Theoretical Aspects of Massive Gravity

TL;DR

The work surveys the theoretical structure of Lorentz-invariant massive gravity, tracing the path from the Fierz-Pauli linear theory through nonlinearities, the vDVZ discontinuity, and the Vainshtein mechanism, to ghost-free nonlinear realizations such as the $\Lambda_3$ theory. It develops the Stückelberg formalism to expose hidden degrees of freedom, analyzes decoupling limits revealing galileon interactions, and demonstrates how carefully tuned nonlinearities raise the EFT cutoff and suppress unwanted modes. The review also connects four-dimensional theories to higher-dimensional embeddings (Kaluza-Klein, DGP) and discusses massive gravity in three dimensions, highlighting both the prospects for addressing cosmological constant naturalness and the challenges of UV completion and potential superluminality. Overall, it presents a coherent framework in which a technically natural, weakly coupled massive graviton could reconcile infrared modifications of gravity with solar-system tests, while outlining major open questions in UV completion and phenomenology.

Abstract

Massive gravity has seen a resurgence of interest due to recent progress which has overcome its traditional problems, yielding an avenue for addressing important open questions such as the cosmological constant naturalness problem. The possibility of a massive graviton has been studied on and off for the past 70 years. During this time, curiosities such as the vDVZ discontinuity and the Boulware-Deser ghost were uncovered. We re-derive these results in a pedagogical manner, and develop the Stükelberg formalism to discuss them from the modern effective field theory viewpoint. We review recent progress of the last decade, including the dissolution of the vDVZ discontinuity via the Vainshtein screening mechanism, the existence of a consistent effective field theory with a stable hierarchy between the graviton mass and the cutoff, and the existence of particular interactions which raise the maximal effective field theory cutoff and remove the ghosts. In addition, we review some peculiarities of massive gravitons on curved space, novel theories in three dimensions, and examples of the emergence of a massive graviton from extra-dimensions and brane worlds.

Figures (5)

• Figure 1: Degrees of freedom and their stability for values in the $R,m^2$ plane for massive gravity on an Einstein space (shown for $D=4$, the other dimensions follow similarly). The line $R=6m^2, \ m^2\not=0$ is where a scalar gauge symmetry appears, reducing the number of degrees of freedom by one. The line $m^2=0$ is where the vector gauge symmetries appear, reducing the number of degrees of freedom by three.
• Figure 2: Regimes for GR.
• Figure 3: Regimes for massive gravity with cutoff $\Lambda_5=(M_Pm^4)^{1/5}$, and some values within the solar system, for which $\Lambda_5^{-1}\sim 10^{11}\ {\rm km}$. Note that $r_Q$ is a bit larger than the observable universe, i.e. this theory makes no observable predictions within its range of validity.
• Figure 4: Regimes for massive gravity with cutoff $\Lambda_3=(M_Pm^2)^{1/3}$, and some values within the solar system. The values are much more reasonable than those of the $\Lambda_5$ theory.
• Figure 5: Splitting the DGP action