Komar energy and Smarr formula for noncommutative Schwarzschild black hole

TL;DR

This paper computes the Komar energy for noncommutative Schwarzschild black holes, revealing deviations from classical relations and showing non-zero energy at extremal points, thus extending black hole thermodynamics to noncommutative geometries.

Contribution

It introduces a corrected Komar energy and Smarr formula for noncommutative black holes, highlighting deviations from classical relations due to noncommutative effects.

Findings

Deformation of the relation E=2ST_H in noncommutative case

Nonvanishing Komar energy at extremal black holes

Breakdown of the area law in noncommutative geometry

Abstract

We calculate the Komar energy $E$ for a noncommutative Schwarzschild black hole. A deformation from the conventional identity $E=2ST_H$ is found in the next to leading order computation in the noncommutative parameter $\theta$ (i.e. $\mathcal{O}(\sqrt{\theta}e^{-M^2/\theta})$) which is also consistent with the fact that the area law now breaks down. This deformation yields a nonvanishing Komar energy at the extremal point $T_{H}=0$ of these black holes. We then work out the Smarr formula, clearly elaborating the differences from the standard result $M=2ST_H$, where the mass ($M$) of the black hole is identified with the asymptotic limit of the Komar energy. Similar conclusions are also shown to hold for a deSitter--Schwarzschild geometry.