Area (or entropy) products in modified gravity and Kerr-MG/CFT correspondence

Abstract.

We examine the thermodynamic features of 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 not mass-independent because they depend on the ADM mass parameter while, in Einstein gravity, this formula is mass-independent (universal). We also explicitly verify the 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 Smarr-like mass formula for this kind of black hole (BH) in MOG. Moreover, we derive the area bound for both 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 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 for 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 the dual CFT temperature of the Frolov-Thorne thermal vacuum state. In the limit $\alpha = 0$, we find the dimensionless temperature of a Kerr BH. Consequently, the Cardy formula gives us microscopic entropy for extreme Kerr-MOG BH, $S_{\rm micro}=\frac{\alpha+2}{\sqrt{1+\alpha}} \pi J$, for the CFT, which is completely in agreement with the macroscopic Bekenstein-Hawking entropy. Therefore we may conjecture that, in the extremal limit, the Kerr-MOG BH is holographically dual to a chiral 2D CFT with central charge $c_{L}=12J$. Finally, we derive the mass-independent area (or entropy) product relations for regular MOG BH.