  Recently Pikovski et al. have proposed in [ Pikovski I et al. 2015 Nature
Phys. 11, 668] an intriguing universal decoherence mechanism, suggesting that
gravitation may play a conceptually important role in the quantum-to-classical
transition, albeit vanishingly small in everyday situations. Here we analyze
information transfer induced by this mechanism. We show that generically on
short time-scales, gravitational decoherence leads to a redundant information
encoding, which results in a form of objectivization of the center-of-mass
position in the gravitational field. We derive the relevant time-scales of this
process, given in terms of energy dispersion and quantum Fisher information. As
an example we study thermal coherent states and show certain robustness of the
effect with the temperature. Finally, we draw an analogy between our
objectivization mechanism and the fundamental problem of point individuation in
General Relativity as emphasized by the Einstein's Hole argument.
