  Modifications of General Relativity usually include extra dynamical degrees
of freedom, which to date remain undetected. Here we explore the possibility of
modifying Einstein's theory by adding solely nondynamical fields. With the
minimal requirement that the theory satisfies the weak equivalence principle
and admits a covariant Lagrangian formulation, we show that the field equations
generically have to include higher-order derivatives of the matter fields. This
has profound consequences for the viability of these theories. We develop a
parametrization based on a derivative expansion and show that - to next to
leading order - all theories are described by just two parameters. Our approach
can be used to put stringent, theory-independent constraints on such theories,
as we demonstrates using the Newtonian limit as an example.
