80

РЕДАКЦИЯ АЛҚАСЫ:

Буфенди Лайфа,профессор (Франция)

Давлетов А.Е., ф.м.ғ.д., профессор – ғылыми редак-

тор (Қазақстан)

Иващук В.Д., ф.м.ғ.д., профессор (Ресей)

Лаврищев О.А.,ф.м.ғ.к. – ғылыми редактордың

Ишицука Эцуо, доктор (Жапония)­

орынбасары (Қазақстан)

Лунарска Элина, профессор (Польша)

Әбишев М.Е.,ф.м.ғ.д., профессор(Қазақстан)

Сафарик П., доктор (Чехия)­

Асқарова Ә.С.,ф.м.ғ.д., профессор(Қазақстан)

Тимошенко В.Ю., ф.м.ғ.д., профессор (Ресей)­

Буртебаев Н., ф.м.ғ.д., профессор(Қазақстан)

Кеведо Эрнандо, профессор (Мексика)

Дробышев А.С., ф.м.ғ.д., профессор (Қазақстан)

Жаңабаев З.Ж., ф.м.ғ.д., профессор (Қазақстан)

ТЕХНИКАЛЫҚ ХАТШЫ

Косов В.Н., ф.м.ғ.д., профессор (Қазақстан)

Дьячков В.В., ф.м.ғ.к. (Қазақстан)

When

1. Introduction

kn

nm

mk

.

According to the ideas of modern theoretical

+

+

= 0

astrophysics [1-2], neutron stars have magnetic

studying the laws of propagation for weak

dipole fields, which on their surface reach values

electromagnetic waves in a strong external field

ik

comparable with quantum electrodynamic induction

the eikonal equation was used. Calculations

have

Bq = 4.41 · 10

13

Gs. In such fields, the nonlinearity

shown that the propagation of a weak

of electrodynamics

in

a

vacuum

must

appear,

electromagnetic wave

according

to

the laws

of

leading to the appearance of various physical effects

nonlinear electrodynamics (1) in space-time with a

[3-8]. Theoretical studies of such nonlinear

metric tensor

nkand in the presence of an external

electrodynamic processes use the post-Maxwellian

electromagnetic field occurs by geodesic of some

approximation [9]. In this approximation, the

effective pseudo-Riemannian space-time. The

Lagrangian of the nonlinear electrodynamics of

metric tensor of this space-time

nk depends on the

vacuum is written in the parametrized form:

metric tensor

combination of the

nk, the quadratic im

and at

electromagnetic field tensor

ni

mk

1

1

2

m

different polarization

L

2J2

1 2 2

J2

4 2 J4

j

Am ,

it is different for waves of

≠

32

c

(nonlinear electrodynamic birefringence). While for

1/ B2

the first normal wave the tensor

nk

has the form

= − 4

where

J

2

Fnk F kn and

J 4 Fnk F km Fmi F in

are

( )

im

mk,

(2)

invariants of the electromagnetic field tensor

Fkn ,

nk

ni

nk

q ,

1,2 are

postmaxwellian

parameters

for the second normal wave, having orthogonal

whose magnitude is different in different theoretical

polarization to the polarization of the first wave, the

models of nonlinear electrodynamics of vacuum.

tensor differs by the second term coefficient:

2. The equations of the electromagnetic field

nk

nk

ni

im

mk

(3)

−4

.

In

the

Heisenberg-Euler theory,

which

is a

( ) =

consequence

of quantum

electrodynamics,

the

According to the Lagrange-Charpy theorem, this

means that in order to find the trajectories along

=

⁄(

) = 5 1

and

=

which the momentum of a weak electromagnetic

numerical values for the parameters

differ

45

.

10

,

wave propagates in the external field and determine

.

10

the laws of its motion along these trajectories, we

180

, while in the nonlinear

Born-Infeld electrodynamics they are equal to each

need to solve the equations of isotropic geodesic

⁄(

)

= 9 0

motion in the effective space-time with the metric

other.

The equations of the electromagnetic field with

tensor

nk

:

= 0, ( , ) = 0,

have the form:

mn

ξη

mk

kp

pn

+

(,)

һ = −

+ 4

(1)

dK

mn

( )

nm

(4)

( )

1 + ( − )

.

where

mn are Christoffel symbols of the space-time

with a

metric tensor

nk or

nk

, depending on the

⁄

studied,

affine

parameter,

The second pair of equations of the

mode

being

is

electromagnetic

field

coincides

with

the

dx

is a four-vector tangent to the

corresponding equations of Maxwell's theory:

corresponding

=

isotropic geodesic.

ISSN 1563-0315

Recent Contributions to Physics. №3 (66). 2018

5

= −

BR

(1 + 3cos2 ) cos − 3 sin2 cos sin cos

+

3 sin cos2 sin sin ,−3 sin2cos sin cos

+

3 sin cos2 sin sin ,

= −

5

2

+

1

2

−

2

sin

cos

sin

BR

10 30 cos

cos

cos

−

6

2

2

,

5

2

5sin

cos

sin

sin

,

=

2

3

sin

sin

cos

−

sin

sin

sin

BR

15 cos

10 sin

where R is the neutron star radius, B is the magnetic

= 1 +2

cos

,

field at the stellar surface, χ1[0,π] and χ2[0,2π] are

two angles specifying the particular geometry of the

1+2

quadrupole magnetic field.

cos

√3

cos

cos

However, for further calculations it is more

convenient for us to

use

a

rectangular

Cartesian

=

sin

,

coordinate system. Re-designating the constants B,

sin

,

=

cos

χ1 and χ2 in accordance with relations

we obtain:

= − 5

+2 5

− + 5 − 3

+10 ,

=

6

−5

+

+ 3

−

−5

,

5

10xzf

10

=

5

3

− 5

+2 5

−

− 5

−

,

6

6

Хабаршы. Физика сериясы. №3 (66). 2018

,( ) = ( )+ , ( ) − ( ) ,

, ( ) = ( )+ ,

( ) − ( ) +

( )

( )

( ) ( )

,

( )− ( ) +

( − ) ( ) − ( )

.

,( ) = ( )+ ,

( − )

points r = rs и r = rd, we obtain:

( ) = ( ) = ( ) = 0,

Since the electromagnetic pulse at time t = ts was

at the point r = rs, and the ray must pass through the

( ) = ( ) = , ( ) =

dz

−

dz

−

dz

( )

= 1

dt

( )

dx

( )

dy

= ( ) = 0, ( ) = .

.

dz

dz

dz

From these equations it follows that:

have:

We ( ) =

+

, ( ) =

, ( ) = 0

−

Then in the Maxwellian approximation we will

( )

2

have:

dt

=

+

=

4

( − )+

(1 − 2 − ) +

2514

zf

dz

+

( − )+ (

+ +

− 2) +

4

+

20

zf

12

+

2

(2 − )+ (4 − 4 −

15

− ) ,

= + .

410

zf

under consideration

dependence of ():

equation,

we find

the

where in the approximation

Integrating

this

ISSN 1563-0315

Recent Contributions to Physics. №3 (66). 2018

7

( ) =

( − )−

+

(5 − ) + (2 − 4 ) +

(13)

50