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Recently, interest has increased in the entanglement of remote quantumparticles through the Newtonian gravitational interaction, both from afundamental perspective and as a test case for the quantization of gravity.Likewise, post-Newtonian gravitational effects in composite nonrelativisticquantum systems have been discussed, where the internal energy contributes tothe mass, promoting the mass to a Hilbert space operator. Employing a modifiedversion of a previously considered thought experiment, it can be shown thatboth concepts, when combined, result in inconsistencies, reinforcing thearguments for the necessity of a rigorous derivation of the nonrelativisticlimit of gravitating quantum matter from first principles.

Gravitational entanglement and the mass contribution of internal energy in nonrelativistic quantum systems