Uncertainty in Measurements of Distance

Abstract

Ng and van Dam have argued that quantum theory and general relativity give a lower bound of L^{1/3} L_P^{2/3} on the uncertainty of any distance, where L is the distance to be measured and L_P is the Planck length. Their idea is roughly that to minimize the position uncertainty of a freely falling measuring device one must increase its mass, but if its mass becomes too large it will collapse to form a black hole. Here we show that one can go below the Ng-van Dam bound by attaching the measuring device to a massive elastic rod. Relativistic limitations on the rod's rigidity, together with the constraint that its length exceeds its Schwarzschild radius, imply that zero-point fluctuations of the rod give an uncertainty greater than or equal to L_P.