arXiv subjects

Parthapratim Pradhan

Publications and source records attributed to Parthapratim Pradhan.

At least 19 recordsLinked to original sources

arXiv2026

Monodromy, Hidden Conformal Symmetry and Entropy Product Formula For Kerr-MOG Black Hole

We examine the two-dimensional conformal field theory (2D CFT) dual of the four-dimensional Kerr-MOG (Kerr-modified gravity) black hole using the black hole thermodynamics and monodromy techniques. From these independent procedures, we derive consistent expressions for the left- and right-moving temperatures and the central charge of the dual CFT. Finally, we derive the entropy product using these approaches. We show that the product is \emph{not} universal as well as not quantized. We further explore the hidden conformal symmetry of the non-extremal Kerr-MOG black hole by analyzing the near-region dynamics of a massless scalar field. The resulting Kerr-MOG/CFT correspondence identifies the near-region geometry with a two-dimensional CFT characterized by the temperatures ($T_{L}, T_{R}$) derived in~(\ref{temperatures}). Moreover, using the Cardy formula, we reproduce the microscopic entropy of the dual CFT and find exact agreement with the Bekenstein-Hawking entropy of the Kerr-MOG black hole. We also show that the low-frequency scalar absorption cross section~(greybody factor) in the near-region Kerr-MOG geometry precisely matches the finite-temperature absorption cross section of the dual 2D CFT. In addition, the near-region scalar wave equation exhibits a hidden $(SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R)$ conformal symmetry. These independent results provide strong evidence for the existence of a Kerr/CFT-type holographic duality for the Kerr-MOG black hole, establishing a consistent correspondence between the four-dimensional Kerr-MOG spacetime and a two-dimensional conformal field theory. horizon.

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arXiv2026

Inertial Frame Dragging as a Probe to Differentiate Kerr-Newman Naked Singularities from Black Holes

We study the spin precession of a test gyroscope attached to a stationary observer in Kerr-Newman spacetime to distinguish a naked singularity from a black hole. Extending earlier work on Kerr, we examine how the electric charge $Q$ affects precession in both cases. For gyroscopes with nonzero angular velocity $Ω$, we derive closed-form expressions for the general spin-precession, Lense-Thirring, and geodetic precession frequencies. For Kerr-Newman black holes, the spin-precession frequency generically diverges as the event horizon is approached from any direction, remaining finite only for zero-angular-momentum observers (ZAMOs). By contrast, for Kerr-Newman naked singularities, it remains finite everywhere except at the ring singularity on the equatorial plane. We show that $Q$ systematically modifies these features, especially in rapidly rotating regimes, and that the acceleration scalar further sharpens the black hole/naked singularity distinction. We also investigate the Lense-Thirring (nodal) precession frequency of equatorial circular orbits in accretion disks. For black holes, the nodal frequency decreases monotonically with radius, whereas for naked singularities it rises, attains a finite maximum, and then decreases; for sufficiently large spin and charge it can even change sign, signalling a reversal of the precession direction. We further compute the Keplerian, radial, and vertical epicyclic frequencies, along with the periastron precession frequency, highlighting the role of $Q$ in the ISCO and the orbital-frequency hierarchy. Since these frequencies are closely related to observed quasiperiodic oscillations (QPOs), these features provide a strong-field probe of whether a rotating compact object is a black hole or a naked singularity.

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arXiv2025

Zero Mass limit of Kerr-MOG Black Hole Equals Wormhole

It has been argued in existing literature that the zero mass limit of Kerr spacetime corresponds to either flat Minkowski spacetime or a wormhole exhibiting a locally flat geometry. In this study, we examine that the zero mass limit of the Kerr-MOG black hole is equivalent to a wormhole. Moreover, we derive the Kerr-Schild form of the Kerr-MOG black hole through specific coordinate transformations. We further investigate the physical and topological characteristics of the Kerr-MOG black hole within the framework of modified gravity. Our analysis also includes a discussion of the wormhole using cylindrical coordinates, which comprises two distinct coordinate patches. Furthermore, we extend our analysis to the Kerr-Newman black hole and show that the \emph{zero mass limit of the Kerr-Newman black hole does not yield a wormhole}. However, if we impose an additional criterion such that \emph{both the mass parameter and the charge parameters are equal to zero}, then the Kerr-Newman black hole will be a wormhole.

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arXiv2024

Distinguishing Signature of Kerr-MOG Black Hole and Superspinar via Lense-Thirring Precession

We examine the geometrical differences between the black hole~(BH) and naked singularity~(NS) or superspinar via Lense-Thirring~(LT) precession in spinning modified-gravity~(MOG). For BH case, we show that the LT precession frequency~($Ω_{LT}$) along the pole is proportional to the angular-momentum~($J$) parameter or spin parameter~($a$) and is inversely proportional to the cubic value of radial distance parameter, and also governed by Eq.(1). Along the equatorial plane it is governed by Eq.(2). While for superspinar, we show that the LT precession frequency is inversely proportional to the cubic value of the spin parameter and it decreases with distance by MOG parameter as derived in Eq.(3) at the pole and in the limit $a>>r$ (where $α$ is MOG parameter). For $θ\neq\fracπ{2}$ and in the superspinar limit, the spin frequency varies as $Ω_{LT}\propto \frac{1}{a^3\cos^4θ}$ and by Eq.(38)

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arXiv2024

Bekenstein-Hawking Entropy Products for NUT class of Black Holes in AdS Space

We derive the entropy product rule for Taub-NUT~(Newman-Unti-Tamburino)-de~Sitter black hole~(BH) and Taub-NUT--Anti-de~Sitter BH. We show that the entropy products in terms of both the physical horizons are \emph{mass-independent}. Both \emph{perturbative} approximation and \emph{direct} method have been considered. By introducing the cosmological horizon we show that for Taub-NUT-de~Sitter BH, there exists a mass-independent entropy functional relation in terms of three horizons namely event horizon~(EH), Cauchy horizon~(CH) and cosmological horizon~(CHH) which depends on cosmological parameter~($Λ$) and the NUT parameter~($N$). For Taub-NUT-anti-de~Sitter BHs, we determine the mass-independent entropy functional relations in terms of two physical horizons~(namely EH and CH) which depends on only NUT parameter. Some-times some complicated functions of EH entropy and CH entropy are also strictly mass-independent. This is plausible only due to the new formalism developed in~\cite{wu}~[Phys. Rev. D 100, 101501(R)~(2019)] for NUT class of BHs. The formalism states that a generic four-dimensional Taub-NUT spacetime should be described completely in terms of three or four different types of thermodynamic hairs. They could be defined as the Komar mass~($M=m$), the angular momentum~($J_{n}=mn$), the gravitomagnetic charge ($N=n$), the dual~(magnetic) mass $(\tilde{M}=n)$. Finally, we could say that this universality is mainly due to the presence of \emph{new conserved charges $J_{N}=MN$} which is closely analogue to the Kerr-like angular momentum $J=aM$.

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arXiv2023

Entropy Product Function and Central charges in NUT Geometry

We define an \emph{entropy product function}~(EPF) for Taub-Newman-Unti-Tamburino~(TNUT) black hole~(BH) following the prescription suggested by Wu et al.~\cite{wu} ~[PRD 100, 101501(R) (2019)]. The prescription argues that a generic four-dimensional TNUT spacetime might be expressed in terms of three or four different types of thermodynamic hairs. They can be defined as the Komar mass~($M=m$), the angular momentum~($J_{n}=mn$), the gravitomagnetic charge ($N=n$), the dual~(magnetic) mass $(\tilde{M}=n)$. Taking this prescription and using the \emph{EPF}, we derive the \emph{central charges} of dual CFT~(conformal field theory) via Cardy's formula. Remarkably, we \emph{find} that for TNUT BH there exists a relation between the \emph{central charges and EPF} as $c=6\left(\frac{\partial {\cal F}}{\partial {\cal N}_{i}}\right)$, where ${\cal F}$ is EPF and ${\cal N}_{i}$ is one of the integer-valued charges i.e. the NUT charges~($N$) or any new conserved charges~($J_{N}$). We reverify these results by calculating the exact values of different thermodynamic parameters. We define the EPF~${\cal F}$ from the first law of thermodynamics of both horizons. Moreover, we write the first laws of both the horizons for left-moving and right-moving sectors. Introducing the Bézout's identity, we show that for TNUT BH one can generate more holographic descriptions described by a pair of integers $(a,b)$. More holographic pictures have a great significance in understanding the holographic nature of quantum gravity. Furthermore, using the \emph{EPF} we derive the central charges for Reissner-Nordström-NUT~(RNNUT) BH, Kerr-Taub-NUT~(KNUT) BH and Kerr-Newman-NUT~(KNNUT) BH. Finally, we prove that they are equal in both sectors provided that the EPF is mass-independent~(or universal).

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arXiv2023

Energy Formula, Surface geometry and Energy Extraction for Kerr-Sen Black Hole

We evaluate the \emph{surface energy~(${\cal E}_{s}^{\pm}$), rotational energy~(${\cal E}_{r}^{\pm}$) and electromagnetic energy~(${\cal E}_{em}^{\pm}$)} for a \emph{Kerr-Sen black hole~(BH)} having the event horizon~(${\cal H}^{+}$) and the Cauchy horizon~(${\cal H}^{-}$). Interestingly, we find that the \emph{sum of these three energies is equal to the mass parameter i.e. ${\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}={\cal M}$}. Moreover in terms of the \emph{ scale parameter ~$(ζ_{\pm})$, the distortion parameter~($ξ_{\pm}$) and a new parameter~$(σ_{\pm})$} which corresponds to the area~(${\cal A}_{\pm}$), the angular momentum ~$(J)$ and the charge parameter~($Q$), we find that the \emph{mass parameter in a compact form} ${\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}={\cal M} =\frac{ζ_{\pm} }{2} \sqrt{\frac{1+2\,σ_{\pm}^2}{1-ξ_{\pm}^2}}$ %\begin{eqnarray} %{\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}={\cal M} =\frac{ζ_{\pm} }{2} %\sqrt{\frac{1+2\,σ_{\pm}^2}{1-ξ_{\pm}^2}} \nonumber %\end{eqnarray} which is valid {through all the horizons} (${\cal H}^{\pm}$). We also compute the \emph{equatorial circumference and polar circumference} which is a gross measure of the BH surface deformation. It is shown that when the spinning rate of the BH increases, the \emph{equatorial circumference increases} while the \emph{polar circumference decreases}. Furthermore, we compute the exact expression of \emph{rotational energy that should be extracted from the BH via the Penrose process}. The maximum value of rotational energy which is extractable should occur for \emph{extremal Kerr-Sen BH} i.e. ${\cal E}_{r}^{+}%=\left(\frac{\sqrt{2}-1}{2}\right)\sqrt{2{\cal M}^2-Q^2} =\left(\sqrt{2}-1\right)\sqrt{\frac{J}{2}}$.

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arXiv2021

Energy Formula for Newman-Unti-Tamburino class of Black Holes

We compute the \emph{surface energy~(${\cal E}_{s}^{\pm}$), the rotational energy~(${\cal E}_{r}^{\pm}$) and the electromagnetic energy~(${\cal E}_{em}^{\pm}$)} for Newman-Unti-Tamburino~(NUT) class of black hole having the event horizon~(${\cal H}^{+}$) and the Cauchy horizon~(${\cal H}^{-}$). Remarkably, we find that the \emph{mass parameter can be expressed as sum of three energies i. e. $M={\cal E}_{s}^{\pm}+{\cal E}_{r}^{\pm}+{\cal E}_{em}^{\pm}$}. It has been \emph{tested} for Taub-NUT black hole, Reissner-Nordström-Taub-NUT black hole, Kerr-Taub-NUT black hole and Kerr-Newman-Taub-NUT black hole. In each case of black hole, we find that \emph{the sum of these energies is equal to the Komar mass}. It is plausible only due to the introduction of new conserved charges i.e. $J_{N}=M\,N$~(where $M=m$ is the Komar mass and $N=n$ is the gravitomagnetic charge), which is closely analogue to the Kerr-like angular momentum parameter $J=a\,M$.

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arXiv2021

Null Geodesics and QNMs in the field of Regular Black Holes

We analyze the null geodesics of regular black holes. A detailed analysis of geodesic structure both null geodesics and time-like geodesics have been investigated for the said black hole. As an application of null geodecics, we calculate the radius of photon sphere and gravitational bending of light. We also study the shadow of the black hole spacetime. Moreover, we determine the relation between radius of photon sphere~$(r_{ps})$ and the shadow observed by a distance observer. Furthermore, We discus the effect of various parameters on the radius of shadow $R_s$. Also we compute the angle of deflection for the photons as a physical application of null-circular geodesics. We find the relation between null geodesics and quasinormal modes frequency in the eikonal approximation by computing the Lyapunov exponent. It is also shown that~(in the eikonal limit) the quasinormal modes~(QNMs) of black holes are governed by the parameter of null-circular geodesics. The real part of QNMs frequency determines the angular frequency whereas the imaginary part determines the instability time scale of the circular orbit. Next we study the massless scalar perturbations and analyze the effective potential graphically. Massive scalar perturbations also discussed. As an application of time-like geodesics we compute the innermost stable circular orbit~(ISCO) and marginally bound circular orbit~(MBCO) of the regular BHs which are closely related to the black hole accretion disk theory. In the appendix, we calculate the relation between angular frequency and Lyapunov exponent for null-circular geodesics.

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arXiv2020

Black Hole versus Naked Singularity via Axial Perturbation

We differentiate non-extremal black hole, \emph{extremal} black hole and \emph{naked singularity} via metric perturbations for Reissner-Nordström spacetime. First we study the axial perturbations for \emph{extremal} Reissner-Nordström black hole and compute the effective potential due to these perturbations. Then we study the axial perturbations for the naked singularity case and compute the effective potential. We show that for the non-extremal black hole, \emph{the effective potential outside the event horizon~($r_{+}$) is real and positive. While in between Cauchy horizon~($r_{-}$) and event horizon~($r_{-}<r<r_{+}$) the effective potential is negative.} For the \emph{extremal black hole, the effective potential is always positive}. Also for \emph{naked singularity, the effective potential is positive.} From the effective potential diagram, we show that the structure of effective potentials for extremal BH looks like a potential barrier outside the horizon. While for non-extremal BH, the structure of the effective potentials look like a \emph{potential well} rather than a potential barrier. For NS the structure of the effective potentials is \emph{neither a potential barrier nor a potential well. Preferably it looks like an exponential decay function}. We observe that the geometric construction of an effective potential barrier due to axial perturbations could allow us to distinguish between the non-extremal black hole, extremal black hole, and naked singularity. Stability of extremal BH has been discussed.

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arXiv2020

Geodesic stability and Quasi normal modes via Lyapunov exponent for Hayward Black Hole

We derive proper-time Lyapunov exponent $(λ_{p})$ and coordinate-time Lyapunov exponent $(λ_{c})$ for a regular Hayward class of black hole. The proper-time corresponds to $τ$ and the coordinate time corresponds to $t$. Where $t$ is measured by the asymptotic observers both for for Hayward black hole and for special case of Schwarzschild black hole. We compute their ratio as $\frac{λ_{p}}{λ_{c}} = \frac{(r_σ^{3} + 2 l^{2} m )}{\sqrt{(r_σ^{2} + 2 l^{2} m )^{3}- 3 m r_σ^{5}}}$ for time-like geodesics. In the limit of $l=0$ that means for Schwarzschild black hole this ratio reduces to $\frac{λ_{p}}{λ_{c}} = \sqrt{\frac{r_σ}{(r_σ-3 m)}}$. Using Lyponuov exponent, we investigate the stability and instability of equatorial circular geodesics. By evaluating the Lyapunov exponent, which is the inverse of the instability time-scale, we show that, in the eikonal limit, the real and imaginary parts of quasi-normal modes~(QNMs) is specified by the frequency and instability time scale of the null circular geodesics. Furthermore, we discuss the unstable photon sphere and radius of shadow for this class of black hole.

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arXiv2020

Distinguishing Black Hole and Naked Singularity in MOG via Inertial Frame Dragging Effect

We analyze the generalized spin precession of a test gyroscope around a stationary spacetime i.e. for Kerr-MOG black hole~(BH) in scalar-tensor-vector gravity or modified gravity~(MOG). A detailed study of generalized spin frequency has been done for \emph{non} extremal Kerr-MOG BH, \emph{extremal} Kerr-MOG BH and \emph{naked singularity~(NS)} in comparison to non-extremal BH, extremal BH, and NS of Kerr spacetime. The generalized spin frequency that {we have} computed could be expressed in terms of {the} BH mass parameter, the angular momentum parameter, and the MOG parameter. Moreover, we differentiate the non-extremal BH, extremal BH, and NS via computation of the said precession frequency. The Lense-Thirring~(LT) frequency {can} obtain from generalized spin frequency by taking the limit as $Ω=0$ i. e. {when the} angular frequency is set to zero limit. Furthermore, we compute the LT frequency for various {values of} angular coordinates i.e. starting from polar to {the} equatorial plane. We show that the LT frequency diverges at the horizon for extremal BH. Finally, we study the accretion disk physics by computing three epicyclic frequencies namely the Keplerian frequency, {the} radial epicyclic frequency and {the} vertical epicyclic frequency. We also compute the periastron frequency and nodal frequency. With the aid of these frequency profiles, {one} can distinguish three compact objects i. e. \emph{non-extremal BH, extremal BH} {and} \emph{NS}.

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arXiv2020

Area (or entropy) products for Newman-Unti-Tamburino class of Black Holes

We compute area (or entropy) product formula for Newman-Unti-Tamburino (NUT) class of black holes. Specifically, we derive the area product of outer horizon and inner horizon (${ \mathcal{H}}^{\pm }$) for Taub-NUT, Euclidean Taub-NUT black hole, Reissner-Nordström--Taub-NUT black hole, Kerr-Taub-NUT black hole and Kerr-Newman-Taub-NUT black hole under the formalism developed very recently by Wu et al. \cite{wu} [PRD 100, 101501(R) (2019)]. The formalism is that a generic four dimensional Taub-NUT spacetime should be described completely in terms of three or four different types of thermodynamic hairs. They are defined as the Komar mass ($M=m$), the angular momentum ($J_{n}=m\,n$), the gravitomagnetic charge ($N=n$), the dual (magnetic) mass $(\tilde{M}=n)$. After incorporating this formalism, we show that the area (or entropy) product of both the horizons for NUT class of black holes are \emph{mass-independent}. Consequently, the area product of ${\mathcal{H}}^{\pm }$ for these black holes are \emph{universal}. Which was previously known in the literature that the area product of said black holes are \emph{mass-dependent}. Finally, we can say that this universality is solely due to the presence of \emph{new conserved charges $J_{N}=M\,N$} which is closely analogue to the Kerr like angular momentum $J=a\,M$.

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arXiv2019

Study of energy extraction and epicyclic frequencies in Kerr-MOG~(Modified Gravity) black hole

We investigate the energy extraction by the Penrose process in Kerr-MOG black hole~(BH). We derive the gain in energy for Kerr-MOG as \begin{eqnarray} Δ{\cal E} \leq \frac{1}{2}\left(\sqrt{\frac{2}{1+\sqrt{\frac{1}{1+α}-\left(\frac{a}{\cal M}\right)^2}} -\fracα{1+α} \frac{1}{\left(1+\sqrt{\frac{1}{1+α}-\left(\frac{a}{\cal M}\right)^2} \right)^2}}-1\right) \nonumber \end{eqnarray} Where $a$ is spin parameter, $α$ is MOG parameter and ${\cal M}$ is the Arnowitt-Deser-Misner(ADM) mass parameter. When $α=0$, we obtain the gain in energy for Kerr BH. For extremal Kerr-MOG BH, we determine the maximum gain in energy is $Δ{\cal E} \leq \frac{1}{2} \left(\sqrt{\frac{α+2}{1+α}}-1 \right)$. We observe that the MOG parameter has a crucial role in the energy extraction process and it is in fact diminishes the value of $Δ{\cal E}$ in contrast with extremal Kerr BH. Moreover, we derive the \emph{Wald inequality and the Bardeen-Press-Teukolsky inequality} for Kerr-MOG BH in contrast with Kerr BH. Furthermore, we describe the geodesic motion in terms of three fundamental frequencies: the Keplerian angular frequency, the radial epicyclic frequency and the vertical epicyclic frequency. These frequencies could be used as a probe of strong gravity near the black holes.

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arXiv2019

Horizon Areas and Logarithmic Correction to the Charged Accelerating Black Hole Entropy

It has been shown by explicit and exact calculation that the geometric product formula i.e. horizon area (or entropy) product formula of outer horizon (${\cal H}^{+}$) and inner horizon (${\cal H}^{-}$) for charged accelerating black hole (BH) should \emph{ neither be mass-independent nor it be quantized}. This implies that the horizon area (or entropy ) product is mass-independent conjecture has been~\emph{broken down} for charged accelerating BH. This also further implies that the mass-independent feature of the area product of ${\cal H}^{\pm}$ is \emph{not} a generic feature at all. We also compute that the \emph{Cosmic-Censorship-Inequality} for this BH. Indeed it is violated for this BH. Moreover, we compute the specific heat for this BH to determine the local thermodynamic stability of this BH. Under certain criterion, the BH shows the second order phase transition. Furthermore, we compute logarithmic corrections to the entropy for the said BH due to small statistical fluctuations around the thermal equilibrium.

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arXiv2018

Extended Phase Space Thermodynamics of Black Holes in Massive Gravity

We study the extended phase space thermodynamics of black holes in massive gravity. Particularly, we examine the critical behaviour of this black hole using the extended phase space formalism. Extended phase space in a sense that in which the cosmological constant should be treated as a thermodynamic pressure and its conjugate variable as a thermodynamic volume. In this phase space, we derive the black hole equation of state, the critical pressure, the critical volume and the critical temperature at the critical point. We also derive the critical ratio of this black hole. Moreover, we derive the black hole reduced equation of state in terms of the reduced pressure, the reduced volume and the reduced temperature. Furthermore, we examine the Ehrenfest equations of black holes in massive gravity in the extended phase space at the critical point. We show that the Ehrenfest equations are satisfied of this black hole and the black hole encounters a second order phase transition at the critical point in the said phase space. This is re-examined by evaluating the Pregogine-Defay ratio~($\varPi$). We determine the value of this ratio is $\varPi=1$. The outcome of this study is completely analogous to the nature of liquid-gas phase transition at the critical point. This investigation also further gives us the profound understanding between the black hole of massive gravity with the liquid-gas system.

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arXiv2018

Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric Spacetime

In this Letter, we have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetimes in the framework of \emph{extended phase space}. Such volume products usually formulated in terms of the outer horizon~(${\cal H}^{+}$) and the inner horizon~(${\cal H}^{-}$) of black hole ~ (BH) spacetime. Besides volume product, the other thermodynamic formulations like \emph{volume sum, volume minus and volume division} are considered for a wide variety of spherically symmetric spacetime and axisymmetric spacetimes. Like area~(or entropy) product of multihorizons, the mass-independent~(universal) feature of volume products are sometimes also \emph{fail}. In particular for a spherically symmetric AdS spacetimes the simple thermodynamic volume product of ${\cal H}^{\pm}$ is not mass-independent. In this case, more complicated combinations of outer and inner horizon volume products are indeed mass-independent. For a particular class of spherically symmetric cases i.e. Reissner Nordström BH of Einstein gravity and Kehagias-Sfetsos BH of Hořava Lifshitz gravity, the thermodynamic volume products of ${\cal H}^{\pm}$ is indeed \emph{universal}. For axisymmetric class of BH spacetime in Einstein gravity all the combinations are \emph{mass-dependent}. There has been no chance to formulate any combinations of volume product relation is to be mass-independent. Interestingly, \emph{only the rotating BTZ black hole} in 3D provides the volume product formula is mass-independent i.e. \emph{universal} and hence it is quantized.

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arXiv2018

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

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 $T_{L} = \frac{1}{4π} \frac{α+2}{\sqrt{1+α}}$, where $α$ is a MOG parameter. This is actually dual CFT temperature of the Frolov-Thorne thermal vacuum state. In the limit $α=0$, we find the dimensionless temperature of Kerr BH. Consequently, Cardy formula gives us microscopic entropy for extreme Kerr-MOG BH, $S_{micro} = \frac{α+2}{\sqrt{1+α}} πJ $ for the CFT which is completely in agreement with macroscopic Bekenstein-Hawking entropy.

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