Violation of the Third Law of Thermodynamics by Black Holes, Riemann Zeta Function and Bose Gas in Negative Dimensions

TL;DR

This paper explores the exotic thermodynamic behavior of black holes, showing they violate the third law, and models this using Bose gases in negative dimensions with the Riemann zeta function.

Contribution

It introduces a novel quantum statistical model linking black hole entropy to Bose gases in negative dimensions using the Riemann zeta function.

Findings

Black holes violate the third law of thermodynamics.

Schwarzschild black hole entropy is inversely proportional to the square of temperature.

Black hole behavior corresponds to Bose gases in negative spatial dimensions.

Abstract

Black holes violate the third law of thermodynamics in its standard formulation. Schwarzschild black hole entropy is inverse proportional to the square of the temperature $S=1/(16 \pi T^2) $ and tends to infinity rather than zero when the temperature goes to zero. We search for quantum statistical models with such exotic thermodynamic behaviour. It is shown that the Schwarzschild black hole in $D=4 $ spacetime dimensions corresponds to a Bose gas in a space with $d=-\, 4$ negative spatial dimensions. The Riemann zeta function is used to define the entropy of the Bose gas in negative dimension. The correspondence between black holes in higher dimensions and de Sitter spacetime with Bose gases is considered.