New Kinetic Interactions for Massive Gravity?

TL;DR

This work addresses whether new kinetic terms for a massive graviton can exist in four-dimensional metric gravity beyond the Einstein–Hilbert action. Using dimensional deconstruction and a combination of decoupling-limit and ADM analyses, the authors show that any candidate with two derivatives either reduces to EH or introduces ghosts at higher scales, and that higher-derivative candidates are ruled out by general consistency arguments. Consequently, up to total derivatives, the Einstein–Hilbert term is the unique ghost-free, two-derivative kinetic term in $D=4$, with significant implications for the structure and viability of massive gravity, bi-gravity, and multi-gravity theories. The results hinge on a careful treatment of nonlinear Stückelberg expansions, minisuperspace ghosts, and perturbative constraint analyses, and they motivate further scrutiny of kinetic-sector possibilities in extended formalisms such as the vielbein approach. Overall, the paper establishes a robust no-go for new kinetic interactions in four-dimensional metric gravity and reinforces the special status of General Relativity’s kinetic structure.

Abstract

We show that there can be no new Lorentz invariant kinetic interactions free from the Boulware-Deser ghost in four dimensions in the metric formulation of gravity, beyond the standard Einstein-Hilbert, up to total derivatives. We use dimensional deconstruction as a way to motivate a non-linear ansatz for potential new ghost free kinetic interactions for massive gravity, bi-gravity and multi-gravity in four and higher dimensions. These interactions descend from Lovelock terms, and so naively one might expect the interactions to be ghost free. However we show that these new interactions inevitably lead to more than five propagating degrees of freedom. We then perform a general perturbative analysis in four dimensions, and show that the only term with two derivatives that does not introduce a ghost is the Einstein-Hilbert term. This result extends to all orders in perturbations.