Dynamics of charged particles near a black hole in a magnetic field

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Dynamics of charged particles near a black hole in a magnetic field

Valdimír Karas, David Vokrouhlický

To cite this version:

Valdimír Karas, David Vokrouhlický. Dynamics of charged particles near a black hole in a magnetic

field. Journal de Physique I, EDP Sciences, 1991, 1 (7), pp.1005-1012. �10.1051/jp1:1991184�. �jpa-

00246380�

Classification

Physics

Abstracts

04.20 97.60L

Dynamics of charged particles

_{near a}

black hole in

_a

magnetic

field

Vladimir _Karas and David

Vokrouhlick§

Department

of

Astronomy

and

Astrophysics,

Charles

University, (vbdskh8,

15000

Prague,

Czechoslovakia

(Received12December1990, accepted

in

final form

1March

1991)

Abstract. Flow lines of the

plasma

_{near a}

magnetised

Kerr-Newman black hole are studied within the framework of the

guiding

centre

approximation.

Surfaces of constant

magnetic

and electric flux _are constructed and the

shape

of the

plasma

horizon is discussed.

1. Introduction.

Black holes immersed in cosmic environments _are of

longstanding

interest to

astrophysicists.

Effects

accompanying

the accretion of matter onto a black hole could

explain

violent processes, which _occur in quasars and

galaxies

with active nuclei

[lj.

The electro-vacuum solution

describing

_an

isolated, rotating, electrically charged

black hole

(Kerr-Newman solution)

is well understood and its

uniqueness

is proven

[2, 3]. (Hereafter,

we do not take _a

theoretical

possibility

of _{a nonzero}

magnetic charge

of the hole into

account.) However,

realistic solutions with external

magnetic

fields and

complicated

motions of

plasma

_are much

more difficult to handle.

Analytic models,

no matter how

inspiring they

_are,

require

further

assumptions (on

the symmetry ^{of the}

problem,

the

equation

of state,

etc.)

to become tractable

[4, 5].

Self-consistent

description

of _a black hole

magnetosphere

is constructed

numerically (see,

_e-g-,

Refs.[6-9]

and references cited

therein)

and _{even so} the current models suffer from serious inconsistencies

[10].

This paper extends the

approach

of Damour et al.

[I Ii

who

investigated

the

regions

of

magnetic

_support of

plasma

_{near a} black hole and

constructed flow lines with the framework of the

guiding

centre

approximation.

We _assume that the

magnetic

field is

generated

far from the black

hole,

_so that _near the

hole it _{appears as} uniform.

Therefore,

_{we can}

employ

_a

special exacj

^{solution of the Einstein-}

Maxwell

equations describing

_a

magnetised

Kerr-Newman

(MKN)

black hole

[12].

This

solution is

stationary

and

axially symmetric, magnetic

field

being asymptotically (far

from the

hole) parallel

to the rotation axis. In section

2,

to elucidate the structure of the

electromagne-

tic

field,

_we construct surfaces of constant

magnetic

and electric flux. Then _we discuss

potentially

stable

regions

of the

magnetosphere

where the

magnetic

force _can balance the electric _one. The

boundary

of this

region

is referred _{to as a}

plasma

horizon

[13].

The

shape

of the

plasma

horizons in various

configurations

with

uncharged

_or

weakly charged

black holes

1006 JOURNAL DE

PHYSIQUE

I M 7

were

plotted by

Hanni

[14]

and Damour _et al.

[llj. However,

the existence of the

plasma

horizon is

only

_{a necessary} condition for the

magnetosphere

to be stable and

physical significance

of this

approach

is very limited

(for

critical remarks about the

concept

of the

filasma

horizon _see Burman

[15]). Indeed,

with _a

large

value of the

angular

momentum or

charge

of the hole _{we can see} that rather

complicated topology

of the

plasma

horizon does _not meet characteristic features of the

plasma

motion.

Flow lines of the

plasma

which _are constructed in section 3 then describe the structure of the

magnetosphere

_more

appropriately,

_as described

originally

in reference

[I Ii.

We compare the flow lines _{near a} MKN black hole with those constructed within the

approximation

of the

electromagnetic

test fields in the fixed

background

metric of _an isolated

rotating (Kerr)

black hole. For

astrophysically

relevant values of the

magnetic

field the

approximate

flow lines coincide with _{exact ones.} In

Appendix

we estimate corrections to the flow due to the drift of the

particles

caused

by

the strong

gravitational

attraction _near the horizon of the hole.

2.

Magnetitsed

Kerr-Newman bIack hole _: the

eIectromagnetitc

field and

plasma

horizon.

Four parameters describe the

space-time

of the MKN black hole

completely

the _mass

M, specific angular

momentum a and

charge

_e of the

hole,

and the

magnetic

field parameter

Bo.

We _use

geometric

units with the

velocity

of

light

and

gravitational

_constant

equal unity.

Thus,

for the black hole _mass parameter we have

M[cm]

+ 1.5 _x

10~(M/M~ (M~ being

the

solar

mass);

^for

specific angular

momentum

a[cmj

+3.3

x10~'~ fi[cm~/s];

for electric

charge e[cm]

+2.9

x10~~~ I[esu]

and for the

magnetic

field

parameter Bo[cm~~]

₊

2.9 _x

10~~~ jio [gauss].

The

space-time

element of the MKN metric in

spheroidal

coordinates

(xi _{x~, x~,} x~)

m

(t,

_r,

o,

_q~

)

_can be

expressed

in the standard form of _an

axially symmetric stationary

metric

ds~

=

e~

~

dt~

₊

e~ *(dq~

_w

dt)~

₊

e~~ dr~+ e~?

_do ^~

(l)

with

e~~

=

[A[~3AA~~, e~*

=

[A[~~A3~~sin~o,

e~~

=

[A[~3A~~, e~?

=

[A[~3,

3=r~+a~cos~o, A=r~-2Mr+a~+e~,

~ ~~~ ~ ~~~

~~~

_{~~~~ ~}

'

~ _~

~°

~

~~

~

Explicit expressions

for the functions _w, ~P and I _were

given by

Garcia Diaz

[12]. Having astrophysical applications

_on

mind,

_{we assume} weak values of the

magnetic

field

(104 gauss)

and linearize all

expressions

for the metric tensor and

electromagnetic

field components ⁱⁿ a

dimensionless

parameter

^Em

[Bo M[

« I

w m

(2r-e~)aA~~-2Boer(r~+a~)A~~,

A m I ₊

Bo

ear3~

sin~

o

iBo e(r~

₊

_a~)

_3~

cos b

(2)

(The

linearized MKN solution _was also

investigated [16].

In Section 3 _{an error} in

Equation (9)

of this reference is

corrected.)

Rather

complicated

formulas for the components ^{of the}

electromagnetic

field tensor in terms of the functions _~P, I _were

originally

derived

by

Emst and Wild

[17]. Starting

from their

expressions

_we constructed the surfaces df constant

magnetic

and electric

flux,

which _are

given by

constancy ^{of the real and}

imaginary part

^of

~P'mA~~(~P- ~Boi), (3)

r/M

^~

r/M

a)

b)

Fig.

I.- Two

typical configurations

of the

magnetic

and electric lines of force _{near an} extreme

[e=

(M~-a~)~'~], magnetised

Kerr-Newman black hole with p =0.018,

a/M=

0.99 (a), and p =0.071,

a/M=

0.99

(b).

The latter _case

corresponds

to an

uncharged configuration,

e + 2Bo Ma

=

0. Far from the hole the

magnetic

lines of force become uniform and

parallel

to the rotation axis, which _goes

vertically

in the

figure. Figures

are

symmetric

with respect to the

equatorial plane

and

axially symmetric

about the rotation axis. Near the horizon

(denoted by

the

quadrant

r/M ₌

I) they obviously

have _a

dipole-like

structure

(a),

but in _an

uncharged

extreme

configuration

(b) the

magnetic

field lines _are

expelled

out of the horizon

completely.

Electric lines of force _are in both

cases

asymptotically

radial. In

(b)

their

shape, unexpected

_on the basis of _our

experience

with _an

analogous problem from the classical

electrodynamics

of rotating magnetised

spheres (cf.

Thome _et a/.

[23]),

is

produced by

the effects of the

gravitomagnetic

interaction.

respectively (Fig. I).

These surfaces _can

equivalently

be defined _as surfaces where

magnetic (electric)

lines of force reside _:

according

to Christodoulou and Ruffini

[18]

the lines of

magnetic (electric)

force _are tangent to the force

experienced by

_a

magnetic (electric) charge

at rest with

respect

to the

locally nonrotating

frame

(LNRF). (Shapes

of

asymptotically

uniform field lines in the test field

appromixation

_were

plotted by

_a number of authors

[19- 23].)

Orthonormal

components

^{of the field} in the LNRF _are

B~,~ ⁺

iE~,~

= A

'/~sin~

_o

30~P', (4)

B~o~ ⁺

iE~o~

=

(A/A)~/~

sin~ o

3,~P' (5)

It _can be verified that the field of the MKN black hole linearized in

p

and _e coincide with _a

superposition

of the

axisymmetric, asymptotically

uniform

magnetic

test field in the Kerr metric

[24]

and the Kerr-Newman

electromagnetic

field with the

charge

_{e +} 2

Bo

Ma.

~The

term 2

Bo

Ma is

exactly

the Wald's

charge [25]

which is

required

to neutralize the Kerr black hole immersed in the

magnetic

test field. This

correspondence

between the exact solution and the test field

approximations

has been discussed in

[26, 27].)

Now _we turn to the

problem

of

stability

of the

plasma

_near the MKN black hole. The

plasma

horizon is defined _{as a} surface in the

magnetosphere satisfying

the condition

[EXB[ ^=E~ (6)

for the electric and

magnetic

field three-vectors in the local

frame,

_or

equivalently [11, 14]

V~ V~=0, V~WF~~F~P~p, (7)

1008 JOURNAL DE PHYSIQUE I M 7

where

F~~

denotes the

electromagnetic

^field

components

^and _{~ p} ^{is the}

four-velocity

of _{a zero-}

angular-momentum

observer

(ZAMO).

This choice of the

preferred

frame appears natural because

space-time (I)

is

stationary

and

axisymmetric(I). Figure

2 shows the

shape

of the

plasma

horizon around _a few

nearly

extreme

(a~

₊

e~

-

M~) configurations

of the MKN black hole. It _can be _seen

that,

in

general,

stable

regions

of the

magnetosphere

_as defined

by

the

plasma

horizon do not agree with _{a more}

sophisticated approach

^of section 3. In other

words,

flow lines of matter that _cross the

plasma

horizon do not

necessarily

hit the horizon _even in the _case when definition

(7)

of the

plasma

horizon with

preferred

ZAMOS is

accepted.

Moreover,

_{we note} that the

shape

of the

plasma

horizon will be different in another

frame,

what restricts usefulness of this

concept significantly.

0.

Fig,

2. Plasma horizons in the MKN

space-time

with p

= 0.05 and

a/M

= 0.95.

Hatching

denotes the unstable

regions

with V~ V"

< 0 _as defined

by equation (7).

The black hole horizon is denoted by _a

quadrant

in each

figure.

The values of

e/M

_are shown with

figures e/M

~ 0.095

corresponds

to the

positive

total

charge

of the black hole _; this _{case was} discussed by Hanni f14] in detail.

3. Motiton of the

guiding

centre.

In this section _we construct flow lines of the

plasma

in the framework of the

guiding

centre

approximation [28, 29].

First _{we assume} with Damour et al.

[I Ii

that the

electromagnetic

field variations _are small within _a Larmor radius and _a Larmor

period.

We further _assume that all other forces

acting

_on

charged particles except

for the

electromagnetic

_{one can} be

neglected.

This

approximation

is later found to be violated

only

_very close _to the horizon

(r

_{m r~}

= M ₊

(M~ a~

_{e~)~/~; see}

Appendix). Now,

the motion consists of a local

gyration (which

is

averaged

out in the

guiding

center

approximation),

an acceleration

corresponding

to the

hyperbolic

motion in the frame i where the

magnetic

and electric fields _are

parallel,

and _a

(')

The

only

other frame that has been

frequently

used with the Kerr-Newman geometry is the Carter frame

[30]

which is connected with the

principal

null congruence. However, _{one can}

verify

that

(I)

in

Boyer-Lindquist _type coordinates does not allow of the PNC

photons, although they

do exists in the

special

cases Bo = 0

(Kerr-Newman

geometry), or a, e =

0

(magnetised

Schwarzschild geometry).

daft with respect to i. This frame is related _to the LNRF

by

the Lorentz boost in _{a w-} direction to the

velocity

v,

magnitude

of which is

given by

here E and B _are the fields measured in the LNRF. The drift

velocity given by (8) acquires

values in the whole range

(0,1)

_upper ^values near cusps

(where

E.B =0 and

E=B)

and _near the horizon.

Figure

3 shows

projections

of the flow lines _on the

q~ = const.

plane,

I-e- the lines

tangent

to the _common direction of the electric and

magnetic

fields in the frame

orbiting

with the

velocity

u around the hole. Parametric

equation

of the

curves is

dx)dA

=

E~

+ y ^~/~e~~~ u~

BJ,

I,

j

m

1,

2

(9)

where E'

=

Fj

_~

~,

B~

=

*Fj

_{~ ~,} and _y

= g~, goo

g~~. Putting

_a

= e =

0 in the metric

(magnetised

Schwarzschild black

hole)

_we obtain uniform flow lines which

parallel

the o

=

0,

_w

axis,

_as

expected. (One

_can

verify

that our

equations (8), (9)

_are

equivalent

to

Eqs. (6)

and

(8)

of Ref.

[ll].)

~ ~

r/M

^{~ ~ ~}

r/M

^~

a) b)

Fig.

3.-

Typical configurations

of the plasma flow lines _{near an} extreme MKN black hole with p

=

0.018, a/M

= 0.090

(a),

and p ₌ 0.05,

a/M

= 0.933

(b).

Charged particles

_move

along

^the flow lines in _a direction

given by

the direction of the electric field

(El

=

Fib)

in i. To _turn this direction _we need

El

=

0 somewhere _on the flow

line,

which _can

only

be the _case of

perpendicular

electric and

magnetic

fields in the LNRF.

Such

places

do

exist,

_as discussed with

figure

4 below.

It _can be _seen in

figure

3 that

only

_some of the flow lines

(those coming along

the rotation

axis)

_cross the horizon. Matter confined _to the flow lines

crossing

the

equatorial plane (b

=

w/2)

outside te horizon does not fall

(within

the

approximation adopted)

into the black hole. We

computed, numerically,

the _« effective _cross section of the hole

(Fig. 4)

defined _as

a ratio

Jt/JtH

here Jt

=

I (go

o

g~~

)~'~ ^do dq~ is the _area of the surface _r

= const. _m 500

M,

o _w

o~~~ (say)

far from the hole from which the flow lines

eventually

thread the

horizon,

and

A~

=

2 w

Ao

o

[~(r(

₊

a~)

denotes the surface of the

hemisphere

located _on

1010 JOURNAL DE

PHYSIQUE

I M 7

9

~x

'

~

~

O.6

3

ENLARGED

°~-

1~96

0

~~ ~

a/M

Fig. 4. Solid line is _an effective _cross section of _an extreme MKN black hole (p

=

0.05 for the capture of the flow lines (see the text). Broken line denotes the cross section computed with the flow lines

along

which the electric field in lF does not

change

its orientation. Consequently, charged

particles

are accelerated

along

these lines in _one direction from

large

distances into the black hole _or

oppositely, according

to

sign

of their charge.

the horizon. An

important configuration

is that with the

vanishing

_cross

section,

A

=

0,

because the accretion rate then goes ^to zero as well. One _can

verify

that this is the _case of _{an extreme} black hole with _{a zero} total electric

charge

_or, in the limit of _a weak

magnetic field,

e=-

2BoMa, [27].

In other

words,

in

equation(9)

one obtains

dr/dA =0,

do

/dA

_{# 0} at r ₌ r~

just

for the

uncharged

extreme

configurations,

in which the flow lines do

not cross the horizon. This is in agreement ^{with Wald's}

[25]

results about selective accretion

onto a black hole in the test field

approximation. Analogous

results _can be obtained with

exact

expressions

of reference

[12] (nonlinearized

in the

magnetic

field

parameter)

in this

case the effective _cross section vanishes for _e

=

2

Bo Ma/(I El _e~/4).

4. Conclusion.

In this paper we extended

previous

works about accretion of the

plasma

onto a

magnetised

black hole within the

guiding

centre

approximation.

We

employed

the _exact solution of the Einstein-Maxwell

equations,

constructed the flow

lines,

and

computed

the effective _cross section for

capture

^{of the material.}

Compared

to Wald's

analysis,

the present

approach

enables _us to

study asymptotically

non-flat

space-times,

ⁱⁿ which _case energy of

charged particles

at

infinity

cannot be defined.

Acknowledgments.

We would like to thank J. Bibhk for

helpful

discussions

during

this work. We _are also

grateful

to the referee for

pointing

out several omissions in the first version of this _paper.

Appendix.

We estimated the value of the

gravitational

drift which forces

charged particles

to _cross the field lines. We restrict to the

magnetised

Schwarzschild

metric,

in which _case the

equation

of

motion reads

dp/dt

=

rmg

_{+ q}

(E

_{+ v x} B

(io)

m is the rest _mass of the

particle,

_q is its

charge,

r is the Lorentz factor. In the orthonormal frame

~(r)

"

(1

²

M/r)~'~ a,

,

e~o~ ^{= r~} 30

,

e ~~~ ⁼

(r

sin o

)~ _3~ (11)

the

gravitational

acceleration has the

only

_nonzero component ^{in the radial}

direction,

g~~~ ⁼

(1

2

M/r)~

^~/~ Mr~

~, (12)

and the

electromagnetic

field

(4)-(5)

reduces to B~~~ ⁼

Bo

cos o

,

B~o~ =

Bo(1

²

M/r)~/~

sin o

(13)

One _can

verify

that the total force

acting

_{on a}

particle

^will be

parallel

to the

magnetic

field in _a frame

moving

in the

w-direction

with the

velocity

_w

given by

~'~lg

(r)

q~B

(0)

B(r) q~B

_~r)

B(0) (14)

with respect ^{to the frame}

(11).

Far from the hole _we have r

m

I,

g~~~ ^m

M/r~,

and

w m

"~~'/~~

^~

,

(15)

qr

Bo

which is, in

fact,

_a familiar formula for the

gravitational

drift

[29]

_:

w m

"~~

~~~ (16)

qB

In

general, equation (14) gives

Ii

ⁱ

^mMBo

^{sin o 2 1'2 1'2}

~ ~ ~

q~~(B)r))

+

B/0)

^~~~~

The corrections due to the

gravitational

drift _can be

neglected provided

_Hi «

I,

I-e- in the

region

$

»

I (18)

2 M

pq

Since the

electrically charged particles

have

extremely large specific charge q/m

m 10~~ for

electrons), equation (18)

is satisfied for

astrophysically

relevant values of the

magnetic

field

essentially

to the very horizon.

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