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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., Comment: 24 pages, 18 figure panels, Accepted in Int. J. Mod. Phys. D

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-Nordstr\"{o}m-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 $J=a\,M$., Comment: Accepted in General Relativity and Gravitation~(Letter)

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} \Delta {\cal E} \leq \frac{1}{2}\left(\sqrt{\frac{2}{1+\sqrt{\frac{1}{1+\alpha}-\left(\frac{a}{{\cal M}}\right)^2}} -\frac{\alpha}{1+\alpha} \frac{1}{\left(1+\sqrt{\frac{1}{1+\alpha}-\left(\frac{a}{{\cal M}}\right)^2} \right)^2}}-1\right) \nonumber \end{eqnarray} Where $a$ is spin parameter, $\alpha$ is MOG parameter and ${\cal M}$ is the Arnowitt-Deser-Misner(ADM) mass parameter. When $\alpha=0$, we obtain the gain in energy for Kerr BH. For extremal Kerr-MOG BH, we determine the maximum gain in energy is $\Delta {\cal E} \leq \frac{1}{2} \left(\sqrt{\frac{\alpha+2}{1+\alpha}}-1 \right)$. We observe that the MOG parameter has a crucial role in the energy extraction process and it is in fact diminishes the value of $\Delta {\cal E}$ 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., Comment: Accepted in EPJC

We derive proper-time Lyapunov exponent $(\lambda_{p})$ and coordinate-time Lyapunov exponent $(\lambda_{c})$ for a regular Hayward class of black hole. The proper-time corresponds to $\tau$ 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 $\frac{\lambda_{p}}{\lambda_{c}} = \frac{(r_{\sigma}^{3} + 2 l^{2} m )}{\sqrt{(r_{\sigma}^{2} + 2 l^{2} m )^{3}- 3 m r_{\sigma}^{5}}}$ for time-like geodesics. In the limit of $l=0$ that means for Schwarzschild black hole this ratio reduces to $\frac{\lambda_{p}}{\lambda_{c}} = \sqrt{\frac{r_{\sigma}}{(r_{\sigma}-3 m)}}$. 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., Comment: To appear in Mod.Phys.Lett.A, 19 pages and 6 figures

We investigate the entropy product formula for various gravitational instantons. We speculate that due to the mass-independent features of the said instantons they are \emph{universal} as well as they are \emph{quantized}. For isolated Euclidean Schwarzschild black hole, these properties simply \emph{fail}., Comment: Version accepted in AHEP

We derive thermodynamic product relations for four-parametric regular black hole (BH) solutions of the Einstein equations coupled with a non-linear electrodynamics source. The four parameters can be described by the mass ($m$), charge ($q$), dipole moment ($\alpha$) and quadrupole moment ($\beta$) respectively. We study its complete thermodynamics. We compute different thermodynamic products i.e. area product, BH temperature product, specific heat product and Komar energy product respectively. Furthermore, we show that some complicated function of horizon areas that is indeed \emph{mass-independent} and could turn out to be \emph{universal}., Comment: Version accepted in Advances in High Energy Physics

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-Nordstr\"{o}m--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 $J=a\,M$., Comment: Published in PLB

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., Accepted for publication in Modern Physics Letters A (MPLA)

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 Nordstr\"om BH of Einstein gravity and Kehagias-Sfetsos BH of Ho\v{r}ava 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., Comment: 19 pages, Invited Article, Accepted in Advances in High Energy Physics, 2018

We investigate the thermodynamic properties of inner and outer horizons in the background of spinning BTZ(Ba\~{n}ados,Teitelboim and Zanelli) black hole. We compute the \emph{horizon radii product, the entropy product, the surface temperature product, the Komar energy product and the specific heat product} for both the horizons. We observe that the entropy product is \emph{universal}(mass-independent), whereas the surface temperature product, Komar energy product and specific heat product are \emph{not universal} because they all depends on mass parameter. We also show that the \emph{First law} of black hole thermodynamics and \emph {Smarr-Gibbs-Duhem } relations hold for inner horizon as well as outer horizon. The Christodoulou-Ruffini mass formula is derived for both the horizons. We further study the \emph{stability} of such black hole by computing the specific heat for both the horizons. It has been observed that under certain condition the black hole possesses \emph{second order phase transition}., Comment: 9 Pages, Accepted in JETP Letters, 20/08/2015. arXiv admin note: text overlap with arXiv:1506.03173

We examine the possibility of arbitrarily high energy in the Center-of-mass(CM) frame of colliding neutral particles in the vicinity of the horizon of a charged dilation black hole(BH). We show that it is possible to achieve the infinite energy in the background of the dilation black hole without fine-tuning of the angular momentum parameter. It is found that the CM energy $({\cal E}_{cm})$ of collisions of particles near the infinite red-shift surface of the extreme dilation BHs are arbitrarily large while the non-extreme charged dilation BHs have the finite energy. We have also compared the ${\cal E}_{cm}$ at the horizon with the ISCO(Innermost Stable Circular Orbit) and MBCO (Marginally Bound Circular Orbit) for extremal Reissner-Nordstr��m(RN) BH and Schwarzschild BH. We find that for extreme RN BH the inequality becomes ${\cal E}_{cm}\mid_{r_{+}}>{\cal E}_{cm}\mid_{r_{mb}}> {\cal E}_{cm}\mid_{r_{ISCO}}$ i.e. ${\cal E}_{cm}\mid_{r_{+}=M}: {\cal E}_{cm}\mid_{r_{mb}=\left(\frac{3+\sqrt{5}}{2}\right)M} : {\cal E}_{cm}\mid_{r_{ISCO}=4M} =\infty : 3.23 : 2.6 $. While for Schwarzschild BH the ratio of CM energy is ${\cal E}_{cm}\mid_{r_{+}=2M}: {\cal E}_{cm}\mid_{r_{mb}=4M} : {\cal E}_{cm}\mid_{r_{ISCO}=6M} = \sqrt{5} : \sqrt{2} : \frac{\sqrt{13}}{3}$. Also for Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) BHs, the ratio is being ${\cal E}_{cm}\mid_{r_{+}=2M}: {\cal E}_{cm}\mid_{r_{mb}=2M} : {\cal E}_{cm}\mid_{r_{ISCO}=2M} =\infty : \infty : \infty$., Accepted for Publication in Astroparticle Physics, September-2014

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., Comment: version accepted in EPJ Plus

We show that an extremal Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole may act as a particle accelerator with arbitrarily high energy when two uncharged particles falling freely from rest to infinity on the near horizon. We show that the center of mass energy of collision independent of the extreme fine tuning of the angular momentum of the colliding particles. We further show that the center of mass energy of collisions of particles at the ISCO ($r_{ISCO}$) or at the photon orbit ($r_{ph}$) or at the marginally bound circular orbit ($r_{mb}$) i.e. at $r \equiv r_{ISCO}=r_{ph}=r_{mb}=2M$ could be arbitrarily large for the aforementioned spacetimes, which is different from Schwarzschild and Reissner-Nordstr{\o}m spcetimes. For non-extremal GMGHS spacetimes the CM energy is finite and depends upon the asymptotic value of the dilation field ($\phi_{0}$)., Comment: Accepted for publication in Astrophysics and Space science

We present \emph{area (or entropy) functional relation} for multi-horizons five dimensional (5D) Einstein-Maxwell-Gauss-Bonnet-AdS Black Hole. It has been observed by exact and explicit calculation that some complicated function of two or three horizons area is \emph{mass-independent} whereas the entropy product relation is \emph{not} mass-independent. We also study the local thermodynamic stability of this black hole. The phase transition occurs at certain condition. \emph{Smarr mass formula} and \emph{first law} of thermodynamics have been derived. This \emph{mass-independent} relation suggests they could turn out to be an \emph{universal} quantity and further helps us to understanding the nature of black hole entropy (both interior and exterior) at the microscopic level. In the \emph{Appendix}, we have derived the thermodynamic products for 5D Einstein-Maxwell-Gauss-Bonnet black hole with \emph{vanishing} cosmological constant., Version accepted for publication in General Relativity and Gravitation

The geodesic structure of the Teo wormhole is briefly discussed and some observables are derived that promise to be of use in detecting a rotating traversable wormhole indirectly, if it does exist. We also deduce the exact Lense-Thirring (LT) precession frequency of a test gyroscope moving toward a rotating traversable Teo wormhole. The precession frequency diverges on the ergoregion, a behavior intimately related to and governed by the geometry of the ergoregion, analogous to the situation in a Kerr spacetime. Interestingly, it turns out that here the LT precession is inversely proportional to the angular momentum ($a$) of the wormhole along the pole and around it in the strong gravity regime, a behavior contrasting with its direct variation with $a$ in the case of other compact objects. In fact, divergence of LT precession inside the ergoregion can also be avoided if the gyro moves with a non-zero angular velocity in a certain range. As a result, the spin precession frequency of the gyro can be made finite throughout its whole path, even very close to the throat, during its travel to the wormhole. Furthermore, it is evident from our formulation that this spin precession not only arises due to curvature or rotation of the spacetime but also due to the non-zero angular velocity of the spin when it does not move along a geodesic in the strong gravity regime. If in the future, interstellar travel indeed becomes possible through a wormhole or at least in its vicinity, our results would prove useful in determining the behavior of a test gyroscope which is known to serve as a fundamental navigation device., LaTex, 18 pages including 7 figures, extensively revised version, one new section added, accepted for publication in JCAP

{We examine the critical behaviour i. e. $P-V$ criticality of conformal gravity~(CG) in an extended phase space in which the cosmological constant should be interpreted as a thermodynamic pressure and the corresponding conjugate quantity as a thermodynamic volume.} The main potential point of interest in CG is that there exists a {non-trivial} \emph{Rindler parameter ($a$)} in the {spacetime geometry. This geometric parameter has an important role to construct a model for gravity at large distances where the parameter "$a$" actually originates}. We also investigate the effect of the said parameter on the {black hole~(BH) \emph{thermodynamic} equation of state, critical constants, Reverse Isoperimetric Inequality,} {first law of thermodynamics, Hawking-Page phase transition and Gibbs free energy} for this BH. We speculate that due to the presence of the said parameter, there has been a deformation {in the shape} of {the} isotherms in the $P-V$ diagram in comparison with {the} charged-AdS~(anti de-Sitter) BH and {the} chargeless-AdS BH. Interestingly, we find {that} the \emph{critical ratio} for this BH is $\rho_{c} = \frac{P_{c} v_{c}}{T_{c}}= \frac{\sqrt{3}}{2}\left(3\sqrt{2}-2\sqrt{3}\right)$, which is greater than the charged AdS BH and Schwarzschild-AdS BH {i.e.} $\rho_{c}^{CG}:\rho_{c}^{Sch-AdS}:\rho_{c}^{RN-AdS} = 0.67:0.50:0.37$. The symbols are defined in the main work. Moreover, we observe that \emph{{the} critical ratio {has a constant value}} and {it is} independent of the {non-trivial} \emph{Rindler parameter ($a$)}. Finally, we derive {the} \emph{reduced equation of state} in terms of {the} \emph{reduced temperature}, {the} \emph{reduced volume} and {the} \emph{reduced pressure} respectively., Comment: Accepted in MPLA, Modern Physics Letters A, 2018

We investigate the properties of inner and outer horizon thermodynamics of Sen black hole(BH) both in \emph{Einstein frame(EF)} and \emph{string frame(SF)}. We also compute area(or entropy) product, area(or entropy) sum of the said BH in EF as well as SF. In the EF, we observe that the area(or entropy) product is \emph{ universal}, whereas area (or entropy) sum is \emph{not} universal. On the other hand, in the SF, area(or entropy) product and area(or entropy) sum don't have any universal behavior because they all are depends on ADM(Arnowitt-Deser-Misner) mass parameter. We also verify that the \emph{first law} is satisfied at the Cauchy horizon(CH) as well as event horizon(EH). In addition, we also compute other thermodynamic products and sums in the EF as well as in the SF. We further compute the \emph{Smarr mass formula} and \emph{Christodoulou's irreducible mass formula} for Sen BH. Moreover, we compute the area bound and entropy bound for both the horizons. The upper area bound for EH is actually the Penrose like inequality, which is the first geometric inequality in BHs. Furthermore, we compute the central charges of the left and right moving sectors of the dual CFT in Sen/CFT correspondence using thermodynamic relations. These thermodynamic relations on the multi-horizons give us further understanding the microscopic nature of BH entropy(both interior and exterior)., Accepted for publication in EPJC

It has been shown by explicit and exact calculation that whenever we have taken the \emph{effects of stable thermal fluctuations} the entropy product formula should \emph{not be mass-independent} nor \emph{does it quantized}. It has been examined by giving some specific examples for non-rotating and rotating black hole., Comment: 10 pages, Version accepted for publication in GRG

We examine the logarithmic corrections to the black hole (BH) entropy product formula of outer horizon and inner horizon by taking into account the \emph{effects of statistical quantum fluctuations around the thermal equilibrium} and \emph{via conformal field theory} (CFT). We argue that logarithmic corrections to the BH entropy product formula when calculated using CFT and taking into the effects of quantum fluctuations around the thermal equilibrium, the formula should \emph{not be universal} and it should also \emph{not be quantized}. These results have been explicitly checked by giving several examples., Comment: Version accepted in AHEP

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., Comment: Accepted in Universe,2019, Invited article

In this work we investigate the thermodynamic properties of conformal gravity in four dimensions. We compute the \emph{area(or entropy) functional} relation for this black hole. We consider both de-Sitter (dS) and anti de-Sitter (AdS) cases. We derive the \emph{Cosmic-Censorship-Inequality} which is an important relation in general relativity that relates the total mass of a spacetime to the area of all the black hole horizons. Local thermodynamic stability is studied by computing the specific heat. The second order phase transition occurs at a certain condition. Various type of second order phase structure has been given for various values of $a$ and the cosmological constant $��$ in the Appendix. When $a=0$, one obtains the result of Schwarzschild-dS and Schwarzschild-AdS cases. In the limit $aM<, Accepted in MPLA

If the negative cosmological constant is treated as a dynamical pressure and if the volume be its thermodynamically conjugate variable then the gravitational mass can be expressed as the total gravitational enthalpy rather than the energy. Under these circumstances, a new phenomena emerges in the context of extended phase space thermodynamics. We \emph{examine} here these features for recently discovered Van-der-Waal (VDW) black hole (BH) \cite{mann15} which is analogous to the VDW fluid. We show that the thermodynamic volume is \emph{greater} than the naive geometric volume. We also show that the \emph{Smarr-Gibbs-Duhem} relation is satisfied for this BH. Furthermore, by computing the thermal specific heat we find the local thermodynamic stability criterion for this BH. It has been observed that the BH does \emph{not} possess any kind of second order phase transition. This is an interesting feature of VDW BH by its own right. Moreover, we also derive \emph{Cosmic-Censorship-Inequality} for this class of BH. In addition finally, we compute the \emph{logarithmic correction} to the entropy of this BH due to the quantum fluctuations around the thermal equilibrium., Comment: EPL version, 7 pages, 5 figures

In this work we study the equatorial causal geodesics of the Taub-NUT (TN) space-time in comparison with \emph{mass-less} TN space-time. We emphasized both on the null circular geodesics and time-like circular geodesics. From the effective potential diagram of null and time-like geodesics, we differentiate the geodesics structure between TN spacetime and mass-less TN space-time. It has been shown that there is a key role of the NUT parameter to changes the shape of pattern of the potential well in the NUT spacetime in comparison with \emph{mass-less} NUT space-time. We compared the ISCO (innermost stable circular orbit), MBCO (marginally bound circular orbit) and CPO (circular photon orbit) of the said space-time with graphically in comparison with mass-less cases. Moreover, we compute the radius of ISCO, MBCO and CPO for \emph{extreme} TN black hole (BH). Interestingly, we show that these \emph{three radii} coincides with the Killing horizon i.e. the null geodesic generators of the horizon. Finally in Appendix-A, we compute the center-of-mass (CM) energy for TN BH and mass-less TN BH. We show that in both cases the CM energy is finite. For \emph{extreme} NUT BH, we have found that the diverging nature of CM energy. \emph{First}, we have observed that a non-asymptotic flat, spherically symmetric and stationary extreme BH is showing such feature., Comment: Version accepted in International Journal of Geometric Methods in Modern Physics

We compute the area(or entropy) product formula for a regular black hole derived by Ay\'on-Beato and Garc\'ia in 1998\cite{abg}. By explicit and exact calculation, it is shown that the entropy product formula of two physical horizons strictly \emph{depends} upon the ADM mass parameter that means it is \emph{not} an universal(mass-independent) quantity. But a slightly more complicated function of event horizon area and Cauchy horizon area is indeed a \emph{mass-independent} quantity. We also compute other thermodynamic properties of the said black hole. We further study the stability of such black hole by computing the specific heat for both the horizons. It has been observed that under certain condition the black hole possesses second order phase transition. The pictorial diagram of the phase transition is given., Comment: 11 pages, 7 figures, GRG version

We derive various important thermodynamic relations of the inner and outer horizon in the background of Taub-NUT(Newman-Unti-Tamburino) black hole in four dimensional \emph{Lorentzian geometry}. We compare these properties with the properties of Reissner Nordstr{\o}m black hole. We compute \emph{area product, area sum, area minus and area division} of black hole horizons. We show that they all are not universal quantities. Based on these relations, we compute the area bound of all horizons. From area bound, we derive entropy bound and irreducible mass bound for both the horizons. We further study the stability of such black hole by computing the specific heat for both the horizons. It is shown that due to negative specific heat the black hole is thermodynamically unstable. All these calculations might be helpful to understanding the nature of black hole entropy both \emph{interior} and exterior at the microscopic level., Comment: 10 Pages, Accepted in Journal of Experimental and Theoretical Physics(JETP), 05/08/2015

We investigate the thermodynamics and phase transition for Kiselev black hole and dilaton black hole. Specifically we consider Reissner Nordstrom black hole surrounded by radiation and dust, and Schwarzschild black hole surrounded by quintessence, as special cases of Kiselev solution. We have calculated the products relating the surface gravities, surface temperatures, Komar energies, areas, entropies, horizon radii and the irreducible masses at the Cauchy and the event horizons. It is observed that the product of surface gravities, surface temperature product and product of Komar energies at the horizons are not universal quantities for the Kiselev solutions while products of areas and entropies at both the horizons are independent of mass of the above mentioned black holes (except for Schwarzschild black hole surrounded by quintessence). For charged dilaton black hole, all the products vanish. First law of thermodynamics is also verified for Kiselev solutions. Heat capacities are calculated and phase transitions are observed, under certain conditions., Comment: 21 pages, 4 figures. accepted for publication in Advances in High Energy Physics

We speculate on various thermodynamic features of the inner horizon~(${\mathcal H}^{-}$) and outer horizons~(${\mathcal H}^{+}$) of Kehagias-Sfetsos~(KS) black hole~(BH) in the background of Ho\v{r}ava Lifshitz gravity. We compute particularly the \emph{area product, area sum, area minus and area division} of the BH horizons. We find that they all are \emph{not} showing universal behavior whereas the product is a universal quantity~ [Pradhan P., \textit{Phys. Lett. B}, {\bf 747} (2015) {64}]. Based on these relations, we derive the area bound of all horizons. From the area bound we derive the entropy bound and irreducible mass bound for all the horizons~(${\mathcal H}^{\pm}$). We also observe that the \emph{First law} of BH thermodynamics and \emph {Smarr-Gibbs-Duhem } relations do not hold for this BH. The underlying reason behind this failure due to the scale invariance of the coupling constant. Moreover, we compute the \emph{Cosmic-Censorship-Inequality} for this BH which gives the lower bound for the total mass of the spacetime and it is supported by cosmic cencorship conjecture. Finally, we discuss the KS/CFT~(Conformal Field Theory) correspondence via a thermodynamic procedure., Comment: Version accepted in EPL

We have examined the thermodynamic properties for a variety of spherically symmetric charged-AdS black hole (BH) solutions, including the charged AdS BH surrounded by quintessence dark energy and charged AdS BH in $f(R)$ gravity in \emph{extended phase-space}. This framework involves treating the cosmological constant as thermodynamic variable (for example: thermodynamic pressure and thermodynamic volume). Then they should behave as an analog of Van der Waal (VdW) like systems. In the extended phase space we have calculated the \emph{entropy product} and \emph{thermodynamic volume product} of all horizons. The mass (or enthalpy) independent nature of the said product signals they are \emph{universal} quantities. %Various type of phase diagram of the specific heat has been drawn. The divergence of the specific heat indicates that the second order phase transition occurs under certain condition. In the appendix-A, we have studied the thermodynamic volume products for axisymmetric spacetime and it is shown to be \emph{not universal} in nature. Finally, in appendix-B, we have studied the $P-V$ criticality of Cauchy horizon for charged-AdS BH and found to be an universal relation of critical values between two horizons as $P_{c}^{-} = P_{c}^{+}$, $v_{c}^{-}=v_{c}^{+}$, $T_{c}^{-} = -T_{c}^{+}$, $\rho_{c}^{-} = -\rho_{c}^{+}$. The symbols are defined in the main work., Comment: Accepted in IJMPD

We study the existence and stability criteria for circular geodesics of spherically symmetric tidal charged black hole~(BH). We investigate in details the equatorial causal geodesics of the tidal charged BH in comparison with spherically symmetric Reissner-Nordstr\"{o}m BH spacetime. We particularly focused on both the null circular geodesics and time-like circular geodesics. Using the effective potential diagram, we have compared the geodesic structure between two spacetimes. We have derived the ISCO~(innermost stable circular orbit), MBCO~(marginally bound circular orbit) and CPO~(circular photon orbit) for both the space-times. Moreover, we have derived the \emph{quasi-normal modes~(QNM) frequency} in the \emph{eikonal limit} for both the spacetimes via Lyapunov exponent. In the Appendix section, we have shown that a spherically symmetric tidal-charged BH can act as particle accelerators with ultra-high center-of-mass~(CM) energy in the limiting case of \emph{maximal BH tidal charge~($q$)} and it is possible when two neutral particles are colliding near the horizon., Comment: Version accepted in IJGMMP

We examine properties of the inner and outer horizon thermodynamics of Taub-NUT (Newman-Unti-Tamburino) and Kerr-Taub-NUT (KTN) black hole (BH) in four dimensional \emph{Lorentzian geometry}. We compare and contrasted these properties with the properties of Reissner Nordstr{\o}m (RN) BH and Kerr BH. We focus on "area product", "entropy product", "irreducible mass product" of the event horizon and Cauchy horizons. Due to mass-dependence, we speculate that these products have no beautiful quantization feature. Nor does it has any universal property. We further observe that the \emph{First law} of BH thermodynamics and \emph {Smarr-Gibbs-Duhem} relations do not hold for Taub-NUT (TN) and KTN BH in Lorentzian regime. The failure of these aforementioned features are due to presence of the non-trivial NUT charge which makes the space-time to be asymptotically non-flat, in contrast with RN BH and Kerr BH. The another reason of the failure is that Lorentzian TN and Lorentzian KTN geometry contains \emph{Dirac-Misner type singularity}, which is a manifestation of a non-trivial topological twist of the manifold. The black hole \emph{mass formula} and \emph{Christodoulou-Ruffini mass formula} for TN and KTN BHs are also computed. This thermodynamic product formulae gives us further understanding to the nature of BH entropy (inner and outer) at the microscopic level., Comment: Version accepted for publication in EPL

We argue by explicit computations that, although the area product, horizon radii product, entropy product and \emph {irreducible mass product} of the event horizon and Cauchy horizon are universal, the \emph{surface gravity product}, \emph{surface temperature product} and \emph{Komar energy product} of the said horizons do not seem to be universal for Kerr-Newman (KN) black hole space-time. We show the black hole mass formula on the \emph{Cauchy horizon} following the seminal work by Smarr\cite{smarr} for the outer horizon. We also prescribed the \emph{four} laws of black hole mechanics for the \emph{inner horizon}. New definition of the extremal limit of a black hole is discussed., Accepted for publication in EPJC, May-2014

We compute the proper time Lyapunov exponent for charged Myers Perry black hole spacetime and investigate the instability of the equatorial circular geodesics (both timelike and null) via this exponent. We also show that for more than four spacetime dimensions $(N \geq 3)$, there are \emph{no} Innermost Stable Circular Orbits (ISCOs) in charged Myers Perry black hole spacetime. We further show that among all possible circular orbits, timelike circular orbits have \emph{longer} orbital periods than null circular orbits (photon spheres) as measured by asymptotic observers. Thus, timelike circular orbits provide the \emph{slowest way} to orbit around the charged Myers Perry black hole., Comment: Accepted for publication in EPJC. arXiv admin note: substantial text overlap with arXiv:1212.5758, arXiv:1205.5656

We compute the logarithmic corrections to black hole (BH) entropy product of ${\cal H}^{\pm}$ \footnote{ ${\cal H}^{+}$ and ${\cal H}^{-}$ denote outer (event) horizon and inner (Cauchy) horizons} by using \emph{Cardy prescription}. We particularly apply this formula for \emph{BTZ BH}. We speculate that the logarithmic corrections to entropy product of ${\cal H}^{\pm}$ when computed \emph{via Cardy formula} the product should be neither \emph{mass-independent (universal)} nor be \emph{quantized}., Comment: EPL Style, 4 pages

The causal geodesics in the equatorial plane of a static extremal charged black holes in heterotic string theory are examined with regard to their geodesic stability, and compared with similar geodesics in the non-extremal situation. Extremization of the effective potential for time-like and null circular geodesics implies that in the extremal limit, the radius of ISCO(Inner-most Stable Circular Orbit) $(r_{ISCO})$, circular photon orbit (CPO) $(r_{ph})$ and marginally bound circular orbit (MBCO) $(r_{mb})$ are coincident with the event horizon $(r_{hor})$ i.e. $r_{ISCO}=r_{ph}=r_{mb}=r_{hor}=2M$. Since the proper radial distance on a constant time slice both in Schwarzschild and Painlev\'{e}-Gullstrand coordinates become zero, thus these three orbits indeed coincide with the \emph{null geodesic generators of the event horizon}. This strange behavior is quite different from the static, spherically symmetric extremal Reissner Nordstr{\o}m black hole., Comment: Accepted for Publication in IJMPD, 26/06/2015

The fact that one must evaluate the near-extremal and near-horizon limits of Kerr space-time in a specific order, is shown to a lead to discontinuity in the extremal limit, such that this limiting space-time differs nontrivially from the precisely extremal space-time. This is established by first showing a discontinuity in the extremal limit of the maximal analytic extension of the Kerr geometry, given by Carter. Next, we examine the ISCO of the exactly extremal Kerr geometry and show that on the event horizon of the extremal Kerr black hole, it coincides with the principal null geodesic generator of the horizon, having vanishing energy and angular momentum. We find that there is no such ISCO in the near-extremal geometry, thus garnering additional support for our primary contention. We relate this disparity between the two geometries to the lack of a trapping horizon in the extremal situation., 17 Pages, Latex 2e, no figures, Published version in EPJC

Circular null geodesic orbits, in extremal Reissner-Nordstrom spacetimes, are examined with regard to their stability, and compared with similar orbits in the near-extremal situation. Extremization of the effective potential for null circular orbits shows the existence of a stable circular geodesic in the extremal spacetime, precisely {\it on} the event horizon, which coincides with its null geodesic generator. Such an orbit also emerges as a global minimum of the effective potential for circular {\it timelike} orbits. This type of geodesic is of course absent in the corresponding near-extremal spacetime, as we show here, testifying to differences between the extremal limit of a generic RN spacetime and the exactly extremal geometry., 13 Pages Latex 2e, no figures; title of paper, abstract and contents modified substantively following critique of anonymous referee

We derive area product, entropy product, area sum and entropy sum of the event horizon and Cauchy horizons for Kerr-Newman-Taub-NUT(Newman-Unti-Tamburino) black hole in four dimensional \emph{Lorentzian geometry}. We observe that these thermodynamic products are \emph{not} universal(mass-independence) for this black hole(BH), whereas for Kerr-Newman(KN) BH such products are universal (mass-independence). We also examine the entropy sum and area sum. It is shown that they all are depends on mass, charge and NUT parameter of the back ground space-time. Thus we can conclude that the universal(mass-independence) behaviour of area product and entropy product, area sum and entropy sum for Kerr-Newman-Taub-NUT(KNTN) BH fails and which is also quite different from KN BH. We further show that the KNTN BH do not possess \emph{first law of BH thermodynamics } and \emph {Smarr-Gibbs-Duhem } relations, and that such relations are unlikely in the KN case. The failure of these aforementioned features are due to presence of the non-trivial NUT charge which makes the space-time to be asymptotically non-flat, in contrast with KN BH. The another reason of the failure is that Lorentzian KNTN geometry contains \emph{Dirac-Misner type singularity}, which is a manifestation of a non-trivial topological twist of the manifold. The BH \emph{mass formula} and \emph{Christodoulou-Ruffini mass formula} for KNTN black holes are also derived. Finally, we compute the area bound which is just Penrose like inequality for event horizon. From area bound we derive entropy bound. These thermodynamic products on the multi horizon playing a crucial role in BH thermodynamics to understand the microscopic nature of BH entropy., Accepted for publication in Modern Physics Letters A

In this paper we investigate the equatorial causal (time-like and null) circular geodesics of the Kerr-Newman-Taub-NUT(Newman-Unti-Tamburino) black hole in four dimensional Lorentzian geometry. The special characteristics of this black hole is that it is of \emph{Petrov-Pirani type D} and the photon trajectories are \emph{doubly degenerate principal null congruence}. We derive the conditions for existence of innermost stable circular orbit, marginally bound circular orbit and circular photon orbit in the background of Kerr-Newman-Taub-NUT(KNTN) space-time. The effective potential for both time-like case and null cases have been studied. It is shown that the \emph{presence of the NUT parameter deforms the shape of the effective potential in contrast with the zero NUT parameter}. We further investigate the energy extraction by the Penrose process for this space-time. It is shown that the efficiency of this black hole depends on both the charge and NUT parameter. It is observed that the energy gain is maximum when NUT parameter goes to zero value and for the maximum spin value. When the value of NUT parameter is increasing the energy-gain is decreasing., Accepted in Classical Quantum Gravity, 18/06/2015, Some 3D figures could not be submitted due to arxiv submission problem this could be found in Journal version