# The generalized and extended uncertainty principles and their implications on the Jeans mass

## TL;DR

This paper investigates how generalized and extended uncertainty principles influence Newtonian gravity and thermodynamics, specifically affecting the Jeans mass and potentially altering gravitational collapse conditions.

## Contribution

It introduces the effects of generalized and extended uncertainty principles on the Jeans mass within the Rényi entropy framework, highlighting differences from standard Newtonian predictions.

## Key findings

- Extended uncertainty principle and Rényi entropy increase the Jeans mass.
- Generalized uncertainty principle can decrease the Jeans mass.
- A cloud smaller than the standard Jeans mass may undergo collapse under GUP effects.

## Abstract

The generalized and extended uncertainty principles affect the Newtonian gravity and also the geometry of the thermodynamic phase space. Under the influence of the latter, the energy-temperature relation of ideal gas may change. Moreover, it seems that the Newtonian gravity is modified in the framework of the R\'{e}nyi entropy formalism motivated by both the long-range nature of gravity, and the extended uncertainty principle. Here, the consequences of employing the generalized and extended uncertainty principles, instead of the Heisenberg uncertainty principle, on the Jeans mass are studied. The results of working in the R\'{e}nyi entropy formalism are also addressed. It is shown that unlike the extended uncertainty principle and the R\'{e}nyi entropy formalism which lead to the same increase in the Jeans mass, the generalized uncertainty principle can decrease it. The latter means that a cloud with mass smaller than the standard Jeans mass, obtained in the framework of the Newtonian gravity, may also undergo the gravitational collapse process.