Rainbow Spacetime from a Nonlocal Gravitational Uncertainty Principle

Occurrence of spacetime singularities is one of the peculiar features of Einstein gravity, signalling limitation on probing short distances in spacetime. This alludes to the existence of a fundamental length scale in nature. On contrary, Heisenberg quantum uncertainty relation seems to allow for probing arbitrarily small length scales. To reconcile these two conflicting ideas in line with a well known framework of quantum gravity, several modifications of Heisenberg algebra have been proposed. However, it has been extensively argued that such a minimum length would introduce nonlocality in theories of quantum gravity. In this Letter, we analyze a previously proposed deformation of the Heisenberg algebra (i.e. $p rightarrow p (1 + lambda p^{-1})$) for a particle confined in a box subjected to a gravitational field. For the problem in hand, such deformation seems to yield an energy-dependent behavior of spacetime in a way consistent with gravitys rainbow, hence demonstrating a connection between non-locality and gravitys rainbow.