Rainbow spacetime from a nonlocal gravitational uncertainty principle

¹ Department of Physics - Zewail City, Giza, Egypt
² Department of Physics, Zhejiang University - Hangzhou 310027, China
³ Department of Physics, Faculty of Science, Benha University - Benha, 13518, Egypt

^(a) s-omarelrefy@zewailcity.edu.eg
^(b) masood@zju.edu.cn
^(c) sxwlg@yahoo.com
^(d) ahmed.ali@fsc.bu.edu.eg

Received: 6 August 2020
Accepted: 7 September 2020

Abstract

The 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 the contrary, the 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.