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the study of accretion dynamics of charged particles in the vicinity of a compact gravitational source is of observational as well as theoretical interest ; particularly for the information it reveals about the nature and influence of the background spacetime on the various physical processes occurring in the star s vicinity . whereas a compact star is formed largely of a degenerate plasma@xmath0 , matter surrounding the star exists in a highly ionized , though  a less dense , state@xmath1 . in this region
the charged particles are more free to move while not accreted to relativistic velocities , and the dynamics of these charged particles is governed not only by the gravitational but also by the electromagnetic field of the star .
however the electromagnetic field is itself  effected by the geometry of the background spacetime .
therefore it is of importance and of astrophysical relevance that the two - fold effects ; namely , the coupling of the gravitational and the electromagnetic field of the star , as well as the direct effects of the background spacetime on the motion of a charged particle , must be addressed together@xmath2 .    here
we present an investigation of these effects on the dynamics of an accreted  charged particle lying in the star s plasma atmosphere . to pose the problem quantitatively
we consider a slowly rotating isolated compact star as the source of gravitational field which defines the geometry of the background spacetime . an axially symmetric , highly conducting plasma with a stationary , axially symmetric electromagnetic field is assumed to constitute the star s atmosphere .
we regard the electromagnetic field as perturbed by the background spacetime , however the back - reaction of the electromagnetic field on the spacetime curvature is neglected and the standard kerr metric ( in its linearized form ) is used for the fixed background spacetime . for the plasma surrounding the star the magnetohydrodynamic ( mhd ) approximation
is assumed , as such the plasma is considered as an ideal highly conducting mhd fluid@xmath3 .
the formulation of the electromagnetic field equations is based on a generalized definition of the electromagnetic field tensor for an ideal mhd fluid in a curved spacetime@xmath4 .
the direct effects of the spacetime dragging on the particle dynamics are discussed in the gravitoelectromagnetic ( gem ) approximation to the geodesic equation@xmath5 .
the gem approximation , which assumes the charged particle motion to be slow , is generally valid in the case of stable compact stars ( i.e. non - collapsing objects ) .
also for the case of a magnetized compact star with a dipole field the gem approximation is applicable except at or very close to the poles where particle velocities can be relativistic due to intense magnetic field .
generally the accretion occurs well below the relativistic limit and so the gem approximation can be used . throughout
we take the velocity four vector as defined by a comoving observer zamo ( zero angular momentum observer ) .
the zamo is a useful class of observers , relevant to astrophysics , circling the gravitational source with a given angular velocity at a fixed radial distance and polar angle@xmath6 .
the paper is organized as follows . in the first part of our analysis ,
comprising section ii , we obtain exact solutions of the general relativistic maxwell equations using the generalized definition of the electromagnetic field tensor for an ideal , highly conducting mhd fluid in linearized kerr spacetime .
the solution obtained shows that in the slow rotation approximation the effects of rotation on the electromagnetic field are absent .
then , in section iii , assuming the star to be a sphere of homogeneous mass density , we study numerically the dynamics of the charged particle using the gem approximation . in this approximation
the direct effects of rotation ( i.e. via frame - dragging ) on particle trajectories are found to oppose that of the electromagnetic field , hence the particle motion is a result of balancing and counter - balancing of these effects .
lastly , in section iv , we give a discussion and a summary of the main conclusions of the paper .
the exterior spacetime to a slowly rotating compact gravitational source of mass @xmath7 is described by the linearized kerr metric . in the boyer - lindquist
coordinates the metric can be written as the general line element :    @xmath8    where    @xmath9    and @xmath10 is the angular velocity of a free falling frame brought into rotation by the frame dragging of the spacetime . here and in what follows the greek indices run as @xmath11 and @xmath12 respectively .
also throughout we assume the gravitational units in which @xmath13 .
the general relativistic form of the maxwell equations is @xmath14@xmath15    @xmath16    where @xmath17 represents the determinant of the metric tensor @xmath18 given by the einstein field equations . here
@xmath19 is the generalized electromagnetic field tensor for an ideal mhd fluid given by a unique tensorial expression :    @xmath20    and @xmath21 is current four vector . in general the current four vector
is the sum of two terms corresponding to a convection and to a conduction current :    @xmath22    where @xmath23 is the proper charge density , @xmath24 is the conductivity of the fluid , and @xmath25 is the unit velocity @xmath26-vector .
the volume element 4-form @xmath27 and its dual @xmath28 are defined by    @xmath29    where @xmath30 is the levi -civita symbol , which is @xmath31 or @xmath32 for an even , odd , or non - permutation of @xmath33 respectively .
also the four vectors @xmath34 and @xmath35 , denote the electric and magnetic field components in the four dimensional spacetime . for a zamo @xmath36 and
@xmath37 vanish , and using @xmath38 , the components of the four velocity vector are :    @xmath39    here we employ the often made assumption of a perfectly conducting plasma surrounding the star ( cf e.g. [ 5 ] , [ 14 ] for a discussion of this assumption with respect to the generalized ohm s law ) .
since for plasma the condition of neutrality also holds thus @xmath21 must vanish identically .
the maxwell equations ( 4 ) give :    @xmath40    @xmath41    @xmath42    @xmath43    whereas from expression ( 5 ) we obtain :    @xmath44    @xmath45    @xmath46    @xmath47      the electromagnetic field outside the plasma surrounding the compact star is now determined by equation ( 10 ) to ( 17 ) . to solve this system of equations
let us assume the following separation ansatz for the magnetic field components @xmath48 and @xmath49 :    @xmath50    and similarly for the electric field components @xmath51 and @xmath52 :    @xmath53    substituting from equation ( 18 ) into equation ( 10 ) and simplifying we obtain    @xmath54    for separation we require that @xmath55 and obtain    @xmath56    and    @xmath57    where @xmath58 is the separation constant . solving for @xmath59 the equation ( 21 )
we obtain @xmath60 . notice that because of the exponential function the field at coincident points @xmath61 and @xmath62 is not identical .
we therefore require single valued solution @xmath63 . the solutions to equations ( 21 ) and ( 22 )
are now given by :    @xmath64    and    @xmath65    where @xmath66 and @xmath67 are constants .
therefore we have    @xmath68    where @xmath69 .
and similarly from equation ( 13 ) and ( 18 ) we obtain    @xmath70    and hence    @xmath71    where @xmath72 and @xmath73 .    now to determine the functions @xmath74 we substitute from ( 25 ) and ( 27 ) into equation ( 13 ) . after some simplification
we obtain in a separated form :    @xmath75    as before we have    @xmath76    and    @xmath77    where @xmath78 is a constant
. since ( 30 ) holds for every value of @xmath79 , on comparing coefficients of @xmath80 , @xmath81 , and unity we obtain @xmath82 and @xmath83 .
here are now two possibilities for non - trivial solutions depending on either @xmath84 or @xmath85 ( i.e. @xmath86 )
.    for @xmath84 equation ( 29 ) gives    @xmath87    where @xmath88 is a constant . from equations ( 25 ) , ( 27 ) and ( 17 ) we obtain in a similar manner    @xmath89    hence    @xmath90    and    @xmath91    also from equations ( 11 ) , ( 12 ) and ( 15 ) , ( 16 ) we obtain    @xmath92    @xmath93    for @xmath86 ( and similarly for @xmath94 ) , there is no poloidal component to the magnetic ( electric ) field  and the only non - vanishing components to the electromagnetic field are given by equations ( 35 ) and ( 36 ) .
clearly in both cases the electromagnetic field is independent of @xmath95 .    in figures ( 1 ) and ( 2 )
we give plots for the magnetic ( electric ) field components @xmath49 ( @xmath52 ) and @xmath96 ( @xmath97 ) for @xmath98 ( @xmath99 ) and @xmath100 ( @xmath101
according to the general theory of relativity the trajectory of a particle in the field of a massive object is determined by the geodesic equation :    @xmath102    where @xmath103 is the christoffel symbol . to linearize the geodesic equation we assume that the metric tensor can be expressed as : @xmath104 where @xmath105 @xmath106 is the minkowski metric tensor and @xmath107 is a small perturbation to the spacetime metric such that @xmath108 .
we can take time @xmath109 for the affine parameter @xmath110 . further requiring @xmath111 for a ` slow ' moving particle we obtain@xmath112 @xmath113 where    @xmath114    and    @xmath115    @xmath116 being the mass density and @xmath117 is the volume of the gravitational source .
this approximation , due to its formal analogy to the classical electromagnetic theory , is referred to as the gravitoelectromagnetic ( gem ) approximation to the general theory of relativity ( see reference [ 16 ] and [ 17 ] for different interpretations ) . thus in expression
( 40 ) @xmath118 is called the gravitoelectric ( ge ) force per unit mass whereas the term @xmath119 is called the gravitomagnetic ( gm ) force for a unit mass . in particular the gm force , being independent of the choice of a particular coordinate system used , is general relativistically significant .    in the slow rotation approximation the deformation to the star due to the effects of rotation
are small compared to the  radially attractive ge force ; hence we take the star to be a slowly rotating sphere of homogeneous mass density . then using gem analogy ,
we obtain :    @xmath120    and @xmath121 where @xmath7 is the mass , @xmath122 is the radius and @xmath123 is the angular velocity of the star .
thus in the field of a slowly rotating compact star , the net force acting on a particle of mass @xmath124 and charge @xmath125 is the sum of the gravitational and the electromagnetic forces :    @xmath126    to specify the physical situation we study the motion of the charged particle in a cartesian coordinate system @xmath127 and take @xmath128 to be along the positive @xmath129-axis .
furthermore , since now the effective component of the gm force lies in the @xmath130-plane , we assume that the @xmath130-plane is the equatorial plane of the star and the particle s motion is confined to this plane only .
then for the case of non - vanishing poloidal magnetic field we transform expressions ( 33)-(36 ) and ( 42)-(43 ) into the cartesian coordinates , and obtain the equations of motion in the equatorial plane ( i. e. for @xmath61 ) :    @xmath131    @xmath132    where @xmath133 , @xmath134 and @xmath135 . notice that here the case of vanishing poloidal magnetic field is obtained for @xmath136 whereas requiring @xmath94 does not alter the equations of motion ( 45 ) and ( 46 ) .
we numerically solve the equations of motion ( 45 ) and ( 46 ) for the initial conditions @xmath137 and @xmath138 , and plot ( in figure ( 3 ) to ( 6 ) ) the solutions for various values of the parameters @xmath7 , @xmath139 , @xmath140 , and @xmath141 .
we have considered the dynamics of a charged particle in a highly , conducting plasma surrounding a slowly rotating compact gravitational source .
the effects of background spacetime have been discussed , both directly on the particle s motion and indirectly via an axially symmetric electromagnetic field around the star .
we found ( equations ( 33 ) to ( 36 ) ) that when slow rotation is assumed for the compact source the spacetime dragging does not modify or reduce the electromagnetic field of the star .
hence the field remains , in this approximation , the same as for the schwarzchild spacetime .
however the motion of the particle is directly influenced via gravitational field of the star .
our numerical analysis of the effects of gravitational and electromagnetic field shows ( fig .
( 3 )  to ( 6 ) ) that whereas the electromagnetic effects dominate the overall dynamics of the charged particle at sufficiently large distances , the gravitational field also becomes important closer to the surface of the star especially if the star is rotating fast enough .
firstly we observe that the effects of the electric field is to lift a charge particle from close to the surface of the star ( fig .
( 3 ) ) and make them move along the magnetic field lines . here
are now two possibilities : one , if the magnitude of the magnetic field is larger than the magnitude of the electric field the particle following a helical trajectory falls into the star ( case ( a ) , ( b ) , and ( c ) in fig . ( 4 ) ) ; two , if the magnetic field strength is weaker or is even comparable to the electric field strength , then the electrical effects become dominant and the particle escapes from falling into the star ( case ( d ) , and ( e ) in fig.(4 ) ) .    on the other hand
the effects of gravitational field can be regarded as opposing that of the electromagnetic field . in figure ( 5 ) we notice thatthe ge force attracts the particle towards the star s surface against the electric field . furthermore it is clear from figure ( 6 ) that the gyroradius of the helical trajectory of the particle is increased due to the gm force ; hence the gm force weakens the effects of the magnetic field .    summing up these observations
we note that the accretion of a charged particle in vicinity of a compact star is mainly due to the ge and magnetic effects .
these effects are especially dominant at a sufficiently large distances from the star . however _ close _ to the star the particle trajectory is also effected by the gm force as well as electric field which cause accerted charged particles to follow open lines of force . for a sufficiently fast rotating star and an electric field comparable to the magnetic field the above mechanism ( gm + electric ) may contribute to opposing the in - fall of the plasma surrounding the compact star .
we study the dynamics of a charged particle in the field of a slowly rotating compact star in the gravitoelectromagnetic approximation to the geodesic equation . the star is assumed to be surrounded by an ideal , highly conducting plasma ( taken as a magnetohydrodynamic fluid ) with a stationary , axially symmetric electromagnetic field . 
 the general relativistic maxwell equations are solved to obtain the effects of the background spacetime on the electromagnetic field in the linearized kerr spacetime . 
 the equations of motion are then set up and solved numerically to incorporate the gravitational as well as the electromagnetic effects . 
 the analysis shows that in the slow rotation approximation the frame dragging effects on the electromagnetic field are absent . however the particle is directly effected by the rotating gravitational source such that close to the star the gravitational and electromagnetic field produce contrary effects on the particle trajectories .