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Gravitational effective field theories with nondynamical backgroundsexplicitly break diffeomorphism and local Lorentz invariance. At the same time,to maintain observer independence the action describing these theories isrequired to be mathematically invariant under general coordinatetransformations and changes of local Lorentz bases. These opposing effects ofhaving broken spacetime symmetries but invariance under mathematical observertransformations can result in theoretical inconsistency unless certainconditions hold. The consistency constraints that must hold originate fromNoether identities associated with the mathematical observer invariances in theaction. These identities are examined in detail and are used to investigategravity theories with nondynamical backgrounds, including when a St\"uckelbergapproach is used. Specific examples include gravity theories with fixed scalaror tensor backgrounds, Einstein-Maxwell theory with a fixed external current,and massive gravity.

Noether identities in gravity theories with nondynamical backgrounds and explicit spacetime symmetry breaking