| Summary: | At mesoscopic scales, close to but somewhat larger than Planck length, one can describe the spacetime in terms of an effective geometry. The key feature of such an effective (quantum) geometry is the existence of a zero-point-length which, for example, modifies the propagator for a massive scalar field residing in that spacetime, in a specific manner. Such quantum gravitational effects arise, even in a globally flat spacetime, if one probes the spacetime at length scales close to Planck length. Principle of Equivalence demands that the effects of quantum spacetime observed in a freely-falling-frame (FFF) must be the same as those in a globally flat spacetime. But, in the FFF, gravity disappears and — along with it — the Newtonian gravitational constant G also disappears; therefore, operationally, the Planck length disappears in the FFF! So how can the quantum gravitational effects persist in the FFF, as they must? I show that the answer to this question is interesting and subtle. The Planck length reappears in FFF through the matter sector as the geometric mean LP=λcλg of the Compton wavelength λc=ħ/mic (where mi is the inertial mass) and the Schwarzschild radius λg=Gmg/c2 (where mg is the gravitational mass) when we invoke the Principle of Equivalence again, in the form mi=mg. So the Principle of Equivalence plays a crucial role in making the Planck length disappear and reappear to incorporate the effects of quantum spacetime in a FFF.
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