality in any healthy UV completion of General Relativity.