# Area (or Entropy) Products in Modified Gravity and Kerr-MOG/CFT Correspondence

## TL;DR

This paper explores the thermodynamic properties of black holes in modified gravity (MOG), deriving relations for horizons, verifying the first law, and establishing a Kerr-MOG/CFT correspondence with consistent central charges and entropy calculations.

## Contribution

It introduces thermodynamic product relations for MOG black holes, derives the Kerr-MOG/CFT duality, and computes central charges and temperatures, extending holographic understanding to modified gravity.

## Key findings

- Area product depends on mass in MOG, unlike in Einstein gravity.
- First law of thermodynamics holds at both horizons in MOG.
- Microscopic entropy matches Bekenstein-Hawking entropy via Cardy formula.

## Abstract

We examine the thermodynamic features of \emph{inner} and outer horizons of modified gravity~(MOG) and its consequences on the holographic duality. We derive the thermodynamic product relations for this gravity. We consider both spherically symmetric solutions and axisymmetric solutions of MOG. We find that the area product formula for both cases is \emph{not} mass-independent because they depends on the ADM mass parameter while in \emph{Einstein gravity} this formula is mass-independent~(universal). We also explicitly verify the \emph{first law} which is fulfilled at the inner horizon~(IH) as well as at the outer horizon~(OH). We derive thermodynamic products and sums for this kind of gravity. We further derive the \emph{Smarr like mass formula} for this kind of black hole~(BH) in MOG. Moreover, we derive the area bound for both the horizons. Furthermore, we show that the central charges of the left and right moving sectors are the same via universal thermodynamic relations. We also discuss the most important result of the \emph{Kerr-MOG/CFT correspondence}. We derive the central charges for Kerr-MOG BH which is $c_{L}=12J$ and it is similar to Kerr BH. We also derive the dimensionless temperature of a extreme Kerr-MOG BH which is $T_{L} = \frac{1}{4\pi} \frac{\alpha+2}{\sqrt{1+\alpha}}$, where $\alpha$ is a MOG parameter. This is actually dual CFT temperature of the Frolov-Thorne thermal vacuum state. In the limit $\alpha=0$, we find the dimensionless temperature of Kerr BH. Consequently, Cardy formula gives us microscopic entropy for extreme Kerr-MOG BH, $S_{micro} = \frac{\alpha+2}{\sqrt{1+\alpha}} \pi J $ for the CFT which is completely in agreement with macroscopic Bekenstein-Hawking entropy.