Revisiting particle circular orbits as probes of black hole phase transitions

Abstract

Previous studies suggested that the particle circular orbit can serve as a probe of black hole phase transitions. However, these studies only identified this phenomenon by substituting the horizon radius $r_h$ with the circular orbit radius $r_c$ in thermodynamic state relations. Such simplistic substitution fails to uncover the underlying connection between black hole phase transitions and particle circular orbits. In this work, we successfully establish this profound intrinsic link by deriving a differential relation for $r_c$ that relates to thermodynamic parameters and the first law of black hole thermodynamics. Using this relation, we demonstrate that during a first-order phase transition, if $r_h$ experiences a discontinuous jump (such as in the small/large black hole phase transition), then $r_c$ must simultaneously undergo a discontinuous jump. This finding confirms that particle circular orbits can indeed serve as probes of first-order phase transitions. More importantly, we show that this phenomenon is a direct consequence of the nonzero latent heat inherent to first-order phase transitions. Finally, we demonstrate that the jump sizes $\Delta r_c$ and $\Delta r_h$ across the phase transition share the same critical exponent at the thermodynamic critical point, indicating that $\Delta r_c$ can serve as an order parameter for black hole phase transitions. Notably, this conclusion follows directly from the first law.