Radius Of Innermost Stable Circular Orbit Research Articles

Regular Bardeen Black Holes in Anti-de Sitter Spacetime versus Kerr Black Holes through Particle Dynamics

In this work, test particle dynamics around a static regular Bardeen black hole (BH) in Anti-de Sitter spacetime has been studied. It has been shown for neutral test particles that parameters of a regular Bardeen black hole in Anti-de Sitter spacetime can mimic the rotation parameter of the Kerr metric up to the value a≈0.9 providing the same innermost stable circular orbit (ISCO) radius. We have also explored the dynamics of magnetized particles with a magnetic dipole moment around a magnetically charged regular Bardeen black hole in Anti-de Sitter spacetime. As a realistic astrophysical scenario of the study, we have treated neutron stars orbiting a supermassive black hole (SMBH), in particular, the magnetar PSR J1745-2900 orbiting Sgr A* with the parameter β=10.2, as magnetized test particles. The magnetized particles dynamics shows that the parameter β, negative values of cosmological constant and magnetic charge parameter of the central BH cause a decrease in the ISCO radius. We have compared the effects of the magnetic charge of the Bardeen BH with the spin of rotating Kerr BH and shown that magnetic charge parameter can mimic the spin in the range a/M≃(0,0.7896) when Λ=0 at the range of its values g/M≃(0,0.648).

Particle Motion and Plasma Effects on Gravitational Weak Lensing in Lorentzian Wormhole Spacetime

Here we study particle motion in the specific Lorentzian wormhole spacetime characterized, in addition to the total mass M, with the dimensionless parameter λ. In particular we calculate the radius of the innermost stable circular orbit (ISCO) for test particles and the photonsphere for massless particles. We show that the effect of the dimensionless wormhole parameter decreases the ISCO radius and the radius of the photon orbit. Then, we study plasma effects on gravitational weak lensing in wormhole spacetime and obtain the deflection angle of the light. We show that the effect of λ decreases the deflection angle. We study the effects of uniform and non-uniform plasma on the light deflection angle separately, and show that the uniform plasma causes the deflection angle to be smaller in contrast to the non-uniform plasma.

Charged particle motion around a magnetized Reissner-Nordström black hole

We investigate the dynamics of neutral and charged test particles around axially symmetric magnetized black hole spacetime. We consider its electromagnetic field in the black hole vicinity and study its impact on the dynamics of test particles. We determine the radius of the innermost stable circular orbit (ISCO) for neutral and charged test particles and show that the combined effect of black hole electric charge and magnetic field strongly affects the ISCO radius, thus shrinking its values. We also show that the ISCO radius of positively (negatively) charged particle initially gets increased (decreased) and then gets radically altered with an increase in the value of both black hole electric charge and test particle charge. It turns out that the repulsive (attractive) Coulomb force dominates over the Lorentz force arising from the black hole magnetic field. Typically, black hole rotation causes axially symmetric spacetime case. Similarly, it turns out that a magnetized black hole solution also causes axially symmetric spacetime as a consequence of the presence of magnetic field. We study a degeneracy for the value of the ISCO between the Kerr and the magnetized Reissner-Nordstr\"om black hole geometries and show that the combined effects of black hole charge and magnetic field can be mimicked by Kerr spacetime with the spin parameter up to $a/M\ensuremath{\approx}0.8$. Finally, we consider the center of mass energy of colliding particles and show that an increase in the values of black hole magnetic field and electric charge leads to high center of mass energy extracted by collision of two particles.

Geodesic motion around hairy black holes

In a recent paper Ovalle et al. (2021) it has been obtained new static black hole solutions with primary hairs by the Gravitational Decoupling. In this work we either study the geodesic motion of massive and massless particles around those solutions and restrict the values of the primary hairs by observational data. In particular, we obtain the effective potential, the innermost stable circular orbits, the marginally bounded orbit, and the periastron advance for time-like geodesics. In order to restrict the values taken by the primary hairs we explore their relationship with the rotation parameter of the Kerr black hole giving the same innermost stable circular orbit radius and give the numerical values for the supermassive black holes at Ark 564 and NGC 1365. The photon sphere and the impact parameter associated to null geodesics are also discussed.

Regular nonminimal magnetic black hole as a source of quasiperiodic oscillations

We study the properties of the spacetime around a regular nonminimal magnetic black hole (BH) together with the dynamics of the neutral, magnetized, and magnetically charged particles in its vicinity. The dependence of the values of curvature invariants and the outer event horizon corresponding to the extreme charge of the BH from the coupling parameter are studied in detail. Our study of the test particle motion shows that the innermost stable circular orbit (ISCO) radius decreases with the increase of the magnetic charge of the nonminimal regular magnetic BH. We show that the increase of the ISCO radius with the increase of the coupling parameter is due to the decrease of the extreme value of the BH charge. In particular, when the coupling parameter $q\ensuremath{\rightarrow}\ensuremath{\infty}$ the ISCO radius tends to the value ${r}_{\mathrm{ISCO}}=4.5M$. The effect of the magnetic charge parameter on ISCO radius can mimic the effect due to the spin parameter of the rotating Kerr BH, and the mimicking values of the spin parameter decreases with the increase of the coupling parameter. We also explore the possible mimicking of the parameters through the observations of twin-peak quasiperiodic oscillations (QPOs). Our analysis shows that the Reissner-Nordstr\"om (RN) charge can mimic the spin parameter of the Kerr BH up to $a/M\ensuremath{\simeq}0.51$ provided that there is the same relation between the upper and lower frequencies of the twin-peak QPOs. On the other hand, the nonminimal regular magnetic BH coupling parameter may mimic the spin parameter up to $a/M\ensuremath{\simeq}0.215$ in the relativistic precession model.

Photon emission from inside the innermost stable circular orbit

We consider a situation where a light source orbiting the innermost stable circular orbit (ISCO) of the Kerr black hole is gently falling from the marginally stable orbit due to an infinitesimal perturbation. Assuming that the light source emits photons isotropically, we show that the last radius at which more than 50\% of emitted photons can escape to infinity is approximately halfway between the ISCO radius and the event horizon radius. To evaluate them, we determine emitter orbits from the vicinity of the ISCO, which are uniquely specified for each black hole spin, and identify the conditions for a photon to escape from any point on the equatorial plane of the Kerr spacetime to infinity by specifying regions in the two-dimensional photon impact parameter space completely. We further show that the proper motion of the emitter affects the photon escape probability and blueshifts the energy of emitted photons.

Dynamics of charged particles and magnetic dipoles around magnetized quasi-Schwarzschild black holes

In the present paper, we have investigated the motion of charged particles together with magnetic dipoles to determine how well the spacetime deviation parameter epsilon and external uniform magnetic field can mimic the spin of a rotating Kerr black hole. Investigation of charged particle motion has shown that the deviation parameter epsilon in the absence of an external magnetic fields can mimic the rotation parameter of the Kerr spacetime up to a/M approx 0.5. The combination of an external magnetic field and deviation parameter can do even a better job mimicking the rotation parameter up to a/Msimeq 0.93, which corresponds to the rapidly rotating case. Study of the dynamics of the magnetic dipoles around quasi-Schwarzschild black holes in the external magnetic field has shown that there are degeneracy values of the ISCO radius of test particles at epsilon _{cr}>epsilon ge 0.35 which may lead to two different values of the innermost stable circular orbit (ISCO) radius. When the deviation parameter is in the range of epsilon in (-1, 1), it can mimic the spin of a rotating Kerr black hole in the range a/M in (0.0537, 0.3952) for magnetic dipoles with values of the magnetic coupling parameter beta in [-0.25, 0.25] in corotating orbits.

Innermost stable circular orbit of charged particles in Reissner-Nordström, Kerr-Newman, and Kerr-Sen spacetimes

In this article we study the innermost stable circular orbit (ISCO) of electrically charged particles in the electrically charged Reissner-Nordstr\"om spacetime, the Kerr-Newman spacetime and the Kerr-Sen spacetime. We find that the radius of the ISCO increases with an increasing particle-black hole charge product $|qQ|$ in the case of attractive Coulomb interaction $qQ<0$. For repulsive Coulomb interaction, the ISCO radius first decreases to a minimum and then increases again, until it diverges as the charge product approaches one. If the charge $Q$ of the black hole is very small, the minimum of the ISCO radius lies at $qQ=0$. Repulsive and attractive Coulomb interactions will always increase the ISCO radius in this limit. Stable bound orbits of charged particles cease to exist in the Reissner-Nordstr\"om and Kerr-Newman spacetime for $qQ\geq 1$. In the Kerr-Sen spacetime the limiting case depends on the charge of the black hole and if dilaton coupling is applied to the test particle. We find $qQ\geq 1+Q^2$ without dilaton-coupling and $qQ\geq 1+\frac{3}{2}Q^2$ with dilaton coupling $\alpha=1$.

Test particles dynamics around deformed Reissner-Nordström black hole

In this paper we study the dynamics of neutral, electrically charged and magnetized particles around deformed electrically and magnetically charged Reissner-Nordstr\"om black holes. For the neutral test particles motion, it is shown that the radius of the innermost stable circular orbits (ISCOs) decrease with the increase in both the black hole charge and positive spacetime deformation. It reaches up to $3.85M$ at the value of deformation parameter $\ensuremath{\epsilon}=20.45$ in Schwarzschild spacetime case ($Q=0$). In the extreme charged Reissner--Nordstrom black hole case, the ISCO decreases up to $2.26M$ at the value of deformation parameter $\ensuremath{\epsilon}=6.17$. Moreover, the negative deformation results in an increase in the ISCO radius. Comparing effects of positive deformation and the Reissner-Nordstr\"om black hole charge with spin of rotating Kerr black holes, it is shown that the extremely charged Reissner-Nordstr\"om black hole can mimic rotating Kerr black hole up to the spin parameter $a/M=0.48$, while for the positive deformation with $\ensuremath{\epsilon}=6.17$ the mimic value increases up to $a/M=0.88$ implying that the supermassive black hole M87 cannot be considered as Reissner-Nordstr\"om black hole. Using this comparison, we estimate charge of the supermassive black hole Sagittarius A* as $Q/M\ensuremath{\simeq}0.8287$ without deformation which can mimic the spin of the black hole. When the deformation $\ensuremath{\epsilon}=1$, the mimic charge increases up to $Q/M\ensuremath{\simeq}0.8926$. Our study of the energy extraction from the accretion disk shows that the maximum energy efficiency increases up to 20.02%, which is almost the same for extreme Kerr black hole case (20.6%). We have also considered the behaviour of ISCO of electrically charged particles showing that the attractive (repulsive) electrostatic interactions cause rapid increase (slightly decrease) of the ISCO radius. Finally, we have explored the dynamics of magnetized particles around deformed magnetically charged Reissner-Nordstr\"om black hole. By treating the magnetar PSR J1745-2900, orbiting the supermassive black hole Sagittarius A* as a magnetized particle, showing that the magnetic charge of pure Reissner-Nordstr\"om black hole can mimic spin of a rotating Kerr black hole up to $a/M\ensuremath{\simeq}0.82$ for the value to be in the range ${Q}_{m}/M\ensuremath{\in}(0,0.692)$. Moreover, we find that the positive values of the deformation parameter lead to shift the ISCO for the magnetized particles toward the central object.

Circular motion and the innermost stable circular orbit for spinning particles around a charged Hayward black hole background

The circular orbits of a spinning test particle moving around a charged Hayward black hole is investigated by using the Mathisson–Papapetrou–Dixon equations together with the Tulczyjew spin-supplementary condition. By writing the equations of motion, the effective potential for the description of the test particle is obtained to study the properties of the Innermost Stable Circular Orbit (ISCO). The results show that the ISCO radii for spinning particles moving in the charged Hayward background differ from those obtained in the corresponding Schwarzschild or Reissner–Nordstrom spacetimes, depending on the values of the electric charge and the length-scale parameter of the metric. When the spin of the particle and its orbital angular momentum are aligned, an increase in the spin produces a decrease in the ISCO radius, while in the case in which the spin of the particle and its orbital angular momentum are anti-aligned, an increase in the spin results in an increase of the radius of the ISCO.

Dynamics of test particles around a Bardeen black hole surrounded by perfect fluid dark matter

We study the dynamics of (i) neutral test particles, (ii) magnetically charged test particles, and (iii) test magnetic dipole around a regular Bardeen black hole surrounded by perfect fluid dark matter (PFDM). It has been shown how the magnetic charge of the black hole and the parameter of the surrounding PFDM can influence the innermost stable circular orbit (ISCO) radius of a test particle. We have found that the ISCO radius is strongly affected as a consequence of the combined effect of the magnetic charge parameter and the perfect fluid dark matter. The black hole magnetic charge parameter $g$ and the combined effect of perfect fluid dark matter can mimic the black hole rotation parameter up to $a/M\approx 0.9$. It has been observed that the ISCO for magnetic dipole disappears at the values exceeding the calculated upper value for the magnetic interaction parameter $\beta$. The upper limit decreases with the increase of both the dark matter and magnetic charge parameters. Finally, as an astrophysical application, we have analyzed degeneracy effects of spin of Kerr black holes and magnetic charge of regular Bardeen black holes for the different values of the dark matter parameter providing exactly the same value for ISCO radius of a magnetic dipole with the same value of the parameter $\beta=10.2$ of the magnetar called PSR J1745-2900 orbiting around supermassive black hole Sagittarius A*.

Charged particle motion around non-singular black holes in conformal gravity in the presence of external magnetic field

We consider electromagnetic fields and charged particle dynamics around non-singular black holes in conformal gravity immersed in an external, asymptotically uniform magnetic field. First, we obtain analytic solutions of the electromagnetic field equation around rotating non-singular black holes in conformal gravity. We show that the radial components of the electric and magnetic fields increase with the increase of the parameters L and N of the black hole solution. Second, we study the dynamics of charged particles. We show that the increase of the values of the parameters L and N and of magnetic field causes a decrease in the radius of the innermost stable circular orbits (ISCO) and the magnetic coupling parameter can mimic the effect of conformal gravity giving the same ISCO radius up to omega _{mathrm{B}}le 0.07 when N=3.

Self-gravitating razor-thin discs around black holes via multi-hole seeds

We construct self-gravitating razor-thin discs of counterrotating matter around Schwarzschild black holes (BHs) by applying the ‘displace, cut, and reflect’ method to known seed solutions representing multi-holes. All but one of the sources of the seed solution generate the surrounding annular disc, whereas the remaining BH is mapped onto a Schwarzschild BH which lies at the disc centre after the transformation. The discs are infinite in extent, have annular character, and are linearly stable up to the innermost stable circular orbit (ISCO) of the system. The spacetime is asymptotically flat, having finite Arnowitt–Deser–Misner mass. Moreover, all energy conditions for the disc are satisfied for radii larger than the ISCO radius; the method, however, produces counterrotating streams with superluminal velocities in the vicinity of the central BH. We also comment on charged discs around extremal Reissner–Nordström BHs constructed from a Majumdar–Papapetrou N-BH seed solution. These simple examples can be extended to more general ‘BH + disc’ solutions, obtained by the same method from seeds of the type ‘BH + arbitrary axisymmetric source’. A natural follow-up of this work would be to construct discs around Reissner–Nordström BHs with arbitrary charge-to-mass ratio and around Kerr BHs.

Test particle motion around a black hole in Einstein-Maxwell-scalar theory

In this paper, we explore the test particle motion around a black hole in Einstein-Maxwell-scalar (EMS) theory using three different black hole solutions within this theory. We have first analyzed the spacetime curvature structure of these solutions and shown the existence of two singularities and the first one is at the center $r=0$. In black hole spacetime, there are two regions divided by the critical value of the cosmological parameter $\lambda_0$. The photon sphere around the black hole in EMS theory has also been studied and found that it does not depend on cosmological parameter $\lambda$. We have analyzed the innermost stable circular orbits (ISCO) around the black hole and shown that for all solutions ISCO radius for neutral particle decreases with the increase of black hole charge. We have also studied the charged particle motion around the black hole where charged particle motion is considered in the presence of the gravitational field and the Coulomb potential. It is shown that ISCO radius for charged particles increases depending on the selected value of the coupling parameter which is in contradiction with observations of the inner edge of the accretion disks of the astrophysical black holes and can be used as a powerful tool to rule out the EMS theory from consideration for the gravitational field theory. It is also studied the fundamental frequencies governed by test particle orbiting around the black hole in EMS theory. Finally, as a test of black hole solution in EMS theory ISCO radii is compared with that in Kerr black hole and found that the spin parameter of Kerr can be mimic up to $a/M\simeq 0.936$.

Charged and magnetized particles motion in the field of generic singular black holes governed by general relativity coupled to nonlinear electrodynamics

We study spherically symmetric magnetically charged generic singular black hole solutions of general relativity coupled to nonlinear electrodynamics. For characteristic values of the generic spacetime parameters and the parameter characterizing the ratio of the gravitational and electromagnetic forces acting on an electrically charged particle we study the circular orbits and related epicyclic motion and its frequencies. We demonstrate that the equatorial circular orbits are forbidden in such situations, but off-equatorial circular orbits are possible. We give dependence of the stable circular orbit on the spacetime parameters and intensity of the electromagnetic interaction of the charged particles with magnetically charged black holes. We study the possible resonance phenomena of the epicyclic frequencies and the orbital frequency of the electrically charged particles in order to fit the data of the twin high-frequency quasiperiodic oscillations of x rays observed in microquasars. Moreover, the dynamics of magnetized particles around the magnetically charged generic black hole have also been explored and it is shown that as increasing magnetic charge and magnetic moment parameters, the innermost stable circular orbit (ISCO) radius decreases and disappears at some value of the magnetic moment parameter, inversely proportional to the magnetic charge of black hole. As an astrophysical application we treated the magnetar PSR J1745--2900 orbiting around Sagittarius (Sgr) A* as a magnetized particle and showed that the magnetic charge of black hole can mimic black hole spin up to $a/M=0.865694$ at $\ensuremath{\nu}=2$, and the spin parameter can mimic the magnetic charge parameter up to $q/M=0.578575$ at $\ensuremath{\nu}=1$, providing exactly the same value of the ISCO radius. Finally, we predict that no magnetar with the surface magnetic field of the order of ${10}^{14}--1{0}^{15}\text{ }\text{ }\mathrm{G}$ can follow stable orbits, but it is possible to observe ordinary neutron stars as recycled radio pulsars in the close environment of Sgr A*.

Test particle orbits around regular black holes in general relativity combined with nonlinear electrodynamics

We provide detailed analysis of the curvature structure of the spacetime around regular black holes (RBHs) governed by general relativity (GR) combined with nonlinear electrodynamics (NED), characterized by electric charge $Q$, and degree of nonlinearity $n$. We consider special class of the backgrounds introduced by Toshmatov et al. [Phys. Rev. D 98, 028501 (2018)] that have proper Maxwell weak-field limit of the NED sector. We also study the motion of uncharged and charged particles in the RBH background. We determine shadows of such black holes (BHs) using the effective geometry governing motion of photons in the background of NED RBHs. The analysis of circular motion enables to determine innermost stable circular orbits (ISCO) and marginally bound orbits (MBO). We show that the radius of ISCO and MBO for neutral particles decreases as the parameters of the RBH $Q$ and $n$ increase for the fixed value of the BH mass. We demonstrate that for the electrically charged particles properties of the circular orbits strongly depend on their Coulomb interaction with the RBH charge. The dependence of the ISCO radius on the particle specific charge $q$ and the RBH parameters is rather complex, but for the attractive Coulomb interaction there is a general feature (independent of $n$) giving a limiting value of the intensity of this interaction $q{Q}_{\mathrm{max}}$ behind which no stable circular orbits are allowed around the RBH. Comparison of the RBH with Reissner-Nordstr\"om black holes (RNBH) shows that for the same electric charges of these backgrounds the location of the relevant orbits is at substantially smaller radii for the RBH, demonstrating thus a strong influence of the NED effects. The analysis of ISCO radius of test particles has shown that the charge of RBH $Q$ can mimic the rotation parameter of Kerr BH up to the value $a=0.8M$. We have also shown that the RN BH charge can mimic the rotation parameter up to $a=0.5M$ and the RBH charge up to $Q=0.2M$. Applications from observational data to the parameters of the supermassive black holes (SMBHs) at the galactic center Messier 87 (M87) and Milky Way so called Sagittarius ${\mathrm{A}}^{*}$ (${\mathrm{Sgr}\text{ }A}^{*}$) give the mimic value for the RBH charge parameter for the rotation parameter of M87 to be ${10}^{3}Q\ensuremath{\approx}277.5{6}_{\ensuremath{-}1.26}^{+3.82}$, while for ${\mathrm{Sgr}\text{ }A}^{*}$ there is ${10}^{3}Q\ensuremath{\approx}187.3{3}_{\ensuremath{-}26.4}^{+27.6}$. Moreover, it is shown that the RN BH charge cannot mimic the rotation parameter of the SMBH M87, however, it can mimic the spin parameter of ${\mathrm{Sgr}\text{ }A}^{*}$ at ${10}^{3}{Q}_{\mathrm{RN}}/M=828.7{1}_{\ensuremath{-}60.94}^{+70.58}$.

Observability of the innermost stable circular orbit in a near-extremal Kerr black hole

We consider the escape probability of a photon emitted from the innermost stable circular orbit (ISCO) of a rapidly rotating black hole. As an isotropically emitting light source on a circular orbit reduces its orbital radius, the escape probability of a photon emitted from it decreases monotonically. The escape probability evaluated at the ISCO also decreases monotonically as the black hole spin increases. When the dimensionless Kerr parameter $a$ is at the Thorne limit $a=0.998$, the escape probability from the ISCO is $58.8\%$. In the extremal case $a=1$, even if the orbital radius of the light source is arbitrarily close to the ISCO radius, which coincides with the horizon radius, the escape probability remains at $54.6\%$. We also show that such photons that have escaped from the vicinity of the horizon reach infinity with sufficient energy to be potentially observed because Doppler blueshift due to relativistic beaming can overcome the gravitational redshift. Our findings indicate that signs of the near-horizon physics of a rapidly rotating black hole will be detectable on the edge of its shadow.

Spinning test particles in the γ spacetime

We consider the motion of spinning particles in the field of a well known vacuum static axially-symmetric spacetime, known as $\gamma$ metric, that can be interpreted as a generalization of the Schwarzschild manifold to include prolate or oblate deformations. We derive the equations of motion for spinning test particles by using the Mathisson-Papapetrou-Dixon equations together with the Tulczyjew spin-supplementary condition, and restricting the motion to the equatorial plane. We determine the limit imposed by super-luminal velocity for the spin of the particle located at the innermost stable circular orbits (ISCO). We show that the particles on ISCO of the prolate $\gamma$ spacetime are allowed to have nigher spin than the corresponding ones in the the oblate case. We determine the value of the ISCO radius depending on the signature of the spin-angular momentum, ${\rm s-L}$ relation, and show that the value of the ISCO with respect to the non spinning case is bigger for ${\rm sL}<0$ and smaller for ${\rm sL}>0$. The results may be relevant for determining the properties of accretion disks and constraining the allowed values of quadrupole moments of astrophysical black hole candidates.

Innermost stable circular orbits in the Majumdar-Papapetrou dihole spacetime

We investigate the positions of stable circular massive particle orbits in the Majumdar--Papapetrou dihole spacetime with equal mass. In terms of qualitative differences of their sequences, we classify the dihole separation into five ranges and find four critical values as the boundaries. When the separation is relatively large, the sequence on the symmetric plane bifurcates, and furthermore, they extend to each innermost stable circular orbit in the vicinity of each black hole. In a certain separation range, the sequence on the symmetric plane separates into two parts. On the basis of this phenomenon, we discuss the formation of double accretion disks with a common center. Finally, we clarify the dependence of the radii of marginally stable circular orbits and innermost stable circular orbits on the separation parameter. We find a discontinuous transition of the innermost stable circular orbit radius. We also find the separation range at which the radius of the innermost stable circular orbit can be smaller than that of the stable circular photon orbit.

Charged particle dynamics in the vicinity of black hole from vector-tensor theory of gravity immersed in an external magnetic field

Abstract The dynamics of charged particle in the vicinity of event horizon of a rotating charged black hole (BH) with gravitomagnetic charge from a class of vector-tensor theories of modified gravity immersed in an axially symmetric magnetic field (Bfield) has been studied. The presence of Bfield reduces the radii of innermost stable circular orbit (ISCO) of charged particle and the opposite phenomenon occurs due the parameter β which describes the deviation of modified gravity considered here from the usual Einstein-Maxwell gravity also acts to increase the radius of ISCO. Angular momentum also plays the role regarding the motion of particle which implies that larger the angular momentum easier for a particle to escape. We study the stability of orbits using Lyapunov characteristic exponent which implies more stability in the presence of Bfield. Generally, in the presence of Bfield it is easier for a particle to escape than its absence. We find twist in the trajectories of charged particles close to event horizon due to spin of BH in the presence of Bfield and there is no twist in the absence of it. Finally, we comment on possible utilization of our findings for the relativistic jets and magnetohydrodynamical out flows.