Centre of mass decoherence due to time dilation: paradoxical frame-dependence

The recently proposed centre of mass decoherence of composite objects due to gravitational time-dilation [Pikovski et al., Nat.Phys. 15. June (2015); arXiv:1311.1095] is confronted with the principle of equivalence between gravity and observer's acceleration. In the laboratory frame, a positional superposition vert x_1rangle+vert x_2rangle can quickly decohere whereas in the free-falling frame, as I argue, the superposition can survive for almost arbitrary long times. The paradoxical result is explained by the so far unappreciated feature of the proposed model: the centre of mass canonical subsystem is ambiguous, it is different in the laboratory and the free-falling frames, respectively. As long as the centre-of-mass motion of the composite object is non-relativistic, a simple Galilean-covariant Hamiltonian represents the Pikovski {\it et al} theory with exactly the same physical predictions. We emphasize the power of this Hamiltonian to understand essential features of the Pikovski {\it et al} theory and to moderate a few divergent statements in recent works.