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We provide general guidelines for generalizing dynamical reduction models tocurved spacetimes and propose a class of generally covariant relativisticversions of the GRW model. We anticipate that the collapse operators of ourclass of models may play a r\^ole in a yet-to-be-formulated theory ofsemiclassical gravity with collapses. We show explicitly that the collapseoperators map a dense domain of states that are initially Hadamard to finalHadamard states -- a property that we expect will be needed for theconstruction of such a semiclassical theory. Finally, we provide a simpleexample in which we explicitly compute the violations in energy-momentum due tothe state reduction process and conclude that this violation is of the order ofa parameter of the model -- supposed to be small.

Generally covariant dynamical reduction models and the Hadamard condition