Heat Capacity and the Violation of Scaling Laws in Gravitational System

TL;DR

The paper investigates the universality of scaling laws near critical points in gravitational systems, focusing on isolated critical points with non-standard exponents $(0,1,2,3)$ observed in certain black holes. It uses Landau-type free-energy expansions for van der Waals-like transitions and black-hole thermodynamics, applying the scaling hypothesis to extract critical exponents and identify violations. For the van der Waals case, the exponents are $(\alpha,\beta,\gamma,\delta)=(0,1/2,1,3)$, consistent with scaling, while for special black holes the observed exponents are $(0,1,2,3)$; scaling would predict $(\alpha,\beta,\gamma,\delta)=(-2,1,2,3)$, with the discrepancy traced to $C_V=0$ implying $\alpha=0$. The work further notes symmetry violations between coexisting phases in some black-hole cases and emphasizes that the scaling violation is tied to the thermodynamic feature $C_V=0$, potentially arising from modified gravity or quantum effects, challenging the universality of gravitational critical phenomena.

Abstract

In this paper, we examine the scaling laws in gravitational system from the perspective of free energy landscape and the scaling hypothesis. It has been found that for some special black holes, their critical exponents $(0,1,2,3)$ are beyond the mean field theory, and more surprisingly violate the scaling laws. We find that the main reason for the violation of the scaling laws is that the heat capacity at constant volume $C_V$ is 0, so the critical exponent $α$ is often treated as 0, which can not be derived from the scaling hypothesis. We also find that there is a symmetry violation for the two coexistence states $ω_l$ and $ω_s$.