Finite Hilbert space and maximum mass of Schwarzschild black holes from a Generalized Uncertainty Principle

TL;DR

Implementing a GUP with minimal length and maximal momentum on Schwarzschild black holes yields a finite mass spectrum, an upper mass bound, and regulates Hawking temperature, constraining quantum gravity parameters with astrophysical data.

Contribution

This work introduces a GUP-based approach to black hole physics that results in a finite mass spectrum and bounds, connecting quantum gravity with astrophysical observations.

Findings

Black hole mass spectrum becomes finite and discrete.

An upper bound on black hole mass is established.

Astrophysical data constrains the GUP parameter to $eta\\lesssim 10^{-98}$.

Abstract

We show that implementing a generalized uncertainty principle (GUP) with both minimal length and maximal momentum directly on the reduced phase space of the Schwarzschild black hole (BH) leads to a finite and discrete mass spectrum, a strict upper bound on the BH mass, a bounded entropy, and a fully regulated Hawking temperature. We further construct a GUP-deformed lapse function that preserves the ADM mass and horizon radius while exactly reproducing the GUP temperature through the surface gravity. Using the most massive observed supermassive BHs, we derive the constraint on the GUP parameter, $\beta\lesssim 10^{-98}$, showing that present astrophysical data already impose robust bounds on minimal length quantum gravity.