On the Degrees of Freedom of $R^2$ Gravity in Flat Spacetime

TL;DR

This paper investigates the degrees of freedom in $R^2$ gravity in flat spacetime, revealing that pure $R^2$ gravity has no propagating degrees of freedom, while the full theory has scalar and tensor modes, with strong coupling at high energies.

Contribution

It clarifies the degrees of freedom in pure and full $R^2$ gravity using multiple approaches and shows the apparent discontinuity is a perturbative artifact.

Findings

Pure $R^2$ gravity has no degrees of freedom.

Full $R^2$ gravity propagates one scalar and two tensor modes.

High-energy corrections lead to strong coupling of modes.

Abstract

We study the degrees of freedom of $R^2$ gravity in flat spacetime with two approaches. By rewriting the theory a la Stueckelberg, and implementing Lorentz-like gauges to the metric perturbations, we confirm that the pure theory propagates one scalar degree of freedom, while the full theory contains two tensor modes in addition. We then consider the degrees of freedom by directly examining the metric perturbations. We show that the degrees of freedom of the full theory match with those obtained with the manifestly covariant approach. In contrast, we find that the pure $R^2$ gravity has no degrees of freedom. We show that a similar discrepancy between the two approaches appears also in a theory dual to the three-form, and appears due to the Lorentz-like gauges, which lead to the fictitious modes even after the residual gauge redundancy has been taken into account. At first sight, this implies a discontinuity between the full theory and the pure case. By studying the first-order corrections of the full $R^2$ gravity beyond the linear regime, we show that at high-energies, both scalar and tensor degrees of freedom become strongly coupled. This implies that the apparent discontinuity of pure and full $R^2$ gravity is just an artefact of the perturbation theory, and further supports the absence of degrees of freedom in the pure $R^2$ gravity.