  In generally covariant metric gravity theories with tensor matter fields, the
initial value constraint equations, unlike in general relativity, are in
general not just the 0\mu-components of the metric field equation. This happens
because higher derivatives can occur in the matter stress tensor. A universal
form for these constraints is derived here from a generalized Bianchi identity
that includes matter fields. As an application, the constraints for
Einstein-aether theory are found.
