Quantum Black Hole and the Modified Uncertainty Principle

TL;DR

This paper explores how a modified uncertainty principle with a linear momentum term affects the quantum properties of Schwarzschild black holes, revealing a proportional relationship between black hole mass and quantum number.

Contribution

It introduces a modified Heisenberg algebra into the Wheeler-DeWitt equation, analyzing its impact on black hole mass quantization and confirming Bekenstein's mass spectrum proposal.

Findings

Mass scales with the square root of quantum number n

Mass is proportional to quantum number n with quantum gravity effects

Modified uncertainty reduces black hole mass for given quantum number

Abstract

Recently Ali et al. (2009) proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Plank length). Inspired by this idea we examine the Wheeler-DeWitt equation for a Schwarzschild black hole with a modified Heisenberg algebra which has a linear term in momentum. We found that the leading contribution to mass comes from the square root of the quantum number 'n' which coincides with Bekenstein's proposal. We also found that the mass of the black hole is directly proportional to the quantum number 'n' when quantum gravity effects are taken into consideration via the modified uncertainty relation but it reduces the value of mass for a particular value of the quantum number.