Rainbow Spacetime from a Nonlocal Gravitational Uncertainty Principle

TL;DR

This paper explores how a modified Heisenberg algebra introduces nonlocality and results in an energy-dependent spacetime, linking quantum gravity concepts like gravity's rainbow with nonlocal effects.

Contribution

It demonstrates that a specific deformation of the Heisenberg algebra leads to an energy-dependent spacetime consistent with gravity's rainbow, connecting nonlocality and quantum gravity.

Findings

Deformed Heisenberg algebra induces nonlocality in quantum gravity.

Energy-dependent spacetime behavior aligns with gravity's rainbow.

Connection established between nonlocality and modified quantum uncertainty.

Abstract

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 gravity's rainbow, hence demonstrating a connection between non-locality and gravity's rainbow.