The temperature and free energy of multi-black hole systems

TL;DR

This paper calculates the temperature and free energy of a multi-black hole system, introducing the Hawking average temperature and confirming thermodynamic consistency for Kerr-Newman black holes.

Contribution

It introduces the Hawking average temperature for multi-black hole systems and derives a consistent free energy expression based on horizon properties.

Findings

Defined the Hawking average temperature for multi-black hole systems.

Derived a free energy formula consistent with thermodynamics.

Confirmed the relation between temperature, free energy, and black hole horizon properties.

Abstract

In the present paper, we compute the Euclidean action of a generic system consisting of $N$ arbitrary Kerr-Newman black holes located on the symmetry axis and separated from each other by massless struts. This allows us to introduce the {\it Hawking average temperature} (HAT) $\hat T$ of the multi-black hole system via the condition of vanishing the entire set of terms involving the singular horizons due to periodic time, and the resulting formula for this temperature contains solely the surface gravities $\kappa_i$ and horizon areas $A^H_i$ of the black hole constituents. We also show that the corresponding expression for the {\it free energy} of the system defined by $\hat T$ is consistent with the first law of thermodynamics and Smarr mass relations.