Nonperturbative Continuity in Graviton Mass versus Perturbative Discontinuity

TL;DR

The paper investigates whether gravity can be mediated by a small but nonzero graviton mass without conflicting with observations. It shows that the classic perturbative discontinuity of 4D massive gravity is an artifact of truncated perturbation theory, and that nonperturbative summation resolves it; this is demonstrated both in 4D and in a nonlinear five-dimensional brane-world model. Exact cosmological solutions in the 5D setup reveal a continuous dependence on the graviton mass as $m_g \to 0$, with the helicity-0 mode decoupling in the classical limit. An interpolating solution further connects 4D and 5D tensor structures, bolstering the case that a small graviton mass can be compatible with classical gravity.

Abstract

We address the question whether a graviton could have a small nonzero mass. The issue is subtle for two reasons: there is a discontinuity in the mass in the lowest tree-level approximation, and, moreover, the nonlinear four-dimensional theory of a massive graviton is not defined unambiguously. First, we reiterate the old argument that for the vanishing graviton mass the lowest tree-level approximation breaks down since the higher order corrections are singular in the graviton mass. However, there exist nonperturbative solutions which correspond to the summation of the singular terms and these solutions are continuous in the graviton mass. Furthermore, we study a completely nonlinear and generally covariant five-dimensional model which mimics the properties of the four-dimensional theory of massive gravity. We show that the exact solutions of the model are continuous in the mass, yet the perturbative expansion exhibits the discontinuity in the leading order and the singularities in higher orders as in the four-dimensional case. Based on exact cosmological solutions of the model we argue that the helicity-zero graviton state which is responsible for the perturbative discontinuity decouples from the matter in the limit of vanishing graviton mass in the full classical theory.

Figures (1)

• Figure 1: Scattering of the probe particle at the gravitational field of the heavy source. The bold circle accounts for summation of the higher order iterations over the nonlinearities in the classical equations.