Non-local formulation of ghost-free bigravity theory

TL;DR

This paper demonstrates how a non-local extension of General Relativity, involving the Weyl tensor and inverse d'Alembertian operator, naturally arises from ghost-free bigravity theory when expanded around Minkowski space.

Contribution

It shows that non-local modifications of gravity can emerge from a consistent ghost-free bigravity model at the classical level.

Findings

The remaining metric is governed by Einstein-Hilbert plus a non-local Weyl tensor term.

The non-local term involves the inverse d'Alembertian operator acting on the Weyl tensor.

The derivation is valid to quadratic order in perturbations and all orders in derivatives.

Abstract

We study the ghost-free bimetric theory of Hassan and Rosen, with parameters $\beta_i$ such that a flat Minkowski solution exists for both metrics. We show that, expanding around this solution and eliminating one of the two metrics with its own equation of motion, the remaining metric is governed by the Einstein-Hilbert action plus a non-local term proportional to $W_{\mu\nu\rho\sigma} (\Box-m^2)^{-1}W^{\mu\nu\rho\sigma}$, where $W_{\mu\nu\rho\sigma}$ is the Weyl tensor. The result is valid to quadratic order in the metric perturbation and to all orders in the derivative expansion. This example shows, in a simple setting, how such non-local extensions of GR can emerge from an underlying consistent theory, at the purely classical level.