When studying the consistency of gravitational theories that extend beyond Einstein's General Relativity, one pays attention to two important properties: first, the absence of ghosts (states with negative kinetic energy), and second, the respect of causality, i. e. the time delay experienced by an interacting system due to gravity must be positive. Hassan-Rosen bimetric gravity is currently the only known ghost-free theory with two dynamical interacting metrics. While the phenomenological consequences of the theory have been studied extensively, there remains a risk that higher-order interaction terms between the two metrics can introduce a negative time delay and hence break causality. In this work, we propose to compute the time delay in bimetric gravity. After linearizing the theory to first order around flat background solutions, two propagating modes are identified: the first is massless (identical to that present in General relativity), and the second is massive. Using the gravitational interaction of two massless scalars mediated by the two spin-two modes as a toy model, the time delay is calculated from the eikonal approximation. After adapting the model to the study of one massless and one massive scalar particle, the gravitational interaction of light with the Sun is modeled, and the time delay is calculated. Using the Cassini space mission constraints, the gravitational frequency shift is calculated, and a relationship between the theory parameters, the ratio of the Planck masses of the two interacting metrics, and m₅, the Fierz–Pauli mass of the massive mode, is derived to satisfy these constraints. Finally, in the hope of providing a method to move beyond the eikonal approximation for time delay calculations, we introduce the Wigner-–Smith operator. We show that within the eikonal limit, the two particle cut WS operator reproduces the eikonal time delay result. We then move beyond the two-particle-cut to analyze the influence of soft graviton intermediate states.
Ezzine Mohamed Amine (Thu,) studied this question.