Microstates of position and momentum result in gravitational entropy

Abstract

Measurements of a black hole’s position are limited in four different ways: Absorption of short-wavelength photons by the black hole, gravitational lensing’s interference with geometric diffraction, gravitational redshift decreasing the resolution of interactions close to the event horizon, and the relatively long wavelength of Hawking radiation. These limitations mean that a black hole cannot be localized more precisely than its Schwarzschild radius. Limitations on measuring mass and velocity mean that the position and momentum of a black hole cannot be simultaneously known more precisely than 2 h rs/lP , a value more restrictive than the Heisenberg uncertainty principle. Hidden information about a black hole’s position and momentum results in many possible microstates that are indistinguishable to an observer. One way to interpret the physical meaning of Bekenstein‐Hawking entropy is as a measure of the number of these microstates. This interpretation allows entropy to be generalized to objects in any gravitational field, because gravitational redshift increases uncertainty about position and momentum for objects in all gravitational fields, not just those of black holes.