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

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.

version accepted in EPJ Plus

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• The Bullet Cluster 1E0657-558 evidence shows Modified Gravity in the absence of Dark Matter
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• Thermodynamic Product Formula for Hořava Lifshitz Black Hole

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• Distinguishing Black Hole and Naked Singularity in MOG via Inertial Frame Dragging Effect
• Thermodynamics of Non-Commutative Scalar-Tensor-Vector Gravity Black holes
• Regular Rotating MOG Dark Compact Object
• Kiselev/CFT correspondence and black hole thermodynamics