  The coupling between matter and gravity in General Relativity is given by a
proportionality relation between the stress tensor and the geometry. This is an
oriented assumption driven by the fact that both the stress tensor and the
Einstein tensor are divergenceless. However, General Relativity is in essence a
nonlinear theory, so there is no obvious reason why the coupling to matter
should be linear. On another hand, modified theories of gravity usually affect
the vacuum dynamics, yet keep the coupling to matter linear. In this Letter we
address the implications of consistent nonlinear gravity/matter coupling. The
Eddington inspired Born-Infeld theory recently introduced by Banados and
Ferreira provides an enlightening realization of such coupling modifications.
We find that this theory coupled to a perfect fluid reduces to General
Relativity coupled to a nonlinearly modified perfect fluid, leading to an
ambiguity between modified coupling and modified equation of state. We discuss
observational consequences of this degeneracy and argue that such a completion
of General Relativity is viable from both an experimental and theoretical point
of view through energy conditions, consistency, and singularity-avoidance
perspectives. We use these results to discuss the impact of changing the
coupling paradigm.
