Offshell thermodynamic metrics of the Schwarzschild black hole

TL;DR

This paper extends thermodynamic metrics to Schwarzschild black holes by introducing a new degree of freedom, enabling analysis of their thermodynamic geometry despite the lack of conserved charges.

Contribution

It introduces a novel approach to compute offshell thermodynamic metrics for Schwarzschild black holes using an additional degree of freedom.

Findings

Different offshell metrics are computed for Schwarzschild-like black holes.

The thermal Ricci scalar varies with deformation but reduces to the standard case when deformation is switched off.

The method bridges the gap in thermodynamic geometry analysis for charge-less black holes.

Abstract

Thermodynamic metric usually works only for those black holes with more than one conserved charge, therefore the Schwarzschild black hole was excluded. In this letter, we compute and compare different versions of offshell thermodynamic metric for the Schwarzschild-like black hole by introducing a new degree of freedom. This new degree of freedom could be the running Newton constant, a cutoff scale for regular black hole, a noncommutative deformation, or the deformed parameter in the nonextensive Tsallis-Renyi entropy. The onshell metric of the deformed Schwarzschild solution would correspond to the submanifold by gauge fixing of this additional degree of freedom. In particular, the thermal Ricci scalar for the Schwarzschild black hole, though different for various deformation, could be obtained by switching off the deformation.