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Initial value constraints with tensor matter

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 0μ-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.