Revisiting particle circular orbits as probes of black hole thermodynamics

TL;DR

The paper establishes a general monotonicity criterion for how circular orbit radii $r_c$ map to horizon radii $r_h$, showing the reliability of using $r_c$ to encode black hole phase transitions depends on the ensemble via the first law. In isobaric (fixed pressure) processes, $r_c(r_h)$ is globally monotonic for RN–AdS and thus $r_c$ jumps with the horizon at phase transitions, preserving a robust geometric order parameter. In isothermal processes, $r_c(r_h)$ can become nonmonotonic due to the first law allowing $dM/dr_h$ to vanish or change sign, leading to potential suppression of the $r_c$-based signal in images; photon-sphere radii can nonmonotonically evolve, while ISCOs also exhibit nonmonotonic behavior in several cases. Nevertheless, near the critical point the reparametrization $r_h\to r_c$ is locally monotonic, ensuring the critical exponents (e.g., $\beta$) remain invariant, revealing a deep connection between gravitational geometry and thermodynamics that extends beyond specific metrics to general black holes with $\partial_M \Phi_\varepsilon\neq0$.

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.

Figures (9)

• Figure 1: Behavior of $M(r_h,P)$ in the Sch--AdS spacetime. Thin black solid curves denote contours of constant mass $M$. Along the dashed trajectory from A to B, the horizon radius $r_h$ increases while both $M$ and the photon sphere radius $r_{\rm ps}=3M$ decrease, showing that $r_{\rm ps}(r_h)$ is strictly monotonically decreasing along this path.
• Figure 2: Behavior of $r_{\rm ps}(r_h)$ along isobaric paths in $d=4$. For the Sch--AdS case, the pressure is likewise set to the value $P = 0.8\,P_{\mathrm{c}}$, where $P_{\mathrm{c}}$ is the critical pressure of the RN--AdS solution with $q=1$. The same choice is adopted for subsequent figures and will not be restated hereafter.
• Figure 3: Behavior of $r_{\rm isco}(r_h)$ along isobaric paths in $d=4$
• Figure 4: Behavior of $r_{\rm ps}(r_h)$ along isothermal paths in $d=4$. The marked point on the curve corresponds to $P=0$ at $r_{h,\min}=(d-3)/(4\pi T_0)$, and the unphysical portion with $P<0$ is truncated. For the Sch--AdS case, we likewise set the temperature to $T = 0.8\,T_{\mathrm{c}}$, where $T_{\mathrm{c}}$ is the critical temperature of the RN--AdS solution with $q=1$. The same choice is adopted for subsequent figures and will not be repeated.
• Figure 5: Behavior of $r_{\rm isco}(r_h)$ along isothermal paths in $d=4$.