# The GUP and quantum Raychaudhuri equation

## TL;DR

This paper compares quantum corrections to black hole temperature from different approaches, establishing a connection between deformation parameters and suggesting spacetime has a crystal-like structure.

## Contribution

It links the quantum Raychaudhuri equation with generalized uncertainty principle parameters, revealing a scale-dependent relation and a correspondence between phenomenological models.

## Key findings

- Quantum Raychaudhuri equation relates to GUP deformation parameters.
- The relation between parameters depends on the black hole scale.
- Indicates spacetime may have a crystal-like structure.

## Abstract

In this paper, we compare the quantum corrections to the Schwarzschild black hole temperature due to quadratic and linear-quadratic generalized uncertainty principle, with the corrections from the quantum Raychaudhuri equation. The reason for this comparison is to connect the deformation parameters $\beta_0$ and $ \alpha_0$ with $\eta$ which is the parameter that characterizes the quantum Raychaudhuri equation. The derived relation between the parameters appears to depend on the relative scale of the system (black hole), which could be read as a beta function equation for the quadratic deformation parameter $\beta_0$. This study shows a correspondence between the two phenomenological approaches and indicates that quantum Raychaudhuri equation implies the existence of a crystal-like structure of spacetime.