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

Gravitational effective field theories with nondynamical backgrounds explicitly break diffeomorphism and local Lorentz invariance. At the same time, to maintain observer independence the action describing these theories is required to be mathematically invariant under general coordinate transformations and changes of local Lorentz bases. These opposing effects of having broken spacetime symmetries but invariance under mathematical observer transformations can result in theoretical inconsistency unless certain conditions hold. The consistency constraints that must hold originate from Noether identities associated with the mathematical observer invariances in the action. These identities are examined in detail and are used to investigate gravity theories with nondynamical backgrounds, including when a St\"uckelberg approach is used. Specific examples include gravity theories with fixed scalar or tensor backgrounds, Einstein-Maxwell theory with a fixed external current, and massive gravity.