The quantum geometric limit

Abstract

In Einstein's gedankenexperiment for measuring space and time, an ensemble of clocks moving through curved spacetime measures geometry by sending signals back and forth, as in the global positioning system (GPS). Combining well-known quantum limits to measurement with the requirement that the energy density of clocks and signals be be no greater than the black hole density leads to the quantum geometric limit: the total number of ticks of clocks and clicks of detectors that can be contained in a four volume of spacetime of radius r and temporal extent t is less than or equal to rt/\pi l_P t_P, where l_P, t_P are the Planck length and time. The quantum geometric limit suggests that each event or `op' that takes place in a four-volume of spacetime is associated with a Planck-scale area. This paper shows that the quantum geometric limit can be used to derive general relativity: if each quantum event is associated with a Planck-scale area removed from two-dimensional surfaces in the volume in which the event takes place, then Einstein's equations must hold.