Plasma Modes Along Plasma Modes Along Open Field Lines of Open Field Lines of Neutron Star with Neutron Star with

Gravitomagnetic NUT Gravitomagnetic NUT ChargeChargeJD02-21JD02-21

B. Ahmedov and V. KagramanovaB. Ahmedov and V. Kagramanova

UBAI/UBAI/INP, Tashkent,INP, Tashkent,

Uzbekistan ASUzbekistan AS

P l a n o f t a l kP l a n o f t a l k

Solution of Maxwell equations in the exterior Solution of Maxwell equations in the exterior metric of cylindrical NUT source aroundmetric of cylindrical NUT source around

(i) magnetized star with the dipolar (i) magnetized star with the dipolar magnetic field magnetic field

(ii) electric charge put on cylindrical (ii) electric charge put on cylindrical gravitomagnetic monopole.gravitomagnetic monopole.

Plasma Modes Along Open Field Lines of Plasma Modes Along Open Field Lines of NUT StarNUT Star

ConclusionConclusion

Gravitomagnetic monopole in cylindrical coordinates

The general stationary axial symmetric solution to the vacuum Einstein equations in cylindrical coordinates {t, , , z} is given by

ds2 = f-1[e2(d2+dz2)+2d2]-f(dt- d)2

where f, , are functions of and z only

The explicit functions for f, , have a form

const

How to find expression for z?

A useful class of observers is represented by the “zero angular momentumobservers” or ZAMOs. These are observers that are locally stationary,

but who are “dragged” into rotation with respect to areference frame fixed with respect to distant observers. They have four-velocity

components given by

EM FIELD EQUATIONS:

F[ , ] = 0 F ; = 4 J/c

We will look for stationary and axially symmetric solutions of the Maxwellequations, taking into account that in the vacuum region around the source all

components of electric current are equal to zero. Due to thepossible smallness of the gravitomagnetic mass of the star estimated from someastrophysical observations we may perform calculations to the first order in NUT

parameter.

Electromagnetic fields of nonrotating cylindrical NUT

sourceThe exact solution for the nonvanishing

components of magnetic field has a form:

In the limit of flat spacetime when

and 0these expressions reduce to

which are the Newtonian ordinary expressions for magnetic field of a magnetic dipole

Near to the NUT source the magnetic field will be amplified, then in some intermediate region it will be weakened and in the asymptotically far zone the influence of NUT parameter is negligible, the behavior of field is Newtonian and relation tends to unity. The influence of the NUT parameter is more strong near to the source.

Consider now as a toy model the charged NUT star with the

monopolar electric field configuration

If Q is the electric charge per unit length of the line tube then the solution for the radial electric field admitted by Maxwell equations

is

E = E (, z) 0,

E = E = 0.

E = Qf e-

a

z

The radial dependence of electric field E for different values of the gravitomagnetic monopole l. The effect of the NUT parameter on the electric field is becoming important near to the source.

METRIC OF SLOWLY ROTATING NUT STAR

• EF on the Star Surface: Goldreich & Julian, 1969, Astrophys.J, 157, 869

Cascade generation of electron-positron plasma leads to formation of MS with plasma screening longitudinal EF.Plasma is corotating with the star. Charges along open field lines create plasma

modes.

NS Magnetosphere

11210 1010~~

cmVBc

RE

c

RL

ZAMO OBSERVER:

r) ´ 2aM/r3

Goldreich-Julian charge density in GR

1F2F

r r

M r2/

Plasma modes along open FL

.8

,2

1

2

0

20

m

ne

rc

GM

2 4 6 8 10r

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

GJ

For l0,2step 1,3.22

l0

C o n c l u s i o n

• Analytical general relativistic expressions for the dipolar magnetic field around NUT star are presented. It is shown that in the linear approximation in NUT parameter there is only effect of mass of the star on the value of stationary magnetic field.

• In the case of charged star there is strong influence of NUT parameter on the value of monopolar electric field.

• NUT parameter increases the angle of last opened field line. • Our results in principle could be combined with analysis of

astrophysical data on electromagnetic fields of compact objects in order to get upper limits on the value of NUT parameter.