Revisiting particle circular orbits as probes of black hole thermodynamics

TL;DR

This paper examines how the radius of particle circular orbits can indicate black hole phase transitions, revealing ensemble-dependent reliability and a connection between geometry and thermodynamics.

Contribution

It systematically analyzes the encoding mechanism of phase transitions in particle orbits, highlighting its dependence on thermodynamic ensembles and the role of the first law.

Findings

Reliable in isobaric ensemble

May fail in isothermal ensemble

Orbit radius gap as a criticality indicator

Abstract

Recent studies propose that black hole phase transitions can be encoded in the circular orbit radius of particles. In this paper, we systematically investigate the reliability of this encoding mechanism. We find that this mechanism is highly reliable in the isobaric ensemble, whereas it may break down in the isothermal ensemble. It turns out that the reliability of this mechanism is directly controlled by the first law of black hole thermodynamics. Interestingly, even if this encoding mechanism fails, we prove that, for any black hole exhibiting criticality, the first law can ensure that, near the critical point, the coexistence gap of the circular orbit radius remains a reliable order parameter and yields exactly the same critical exponents as the standard thermodynamic order parameter. Our results provide a potential way to identify the thermodynamic ensemble of a black hole, and reveal a deeper connection between gravitational geometry and thermodynamics.