Effective Field Theory of Gravity in Relativistic Media

Abstract

We develop an effective field theory of gravity in relativistic media. Integrating out the medium leaves vacuum gravity with updated Feynman rules: a graviton propagator dressed by the stress-energy two-point function of the environment, medium-induced bulk graviton vertices from higher correlators, and generalized worldline couplings encoded in matched Wilson coefficients. The diagram topologies are unchanged from vacuum, so different media (perfect fluids, collisionless matter, coherent scalar fields such as wave dark matter) are not different theories but different correlators inserted into the same diagrams. We derive the in-medium rules for a relativistic fluid and obtain the full 1PN Einstein-Infeld-Hoffmann potential, the 1PN Stokes drag, the gravitational self-energy and a generalized Christodoulou memory whose new tensor structure records the orientation of the medium's anisotropy, turning the permanent strain into an $\textit{astrophysical weathervane}$. The same rules activate phenomena forbidden in vacuum: the sound pole converts the symmetry-protected, non-running black-hole Love number into a resonant, running one, and graviton splitting $h\to hh$ is open into the longitudinal branch of the dressed propagator, and gives the transverse graviton a width, $Γ=G_Nω^3(1-c_s^2)^2/120c_s^3$, a $\textit{gravitational opacity}$ set by the sound speed. The same vertex makes a graviton suffer dynamical friction. We assess observational prospects, from dephasing and tidal resonances within reach of the Einstein Telescope and LISA to proof-of-principle memory and opacity signatures.