# Universal ratios of critical physical quantities of charged AdS black holes

## TL;DR

This paper explores universal ratios of critical physical quantities in charged AdS black holes, revealing certain ratios are universal while others are not, and introduces a novel dimensional analysis approach for studying these ratios.

## Contribution

It introduces the use of dimensional analysis to study critical ratios in black hole thermodynamics and identifies which ratios are universal across different critical behaviors.

## Key findings

- The ratio T_c S_c / Q_c is universal.
- The ratio T_c r_c / Q_c is not universal.
- The ratio Φ_c Q_c / T_c is not universal, but two other ratios for Q-Φ criticality are universal.

## Abstract

We investigate the ratios of critical physical quantities related to the $T-S$ criticality of charged AdS black holes. It is shown that the ratio $\frac{T_cS_c}{Q_c}$ is universal while $\frac{T_cr_c}{Q_c}$ is not. This finding is quite interesting considering the former observation that both the $T-S$ graph and $T-r_+$ graph exhibit reverse van der Waals behavior. It is also worth noting that the value of $\frac{T_cS_c}{Q_c}$ differs from that of $\frac{P_cv_c}{T_c}$ for $P-V$ criticality. Moreover, we discuss universal ratios for the $P-V$ criticality and $Q-\Phi$ criticality. We successfully interpret the former finding that the ratio $\frac{\Phi_cQ_c}{T_c}$ is not universal and construct two universal ratios for the $Q-\Phi$ criticality instead. To the best of our knowledge, we are the first to introduce the dimensional analysis technique to study the ratios of critical physical quantities. It is expected that this technique can be generalized to probe the universal ratios for $Y-X$ criticality in future research.