Theoretical Aspects of Massive Gravity (1105.3735v2)

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 St\"ukelberg 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.

• The paper introduces ghost-free massive gravity models by addressing the vDVZ discontinuity and Boulware-Deser ghost via the Stückelberg trick.
• It employs non-linear dRGT interactions to cancel higher-order instabilities, ensuring stability up to the Λ3 energy scale.
• The study connects massive gravity with extra-dimensional theories like Kaluza-Klein and DGP, offering insights into cosmic acceleration alternatives.

Theoretical Aspects of Massive Gravity

The paper explores the intricacies of massive gravity, a compelling modification of general relativity (GR) through the incorporation of a massive graviton. Traditionally, GR has been celebrated for portraying gravity as the curvature of spacetime caused by mass and energy, with the graviton being a massless spin-2 particle. The introduction of mass to the graviton compels a re-evaluation of these principles, stirring interest due to its potential to address the cosmological constant problem by explaining the observed cosmic acceleration without invoking dark energy.

Historically, massive gravity has faced considerable theoretical challenges, notably the vDVZ discontinuity and the Boulware-Deser ghost. The vDVZ discontinuity arises because linear massive gravity predicts different light bending than GR, even as the graviton mass approaches zero. This discontinuity hints at the presence of additional scalar degrees of freedom, which fail to decouple in the massless limit. The Boulware-Deser ghost, discovered when seeking non-linear generalizations of linear massive gravity, involves an unwanted sixth ghost-like degree of freedom that destabilizes the theory.

The paper discusses methodologies to circumvent these problems, primarily through the St\"uckelberg trick, which introduces gauge symmetries to preserve degrees of freedom as the graviton mass approaches zero. This involves adding scalar and vector fields to maintain consistent degrees of freedom, thus elucidating the scalar-mediated force behind the vDVZ discontinuity. Moreover, the Vainshtein screening mechanism is proposed to reconcile linear massive gravity's predictions with GR by utilizing non-linearities to suppress the extra degrees of freedom beyond a characteristic radius, the Vainshtein radius.

Recent advancements have seen the development of ghost-free massive gravity theories, notably the de Rham-Gabadadze-Tolley (dRGT) model, which fine-tunes interactions to cancel higher-order instability-inducing terms, thus raising the UV cutoff to the scale $\Lambda_3 = (M_P m^2)^{1/3}$. This model eliminates the Boulware-Deser ghost in the decoupling limit, ensuring that quantum corrections respect this delicate cancellation. The resultant intricacies reveal a theory that is stable against perturbative expansions at least up to the energy scale $\Lambda_3$.

The paper also extends the discussion to extra-dimensional constructions such as Kaluza-Klein theory and the Dvali-Gabadadze-Porrati (DGP) model, providing examples where massive gravitons naturally occur. These models illustrate how extra dimensions can be compactified or interact with lower-dimensional branes to yield massive graviton spectra or a continuous spectrum of resonance models reminiscent of gravitational soft mass terms. Notably, these approaches highlight the entwined roles of geometry, extra dimensions, and particle masses in formulating coherent massive gravity models.

The implications of this research extend to both theoretical insights and cosmological applications. By offering a theoretically coherent framework that potentially describes massive gravity's modifications to GR, these developments invite further inquiry into the fundamental nature of gravitational interactions at cosmic scales, the puzzle of dark energy, and the true nature of space and time. Speculation persists about future refinements, including potential experimental signatures distinguishable from GR, and the broader quest for a viable UV completion that might yield new insight into unified field theories. As it stands, massive gravity continues to be a fertile ground for exploring alternatives to our classical understanding of gravity and its interaction with the universe's fabric.