Introduction into Black Hole Physics Valeri Frolov

DPG School of Physics, “Galactic Black Hole 2001” Bad Honnef, August 26–31, 2001 0.1

Lectures Outline: I. Particle Motion near a Non-rotating Black Hole ➠ ➠ ➠ ➠ ➠ ➠

Equation of motion Symmetries and integrals of motion First integals of the equations of motion Types of trajectories ‘Tilted’ spherical coordinates Motion of ultra-relativistic particles

➠ Gravitational capture

II. Particles Motion near a Rotating Black Hole ➠ ➠ ➠ ➠

Gravitational field of a rotating black hole Equations of motion of a free test particle Motion in the equatorial plane Motion off the equatorial plane

➠ Gravitational capture

III. Fields in the Black Hole Spacetime ➠ Scalar massless field in the Schwarzschild metric ➠ Scalar field evolution ➠ Wave fields in the Kerr metric ➠ Effects connected with a black hole rotation

IV. Black Hole Electrodynamics ➠ ➠ ➠ ➠

Electrodynamics in a homogeneous grav. field Membrane interpretation Electromagnetic fields of a charge near a black hole Black hole in the magnetic field

➠ Mechanism of the power generation 0.2

1

Particles near a Non-rotating BH

1.1

Equation of motion

1.1.1 Particle moton à Black hole physics studies black holes, their properties and their interaction with surrounding matter and fields. à Astrophysical information is encoded in the properties of fields (electromagnetic, gravitational) or particles (cosmic rays, etc) reaching an observer. à A test particle moving near a black hole is the simplest and very important for applications problem. à Examples: planet like objects in their Keplerian motion near a black hole, small elements of accreting matter, stellar black holes and neutron stars falling into a massive black hole and so on. à In the absence of other forces a particle is moving along a geodesic line





 
 



       

        is four-velocity and 1.0

is proper time.

1.1.2 Schwarzschild metric à The geometry of a static non-rotating black hole is spherical symmetric and is described by the Schwarzschild metric

 3254

687 à %#0$ / 

9!:

  .0/1 #   # * +-, +

is the gravitational radius

is a line element on a unit sphere

à

à

determines time and length scales

#%$

 

à

 !" #%$ #

& (')







#$ ;



" 

=' <  

  1 ? >



  0/ A@

is a dimensionless radial coordinate

 ' < # '  # $

is a dimensionless time coordinate à #$

It is sufficient to solve a problem for only one value of

). The solutions for black hole mass (say for #$ other masses can be obtained by simple rescaling. 1.1

1.2

Symmetries and integrals of motion

1.2.1 A Killing vector



à A Killing vector field B is a vector field which satisfies the following (Killing) equation

  B!C E D  à A Killing vector is a generator of symmetry transformation à Killing trajectories are integral lines of the Killing vector field  

(' B

à For regular B





Killing trajectories form a foliation

à In a small spacetime region one can always 'AGIHKJML ' where is a parameter introduce coordinates F HNJ along the trajectory, and are constant on a given trajectory.

1.2

1.2.2 First integrals of motion


  is constant along the à For geodesic motion B world line and hence it is an integral of motion:


 L 

  FOB



   E D P B C SQ R T



  B P UQ R T 

=0: Killing Eq.



=0: geodesic Eq.

1.2.3 Symmetries of the Schwarschild spacetime à The Schwarzschild metric has 4 Killing vector fields à Time symmetry

 B!CWV D B WC V D X  X ' X X

à A 3-parameter group of rotations

B > B >



X   YSZ[]\ X  YSZ`_ [)acb \ X \ ^ X X^ X  B B  X  [)adb \ X  SY Z`_ YSZf[g\ X G \ ^ X X  Xe^ Bh  B!Cdi D B Cdi D X  X  \ X X 1.3

G

j The metric is static and invariant under time 'lk ' ' D  . B CWV is orthogonal to const reflection j B  D  m VnVo  qp Infinite red-shift surface r p CWV Killing horizon r p event horizon.

1.3

First integrals of motion

1.3.1 Orbits are planar à We can always choose coordinates so that at the Es s   ]Luvs 



t 9 and F initial moment one has

^

^

à The only non-vanishing components of ow  are

 w +w

G #

 wixi  [-adb

YUZ[ ^

^

à The -component of the geodesic equation of ^ motion is

   ^  

#

   #  ^  [-adb

^

YUZ[



^

à A solution for given initial data is

^

\  







t 9

Thus the trajectory of a particle is planar and we can  assume it to lie in the equatorial plane t 9 .

^

1.4

1.3.2 Effective potential

'

à The metric is invariant under time and angular coordinate \ translations. The conserved quantities are

y<




B!Cdi D 

z

à

y<

& ( '  1 # $  (specific energy) #




 B!CWV D 



y x{



#



\   (specific angular momentum)

z

, and

}{

|

'

à For motion in the black hole exterior both and \ are  monotonous functions of





à The normalization condition 



à Radial equation of motion

  #

~ F #

L



y< 

 * #$ #

à ~€ is en effective potential 1.5

 ~€ F z & #

#

 

L

1.3.3 Properties of the effective potential à Different types of trajectories can be classified by studying the turning points of its radial motion where

~ F #

L

y< 

is the only scale parameter. Denote by à #%$ < z and | then we have

#$

~ F #

L

 L

‚ „ƒ F

" 





# #$

<  |   

L  „ as a function of and à The effective potential ‚ ƒ F < | is shown at Figure. u < u†… ‡ ˆ it has extrema at à For fixed | < < <  ˆ}Œ  ‰

| Š| ‹ |  à The effective potential as a function of  Ž maximum at and minimum at

1.6



has

6 5 4 3 2

1.7

1 6 5 4 lambda 3 2 1 0

5

10

15 x

20

25

30

1.3 U(x) d 1.2 c 1.1 1 0.9 b 0.8 0.7 a 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

x5

1.8

6

7

8

9 10

1.4

Types of trajectories

1.4.1 Bounded and unbounded trajectories à The specific energy of moving particle is constant

j (a) ‘Elliptic’ motion: y  > ‘

G L , F . Orbits in # > #

general are not not closed. Far from BH, it is an ellipse which slowly rotates in the plane of motion

j (b) ‘Hyperbolic’ motion. ‚

max

’

y

 > ’

j (c) Unbounded motion with capture

, F

# >

G”“ L

j (d) Bounded motion with capture à The trajectories are not conic sections

y • and its à A body can escape to infinity if  velocity – ’ – esc  

9(˜ : – esc — #$ # #

(It is the same as for the Newtonian theory.) à Even if a particle has the escape velocity it can be trapped by the black hole. This effect called the gravitational capture. 1.9

1.1 1.05 1

U(x) c d

b

d

a

0.95 0.9 0.85 1.

2.

4.

7. x

1.10

.2e2 .4e2 .7e2

ORBIT for E2=.95 l=1.9235384 x0=3.1

ORBIT for E2=.95 l=1.923538406 x0=3.1 15 10

10 5

–10

5

0

–5

5

10

–15

–10

–5

5

10

–5

–5

–10

–10

–15 –15

ORBIT for E2=1.05 l=2.095709904 x0=3.1 15

ORBIT for E2=1.05 l=2.121320344 x0=3.1

15

10

10 5

5 –6

–4

–2 0

2

4

6

–5

0

–5

–5

–10

–10

–15

1.11

5

10

15

15

ORBIT for E2=.95 l=2.095709904 x0=1.1

ORBIT for E2=1.05 l=1.962141687 x0=3.1 16

1

14 12

0.5

10 8

–1.5

–1

–0.5

0

6 –0.5

4 –1

2 0

1 2 3

–1.5

–2

1.12

0.5

1

1.5

1.4.2 Circular motion

     (extremum of U) à For circular motion # à Minimum qp stable, maximum qp unstable motion y ™ y For   max , the particle makes many turns near à #

max

before the orbit moves far away

à The maximum and minimum appear on ~ F < # ‡ ˆ when | ’

L

à Circular orbits is possible only if |

<

’

curves

‡ ˆ

à Stable circular orbits thus exist only for

# ’

ˆ

#$

à Even unstable inertial circular motion becomes šœ› impossible at

# ‘

#$

ˆ

particles move at à At the critical circular orbit $ # #  a velocity – — 9 , the energy of a particle being y `ž ™ ž Ÿ ˆ

 K   .  à The motion with the maximum possible binding y ™ {  K¡(›¢ — . energy The maximum efficiency of the energy release by G matter falling into a non-rotating black hole is › ¢!£ . 1.13

1.5

‘Tilted’ spherical coordinates

à Let ¤ -axis be not orthogonal to the orbit plane à à

y<

z¦¥ z

remains the same

#



[-adb



^§

i

– Azimuthal angular momentum

§© ¨

w  [)acb # ª ¬; « § momentum §©¨=­ à

  # ‹

 ^

 §

i & @

 – Total angular

§©¨ &

y< 

z

@ >¯®



G

  1 #$   ; # #   z ¥ ¯ > ® z G   ^ ‹  # [)adb  zn¥ ^ G  \   # [)acb (' y< ^   1 + , + s s t  changes between and , where The angle à s z¥  z ^ ^ ^ [)acb

. The angle between the normal to the s  ^ trajectory plane and ¤ -axis is t 9  ^ 1.14

1.6

Motion of ultra-relativistic particles

1.6.1 Equations of motion

y k “ < k “ ,| while à In the ultra-relativistic limit  ° ° <  y k <° | 

. is the impact parameter at infinity #$ à The equations of motion are

  '<



 ±

1 



\

 '<

<° 

"  

1 

"  ²

<°

 

 h °<    F  à The radial turning point L



1.6.2 Types of trajectories

° ™ °

ˆ‡ ˆ ³



9 , the particle makes a à For min #%$ large number of turns before it flies away to infinity. °

L

à The minimum of F corresponds to (unstable) # at – — motion on a circle of radius šœ›

°

ˆ ‡ ˆ



#

#´$

9 the particle falls into the black à For min #$ hole (gravitational capture) 1.15

7 b/2M 6 5 4 3 2 1 0

1

2

3 x 4

1.16

5

6

7

1.7

Gravitational capture

1.7.1 Gravitational capture cross section

y

à 

max

à If |

< …



qp

|

<

cr

9 with

y ™ 

 qp

à Non-relativistic motion –(µ

#

max

9

#$

9 , gravitational capture takes place

à The angular momentum of a particle moving with ° ° { –fµ , where is the velocity –fµ at the infinity is | an impact parameter

< }{ | | — à 

#$

°

9 qp

cr,nonrel

9

#$

L  —F – µ

à The capture cross-section for a nonrelativistic particle is ¶

°

L  t cr t F — – µ nonrel #$ ° ˆ ‡ ˆ 

9 , and For an ultra-relativistic particle, à cr #$ the capture cross-section is ¶ 9(¢  Ÿ t rel #$ Ÿ

1.17



1.7.2 Critical angle for escape à Not every particle with – ’ – esc flies away to infinity. In addition, it is necessary that the angle · between the direction to the black hole center and the trajectory be greater than · cr

9

F 1 _©¸ b · cr,esc ‹ Ÿ " #%$ The plus sign is chosen for ’ 9 #

 L   #%$ # #%$ F * # #%$ F¹· cr ‘ #´$

# L # s ž L 

à For an ultra-relativistic particle, the critical angle is

_A¸ b ·

cr,rel

‹ 

#$ # The plus sign is taken for

!»› ‘ # # $



"



#$ #  L   3º F ª # #%$

!»› and the minus for # ’ ´# $

1.18

2 2.1

Particles near a Rotating BH Gravitational field of a rotating BH

2.1.1 Kerr metric à The Kerr metric in the Boyer and Lindquist coordinates is

 



"

9!:

(' 

¼ #

¼

  ¼ ¾   #    G ¼ YSZ[   # ½ ^ à : is mass and À ½



Ÿ

:

#!½ ¼

[-adb



^

('N \

 )[ adb     G  ¿ ¼ ^ \ ^ ¾    9š:   # # ½  : is the rotation parameter.

à Scaling property: Besides a global scale parameter G L  ( : ) the Kerr metric depends on only : Á F 

½

2.1.2 Killing vectors The Kerr metric has two Killing vectors

 G D B!CWV B CWV D X  X ' X X

 D B!Cdi B Cdi D X  X \ X X 2.0

The Kerr geometry and its Killing vectors possess the following properties:

j B CWV D is tilted with respect to the section ' const. The tilting angle is different for different

j The event horizon lies at 

:  #

#



and

^ 

 , that is at # # :    ½ ¼

¾

j The infinite red-shift surface  B CWV D  m VnVo 

:

#

:  

irs

½

where

 YSZf[  ^

is an external boundary of the ergosphere.

j The Killing (event) horizon is determined by the  G /    equation   where  B CÃV D  B Cdi D , and /

 ½   # ½ is the angular velocity of the black hole.

j ¼

is everywhere outside the event horizon except two poles. Inside the ergosphere the Killing vector  . field B!CWV D is spacelike, B CWV D irs

‘

2.1

Pseudo Cartesian images of the ergosphere surface ). These images are not and event horizon ( ½ embeddings in Euclidean space. From Pelavas, Neary and Lake, gr-qc/0012052 2.2

Isometric embeddings of the ergosphere surface and ž event horizon ( Ä ).

½

From Pelavas, Neary and Lake, gr-qc/0012052 2.3

2.1.3 Killing tensor à A Killing tensor is a symmetric tensor field B   obeying the equation

 s  B!C  } DOÅ à For a geodesic motion the quantity

Æ



  

B

remains constant along the worldline. à In the Boyer-Lindquist coordinates the non-vanishing components of the Killing tensor B   are

B

s3s

½

 ;

"

9š:

#¼

YSZ[

 ^

@ G

 ¼ YSZf[ G B >I>  ½ ¾ ^

 ¾  ) [ c a b  B  I YUZ[  B h  ½ ¼ ^ Š # ½ ^ #  ¾ [-adb   L

Š )[ adb   B 3h h F ¼ ^ Ÿ # # ½ ½ ª ¼ G



s

2.4



 9

F #



^

Œ

 ½

 LM΂G

2.2

Motion of a free test particle

2.2.1 Integrals of motion à Killing vectors BšCWV D and B(Cdi D qp

y<




 B!CWV D 

¼ #

9š: [-adb
  D

B!Cdi

 #!½ ¼

z¥ y<

9!:

"



conserved quantities

  ( ' 9š: [-adb \ #!½ ¼     ^    ( ' \ -[ adb ^   ¿ ¼ ^  z ¥

y x{

¥ }{

is the specific energy and | à the specific angular momentum of a particle à The Killing tensor qp

Æ

y<

[)adb ½

^



a conserved quantity



z¦¥

[-adb ^



È

   ^

¼ 

à It is often used another form È

z ¥

is

½ y<



YSZ[

 ^

z¦¥L 

 F  ½     SY Zf[  ^  ½ ^ 

 y<   ¼  YUZ`_  YSZ[  ½ ^ ^

2.5



Æ



There exist 4 integrals of motions,
†


 a trivial one,



y

,|

¥ Æ ,

(or È ), and

2.2.2 First integrals of the equations of motion




can be expressed as functions of à Components of the integrals of motion and coordinates and

¼ ( ' ¼  where



\  

z¥

#

  G ¼ > Ê ®  # ‹ É   G ¼  ^ ‹ Ë >¯® y<

y<

^

z¦¥ Œ G ½ # ½ ½ [-adb   < ^  L z¥   L z ¥ Œ G y< y  ¾  ) [ c a b Š  # ½   F F ½ ½ # ½ ½ ^

y<







½¾

Š F





L



z¥ Œ ¾  z¥ ŒG y< L  É Š F    Š  F   È # ½ ½ # z ¥ ½ @   y<  L Ë È  SY Z[  F "  ½ -[ adb ^ ; ^ à The signs ‹ which enter these relations are 

L

independent from one another

k

 these equations coincide with the à In the limit ½ equations of motion in the tilted sherical coordinates. In z  z ¥

 this limit È 2.6

2.2.3 Bounded and unbounded motion à A geodesic worldline is determined by à Consider É

É F

y< 

as the function of

y< 

L

y< z¥ ,

, and È

#

L  z ¥ Œ  h     È  š9 :  ŠF # ª # ½ # z¦¥}L  Œ  y<    !9 : Š È  F È ½ # ½

à The leading term at far distances is positive if y<  case can the motion be ’ . Only in this y  unbounded. For the motion is always bounded

‘

2.2.4 Effective potential à Variety of types of trajectories is wider and their classification is much more involved. We discuss only some important classes of trajectories à Let us rewrite É



as É



Ì

y< 

 Î

G

where

z¦¥ G Ì Š  Í 9 : F  9š: # ª ½ # # ½ # z ¥  L¾  z ¥  F  Î  È ½ # 2.7

L8ŒG

 9Í

y<

à The radial turning points É L y< condition ~ F , where

#

~ ‰

à ~ à ~

#



‰

 are defined by the Í   Ì Î Ì

Í ‹

are effective potentials

:

depends only on dimensionless combinations z¦¥     : , È : , and : ,

½

à The motion is possible only in the regions where y< •  y< … Ž ~ or ~ either à É is invariant under these regions

y< k



y < zn¥ k

z¦¥



,

relating

à In the Schwarzschild geometry the second region y< … Ž ~ is excluded à The limiting values of the effective potentials ~

~

‰

F #

“ L

‹

G

~

‰

F #

y< 

ÏL

½

zn¥ 

9š: #



‰

/*zn¥

there is not more than two turning à For ’ y<  points, while for ‘ there is not more than three turning points 2.8

EFFECTIVE POTENTIAL for a=0. Q=0

6 4 2 0 –2 12

10

8

6 l

4

2

2

8

6

4

18 20 14 16 12 10 r

EFFECTIVE POTENTIAL for a=.99 Q=0

6 4 2 0 –2 12

10

8

6 l

4

2

02

4

6

8

18 20 14 16 12 10 r

EFFECTIVE POTENTIAL for a=.99 Q=20

6 4 2 0 –2 12

10

8

6 l

4

2

2.9

02

4

6

8

18 20 14 16 10 12 r

ORBIT for a=.95 E=.95 l=1.6 x0=5

8

6

6 4

4 2

2 –2

–4

–4

–2

0

2

4

6

–4

ORBIT for a=.95 E=1.2 l=1 x0=.8 1

6 4

0.5 –0.5

6

4

–2

–2

–1

2

0.5

1

2

1.5

–10 –0.5

–5

5 –2 –4

–1

–6

–1.5

–8

2.10

10

15

2.2.5 Motion in the -direction

^

à Consider now properties of the function Ë which determines the motion of a particle in the direction à Since Ë •  only if È

•

 the finite motion with

y< 

^

is possible

‘

à The orbit is characterized by the value È only if it is restricted to the equatorial plane à No nonequatorial bounded orbits with à For È  Ë is positive only if s ‹ -points are defined by

^

^

uÐz¦¥u†…

y< 

’

^

 if and

const . Turning

z ¥  s [)adb

 y<  ^  ½ y<   all the coefficients in É are

à For ½ non-negative, no turning points in , and motion is # unbounded

•

 , there exist both bounded as well as à For È infinite trajectories. They intersect the equatorial plane y< 

 or (for È and ‘ ) are entirely situated in it. The particles with È plane ‘  never cross the equatorial  Ž and move between two surfaces and ^

2.11

^

^

^

2.3

Motion in the equatorial plane

2.3.1 Equations à For particles moving in the equatorial plane one has

  #



y< 

 L h

 š9 : F  # # ½ # ½ ¾ G L¯zÑ¥ y < z¦¥ Ÿ  :  F  9š:  ½ # #  L¯zn¥ y< \  9 : F  9š: ¾  ½  # # h

à These expressions are analogues of equations for a Schwarzschild black hole. An analysis of the peculiarities of motion is performed in the same way as earlier by using effective potential.

2.12

2.3.2 Circular orbits à The most important class of orbits are circular s orbits. The radius of a circular orbit can be found # from the equations

É F

#

sL





G



É



# Ò +ÊÓ Ò to obtain the à One can also use these equations Ò Ò z y<

expressions for circ and circ as the functions of the radius of the circular motion [Bardeen, Press, and # Teukolsky (1972)]

y<

#  F  # # ‡ :

‹ F  # #



 š9 : ˆ : # ÕÔ F # ˆ#  :

#

‹

½ ‡

‡

:

#L

G

 ¯ > ® ½ # ‡ L 9 : z  ½ ‡ # L½  circ ‹ 9 : # ½ zn¥ # >¯® à The upper signs for direct orbits ( ’  ), and the zn¥ lower signs for retrograde orbits ( ‘ ) circ

‹ 9

:

à The angular velocity at the circular orbit is

Ö

circ



\ ('

‹

#

 ‹

2.13

‡

½

:

‡ # : #

2.3.3 Last stable circular orbits à Circular orbits can exist only for those values of for # y< which the denominator in the expressions for circ and z circ is real, so that

#

 

ˆ :

#

‹ 9

‡

½

: #

• 

à The radius of the circular orbit closest to the black hole (the motion along it being at the speed of light) is

#

photon

9!:

 YSZ[ ;

9

ˆ ¸ ×3YEYUZ[Ø

Ô :

& @

½ ˆ

: , while à This orbit is unstable. For  photon ½ # Ÿ

: (direct motion) or photon : for : ½ # photon # (retrograde motion) y< •

are à The circular orbits with ’ photon and # # unstable. A small perturbation directed outward forces this particle to leave its orbit and escape to infinity on an asymptotically hyperbolic trajectory à The unstable circular orbit on which

#

bind

9š:

Ô ½



 9š: >¯® F :

2.14

Ô

y< ½

circ

L

>Ê®



is

à These values of the radius are the minima of periastra of all parabolic orbits. If the orbit of a particle, which comes in the equatorial plane from infinity where its velocity – µ Ù — , passes by the black hole closer than bind , the particle is captured.

#

à The radius of the boundary circle separating stable circular orbits from unstable ones is

#

bound

: Ú

ˆ

L ˆ ML Œ Ü Ô ˆ   Ê> ® ŠF  Û > F  Û >  9 Û  Û

where

Û >

 L h  L hŒ h ¯ > ® ¯ > ® : ŠF  : : >¯®  F *  F " ½ ½ ½ ˆ   L   : Û F  Û ¯> ® > ½ y< 

L

2.3.4 Motion with negative

Inside the ergosphere the Killing vector B WC V D and the y< y<
  B CWV D can be specific energy is defined as  y< negative. Orbits with ‘  make it possible to organize processes that extract the ‘rotational energy’ of the black hole. Such processes were discovered by Penrose (1969). 2.15

2.4

Motion off the equatorial plane

2.4.1 Non-relativistic particles à We consider only special type of motion off the equatorial plane when particles are moving so that the s polar angle remains constant, . For this motion

^

^  ^ sL Ë G Ë F



  ^ ^ ÒÒ w Ó s s Ò

t , and à If we exclude trivial solutions: Ò  , s  ^ ^

t 9 , the relations between the integrals of motion ^

can be written in the form

 y<  L

 )[ acb F ª ½  y<  L

  UY Z[ È F ½ s à The motion with constant y< ^ ^ (infinite motion) when ’ zÑ¥

^

s

ª ^

s

is possible only

à Nonrelativistic particles moving at parabolic velocity L z¥ L F¹– µ  and zero angular momentum F  is a special limiting case. Such particles fall at constant ^ and are dragged into the rotation around the black hole 2.16

2.4.2 Ultra-relativistic particles and light rays à Another important case is photons moving with ° z¥  y<

const. The ratio remains finite and equal °  ^ [-adb to

½

^

à The null vector Ý geodesic is

Ý





tangent to the photon’s null





F# ¾ ½

à By changing  by out-going photons

 G



G G L  ½¾

one obtains a congruence of

à These two null congruencies are known as principle null congruences of the Kerr metric. They are geodesic and shear free and satisfy the following relations

Þ*ߚàšáãâåä à

Þ ßšàšáxä

à á Ý Eæ Ý Ý 

is the Weyl tensor

à The principle null vectors in the Kerr geometry also obey the relation

where

ç

   D B CÃV  Ý ‹ Xç Ý  9 X #  ¼

 m nV V " 9š: 2.17

2.5

Gravitational capture

2.5.1 Capture of non-relativistic particles à Consider the gravitational capture of particles by a rotating black hole

°Uè

of capturing a à The impact parameter nonrelativistic particle moving in the equatorial plane is

°è

‹ 9š:

Ô 

– µ

:

½

à For particles falling perpendicularly to the rotation axis of the black hole with : the cross-section ¶ ½ area is

è

Ÿ



œ9ét F

– µ

L

:



à For particles falling parallel to the rotation axis (for

: )

½

¶

<° ê

ˆ

»(›

–(µ

:

ê G

2.18

Ÿ

¡ët



–µ

:



2.5.2 Gravitational capture of ultra-relativistic particles à Consider now ultra-relativistic particles à For : and photons moving perpendicular to the ½ axis of rotation the cross-section is ¶

è

 Ÿ ˆ

9  t:

à For photons propagating parallel to the rotation axis of the black hole we ¶ have

ê

ˆ ˆ 

9  tì:

à A rotating black hole captures incident particles with lower efficiency that a nonrotating black hole of the same mass does

2.19

3

Fields in Black Hole Spacetime

3.1

Introduction

à Many problems of black hole physics require detailed knowledge of propagation of physical fields in the black hole geometry. In particular, they include

j Radiation from objects falling into a black hole j Gravitational radiation during a slightly non-spherical gravitational collapse

j Scattering and absorption of waves by a black hole j Gravitational radiation from coalescing compact binary systems

j Analysis of stability of black hole solutions j Quantum radiation of black holes à For these and other relevant problems it is often sufficient to consider physical fields (including gravitational perturbations) in a linear approximation and to neglect their back reaction on the background black hole geometry. 3.0

3.2

Scalar massless field in the Schwarzschild metric

3.2.1 Field equation à The electromagnetic field and gravitational perturbations are of the most interest for astrophysical applications. Both of them are massless and have spin à To simplify consideration it is instructive first to consider a zero-spin massless scalar field and to discuss later effects connected with the spin of the field. We also at first consider a simpler case of a non-rotating black hole à A massless scalar field evolves according to the Klein-Gordon equation

L     îñð   ï Ÿ

 t À  F  m ¯> ® F  m >¯® m X X where m is the determinant of the metric m   and À a íî

L

scalar charge density

3.1

3.2.2 Spherical reduction à In a general spherically symmetric spacetime with Góò G L

 ) metric ( F

¿

 

 ô   õ

ÎNôqõ 

#

 0/  G

one can decompose a general solution into the spherical modes


ö A' G L F î÷öMø Mö ø G L

# F \ ^ # ù ö8ø G L F \ are the spherical harmonics. where ^
öMø ù obeys the two-dimensional wave equation à  í

F Î

LŽ

F

 í

 ~

ö L
öMø



Ÿ

ö8ø

t€ú

L  ôqõ Œ Š  ô Î Î õ F is the >Ê® Ê> ® X í X 

two-dimensional “ ” operator for the metric ÎNô.õ , and

~

ö

F 

û û  #

L



à For the Schwarzschild metric 3.2

 í

#  í

#

# %# $h # #

3.2.3 Radial equation and effective potential à The geometry is static. One can decompose a solution them into monochromatic waves


=ü ö
ö

à Function ý F




ö G Ö Lþ Ž J ü V

ý F #

GÖ L is a solution of the equation #

 ö L@
ö GÖ L Ö  ~ F ý F

 X ÿ  # # ; X #

à ÿ is the tortoise coordinate

#

#

ÿ

and ~

ö F

#

L

#



#$




Z

#  # $

is the effective potential

 const

ö

~ F #

L

 1 #%$ #

&

L

F  û û  ; # ö

 # $h #

@

L

à Maximum of effective potential ~ F is near the # ˆ : location of the unstable circular photon orbit (

#

3.3

).

(2M) 2 Vl 1 0.8 0.6

l=2

0.4

l=1

0.2

l=0 00

1

2

3

5 r/(2M)

4

(2M) 2 Vl 1 0.8 0.6

l=2

0.4

l=1

0.2

l=0 0

-4

-2

0

2

4

6

r* /(2M)

à The radial wave equation is similar to the quantum mechanical equation for potential scattering à Waves with  Ù

#´$

are transmitted

à Waves with   reflected

#$

are partly transmitted and

à Waves with  

#$

should be completely reflected 3.4

3.2.4 Basic solutions à Basic IN and UP solutions  Ž Jü G þ 
ö ÿ G Ö L + ü ý F  J L þ #  FÖ +   ¿  ¿  ü J ò ) L þ ò  Ö 

 F G L
ö þ J ü  G +  ý  F ÿ Ö  # +

ÿ k



Ž Jü G # k L þ “ ÖF ÿ  + # Ž Jü G L þ k “ ÿ  FÖ + # k “ ÿ  #

à The complex conjugates of these solutions are also solutions, known as, OUT- and DOWN-modes





 the Wronskian à For any two solutions > and     
G
L



  ÿ   ÿ is constant. F >  >  > # # à Using Wronskians one gets



ò 

ÖF L 

u  u • à ¿

u

uÐ

u  Ðu éG uÐò  uÐ uvò uÐ

 ¿  ¿  L G ò  Ö L L L Ö Ö

 F Ö F   F  F ¿ ¿  ¿  L ò L
ö G
ö L Ö Ö Ö Ö

 F ý ý  9  F 9 F ¿ 
 G
L

qp F ý ö ý ö are liner independent 3.5

“

3.2.5 Interpretation of basic solutions



! 

Ž



! 

! Ž

Ž

in-mode

! Ž



up-mode ! 

Ž



! Ž

Ž

! 

out-mode

! Ž

down-mode

3.6

3.3

Evolution of the scalar massless field around a non-rotating black hole

3.3.1 Retarded Green’s function à The retarded Green’s function is a solution of the equation





ö L@   ~ ˜ ' F X ÿ X  # ; #X X

G G)')L

F ÿ ÿ # ##" G G' obeying the condition ˜ F ÿ ÿ  # # "  µ G G L ˜ ý ret F ÿ ÿ Ö ˜ ret F 's & # ## " # ret

'

$ F 

' L

' L "

$ F ÿ  ÿ

#

L

#%"

' '

 for ‘ . " " Gÿ G)')L)þ J ü V (' ÿ ## "

G GÖ L ret ý ÿ ÿ ˜ is, in fact, a holomorphic function of F à Ö Ö s #   # Ö " for Ö > > ’  à By making the inverse (Laplace) transform one get

G G)' L ˜ ret F ÿ ÿ

# # " 9xt



 J6 µ Ž  J6 ˜ý µ 3.7

Ž Jü  G G L þ V Ö ret F ÿ ÿ Ö # # "

3.3.2 Green’s function representation à ˜ý

ret

obeys the equaion

 

 ö L @ ý ret ÿ G ÿ G Ö L Ö  ~ F ˜ F    ÿ # # # " ; # G GÖ L ret ý can be written as à ˜ F ÿ ÿ # # " 
ö ÿ G 

ý F ý˜ ret F ÿ G H=G Ö L  Ö #H=G L Ö
  ö Ö

 # 9 F ý F ¿ 3.3.3 Analytical properties

L ÿ

$ F  ÿ # # " Ö L
ý ö F H=G L
ö G ý  F ÿ #

Ö L G ÿ H # ‘ H L G Ö ÿ ’ #


ö
ö à ý and ý  are holomorphic functions of complex Ö G L L  Ö Ö

 F has isolated à Wronskian F)(+* ‚ , 9 ¿ zeros in the lower half-plane. They are symmetrically distributed with respect to the imaginary Ö -axis. At G GÖ L ret ý ÿ ÿ these values ˜ F has isolated poles. # # " à Analysis shows that it is necessary to introduce a branch cut along the negative imaginary axis in order
ö to make ý  a single-valued function.

3.8

C

>

>

<

CL CHF

à The radiation produced in response to a perturbation of the black hole can be divided into three components, in accordance with the contributions of different parts of the deformed contour in the lower half of the Ö -plane: 1. radiation emitted directly by the source 2. exponentially damped quasinormal-mode oscillations [contribution of the poles of the Green’s function] 3. a power-law tail [contribution of the branch cut integral]

3.9

3.3.4 Quasinormal modes à The scattering resonances (which are the quantum analogues to quasinormal modes) arise for energies close to the top of a potential barrier - . Ö s ™

™ ~ ö /1032 ˆ‡ ˆ :

û

 9

à This approximation for the fundamental mode is poor for low (the error is something like 30 percent for û

9 ) but it rapidly gets accurate as increases

û

û

à Imaginary part (the lifetime of the resonance) is

‡ ˆ ö >¯®  ~ 465 Ö s ™ ™    ÿ Å 9 9~ !: Ò # Ò + +87:9<; Ò Ò Ò which is accurate to Ò within 10 ÒÒ percent   ö 

à Beam of null rays circling in the unstable photon ˆ

: contains cycles. For slight orbit at û # 465 Ö s perturbation its decay rate  à Schwarzschild black hole is thus a very poor u u ™ ™  > Re Ö?> Im Ö@> oscillator. Its quality factor is = . û  B A !) (Compare with the typical value for an atom: =  3.10

3.3.5 Late-time behavior à The power-law tail is associated with the branch-cut uÖ u integral. The main contribution gives the : ‘ ‘ part of that integral ö D L C L Ÿ  ö : F ÿö  ÿ G G ) ' L L F9  9 > ret ÿ > û

F ˜ õ F ÿ ' #  # "h LCEC Œ  # ##" ŠF9 

û

à If the source of radiation falls down beyond the potential barrier the damping of its radiation that is seen by a distant observer is not purely exponential. The late time behavior of the field is

î

' Ž C ö  h D 

à This power-law behavior is connected with scattering of emitted radiation by the “tail” of the potential barrier (by the spacetime curvature). à Price (1972a) put his conclusions in the following form: “Anything that can be radiated will be radiated.” Consequently, a black hole rids itself from all bumps after it is formed by a non-spherical collapsing star.

3.11

3.4

Wave fields in the Kerr metric

3.4.1 Electromagnetic waves and gravitational perturbations in the Kerr geometry à Teukolsky (1973) demonstrated that equations descibing electromagnetic waves and gravitational perturbations in the Kerr metric, can first be decoupled and then solved by separation of variables. We describe here only the scheme and main result omitting details and long calculations. à Initial equations are of the form F ôqõHG õ  F ôqõ ò is a covariant differential operator, and and à ¿ are collective tensorial indices. à For the electromagnetic field  F      I  I   I I

¿

¿

¿



à For gravitational perturbations ߚàLK ߚà ß ß K    gK ßß F  J 

 I I  I I K K K à ß ß ß ß à K ß`à ß  ß   L  ß à  I I  I I  I I  m FI I 3.12

3.4.2 Decoupling. Teukolsky equation à Decoupling means that there exist three operators  M ô , MON õ , and MQíP ) such that



G

 G

 F q ô õ MRíP MSN õ G ˆ  G u u ‹ 9 ‹ 9 , and is the spin of the M ô

9 ‹ à  ‹ G massless field ô .

õ G õ constructed for any solution of à Scalar M · M N F ôqõ G õ  obeys the equation í M P

M·



à In the Boyer-Lindquist coordinates and for the so-called Kinnersley tetrad the Teukolsky equation is TW Y[Z]\[^@_ c `#^ab^ d ^ ` ^fehgjik^Llm n npo ^ TW cwv T `c ^ d ^ u d n n _ ehgji ^ l \ ^ t n s n TW Y `tZ\ c d r d u d € ~  n e{gji}l n e{gji|l l l n n T W d `#^a r  Z [ \ ^ d‡~€ c d \ d ` eˆl n _ ‚ ƒ… npo TVU TW X

3.13

_ c a Z

`#c r \

^

TW

n _ n oknts k Tbxzy T W d n \ n TW e†l d m n _ ‚„ƒ… ehgji ^ l nts T W X ‹1Œ ^ ^Šl d‡ a ƒ…[‰ q

3.4.3 Field restoration à How to restore a field from a solution of the decoupled equation? Wald’s procedure (1978) à Scalar product of (complex) tensor fields · ô and

G L F¹· G 

G ô

  ·  ô Gëô ª  G Œ  G Œ G G GF G L à Action Š for is Š > F ‡

 m

à =  is an operator which is conjugated to an operator = with respect to the scalar product

à The operator

F

G L G L F¹· = G FŽ=  · G

is self-conjugated

F



F

à Equation obtained by conjugation is F ôqõ  í M ô  M N  õ MP  Ž Ž í ¼ í > M à Direct calculations show that MP  íP  ‘  M ‘ Q M M

 For any solution of scalar equation à  G the tensor function M ô M ô  M‘ is a solution to the field equation. à All such solutions (up to possible gauge transformation) can be presented in this form. 3.14

3.4.4 Separation of Variables. Spin-Weighted Spheroidal Harmonics à Teukolsky equation can be solved by separation of variables Ž Jü ü G L G  L þ M ö ø V ME’ F Ö M Û öMø F \

# ü ^ G L 8ö ø F \ are the spin-weighted spheroidal à MÛ ^ harmonics Ž  ü ü G L L L Jø i M ö ø M ö ø MÛ  F \ F 9xt >¯® M”“ F e ^ ^

à The angular problem reduces to one of solving

[)acb ^





^

[-adb



^ 9



“

{ ^



;½

Ö   YSZf[

{    [-adb ^

^

 Ö  9 SY Z[  ½ ^

YUZ[      @ y “   SY Z`_    ^  [-adb ^ ^ Mö ø L M “ F is Sturm-Liouville eigenvalue problem: à G Œ ^ Š  t regular on the interval à Eigenfunctions form a complete, orthogonal set  { (enumerated by ) for each combination of , Ö and

û

½ L M ö ø F ø coincide with à For  and  functions M “ ½ L ^ö F YSZ[ the associated Legendre polynomials , ^ 3.15

3.4.5 The Radial Equation

ö8ø M ’ à Radial function c v T • • \

c TbxyB•„– • \

Y— ^ d ~  c Z \ d r _ ‚

Z\ ^ _ ` ^ a<™ d `„ ž öMø à Two solutions M’ —

obeys a second order ODE

œ

a˜— _

q€‚

– X ‹ \™ šd › m

› œ Ÿ d ~ „`  ™ _ ` ^ ™ ^ d  Z  _ a u Ž MD’ ö8ø and are related

à Introducing a new dependent variable  L  ¾ M  ME’ ö8ø M¡  ö8ø F  >¯® ®

#

½

the radial equation can be written in the form

 ¡M   8ö ø öMø M¡  öMø

  ÿ   M~ # 8ö ø is à Effective potential M ~ — ^ d ~  Z \ d r a˜— _ ‚ Z\¦ž˜™wa Z\ ^ _ ` ^ c \ § X _ Z\ ^ _ ` ^ a ^ à ÿ is the tortoise coordinate TE¢p£¥¤

X

#

c

• ™  \ d › a d § d § ^ • €q ‚ a^ \€¨  Z \ d r a  \ ^ _ ` ^  ¾ 

#

Z

ÿ

#



½

à This potential is complex and it depends on Ö 3.16

#

k ÿ à In the asymptotic regions ( # M ~ öMø has the form 

Ö F Ö  9   L”G # ª  G



G L M ~ ö8ø F Ö #

ª M « ¯

F



 :

à # Kerr black hole

‹

“

M

ª

L F

  #

¬« G

 °

½+±

ª

L

) the potential

ÿ k # #

“

ÿ k

©

“ 



{ /®­ G

Ö 

is the surface gravity of the

à Two linearly independent solutions have the L µ k “ M .³:´ ¶ Ö µ ÿ F‹  at asymptotic behaviour  p # ² ¾ ‰ M .³t´ L k  µ  F ‹  ª µ ÿ at µ ® ± 3.4.6 Modes

{ Ö Ü , of à Modes are characterized by the set Ú û¯ ‹ ·. quantum numbers, where the spirality , Ž ! Ž à IN-modes vanish at

à UP-modes vanish at

! Ž

à OUT-modes vanish at

but not on



but not on

Ž ! 

but not at !  but not on à DOWN-modes vanish at 3.17



and

3.5

Effects connected with a black hole rotation

3.5.1 Wave evolution in the Kerr spacetime à Emitted radiation consists of the following three components: 1. An initial wave burst that contains radiation emitted directly by the source of the perturbation 2. Exponentially damped ringing at frequencies that do not depend on the source of the perturbation 3. A power-law tail that arises because of backscattering by the long-range gravitational field à Quantitative differences between non-rotating and rotating black holes are of the most astrophysical interest à For nonzero angular momentum, , the azimuthal ½ degeneracy is split. For a multipole there are û ° · distinct modes that approach each consequently ¸ û k ¹ Schwarzschild mode in the limit . These modes ½{ … { … correspond to different values of , where º

û

3.18

û

k

»

) complex à In the limit of extremal black hole ( ½ frequencies of quasinormal modes possess the following properties ÃÀÄ ¾À¿ Im is almost constant  ™ Á X ¼ ½ for  as Æ Ç È Å Re ™ Á increases monotonically ÃÄ ¾À¿ Im Ì Á Ç Í ½ ¼ for É Ê Ë as Æ Ç È Å Re Ì Á Ç ËšÉ Î[Ï à Some quasinormal modes become very long-lived for rapidly rotating black holes. This could potentially be of great importance for gravitational-wave detection

3.19

3.5.2 Gravitational radiation from a particle plunging into the black hole à Consider a particle moving in the gravitational field » of a black hole. When Ð Ñ Ñ , this problem can be viewed as a perturbation problem. The radiation emitted by the particle falling into a black hole is one of the most important astrophysical applications of the perturbation equations à Total energy Ò Ó emitted by the particle is » proportional to Ð [Ô Õ à The figure shows Ò Ó for different values of the × Ø × » Õ angular momentum of the particle Ö and for different values of the black hole rotation parameter Ù . It is assumed that a particle has zero velocity at infinity and is moving in the equatorial plane.

3.20

à For a rotating black hole the curves are clearly × asymmetric for positive and negative Ö » × curves have minima at negative Ö à Ù Ú ÛtÜÞÝ × à Positive values of Ö stands for co-rotating particle, and negative values are for counter-rotation. When an initially counter-rotating particle comes close to the black hole, it will be slowed down because of frame-dragging and it radiates less gravitational waves. An initially co-rotating particle is speeded up, and the radiation increases à When the initial angular momentum is large, it dominates that of the black hole, and the radiated × energy increases with ß Ö ß 3.5.3 Superradiant scattering à In the presence of ergosphere some of inpinging waves can be amplified. This effect is known as superradiance. The condition for superradiant modes » å„æ Ø Ð Ù:Õ#ä is à Ñ Ð áãâ à The maximum amplification of an incoming wave is 0.3% for scalar, 4.4% for electromagnetic, and an impressive 138% for gravitational waves 3.21

4

4.1

Black Hole Electrodynamics

Electrodynamics in a homogeneous gravitational field

4.1.1 Introduction à Black hole electrodynamics is the theory of electrodynamic processes that can occur outside the event horizon à At first glance, black hole electrodynamics is quite trivial. Field of a stationary black hole (of mass È ) is determined by its charge ç and rotation parameter Æ à In astrophysics, the electric charge of a black hole cannot be high. The induced by rotation magnetic field must also be very weak: The induced dipole magnetic moment is èêé}Ê ç®Æ à However, if a black hole is placed in an external electromagnetic field, and if charged particles are present in its surroundings, the situation changes dramatically, and complex electrodynamics does appear à The case important for astrophysics: external magnetic (not electric) fields and rarefied plasma in which a black hole is embedded. In this system a regular magnetic field arises, for example, as it gets cleansed of magnetic loops which fall into a black hole. A regular magnetic field can also be generated in an accretion disk by the dynamo-action 4.0

4.1.2 Electrodynamics in the uniformly accelerated frame à Black hole horizon is a one-way membrane à Let us consider first a uniformly accelerated observer in a flat spacetime ë+ì Ø í ëBî ë„ð ëòñ ëôó Ô Ô|ï Ô}ï Ԛï Ô à In the Rindler coordinates î Ø š õ öø÷Eùûúü)ý þLÿ
ð Ø õòöøúü ý þŠÿ ñ Ø õ Ø

 ý 

ó Ø ý



the accelerated observer is located at . and þ are the acceleration and proper time of the observer ñ î ß ß à Rindler coordinates cover the right wedge of the Minkowsky spacetime. The null surface

Ø î í ñ Ø Û play the role of the event horizon. The observer cannot get any information from the region

Û lying beyond the horizon à The electrodynamics in the accelerated frame is described by the standard Maxwell equations   Ø    Ø   Û

4.1

T

Z

à Since nothing what happens beyond the horizon affects the information available to the observer, let us  simply assume that  vanishes in this region.  Ø   ü ÿ into the Substituting the anzats Ö  Maxwell equations one gets   Ø }ü   ÿ   ! !" #%$ &ü ÿ¶Ø 
 ï Ö  Ö ï Ö  Û   Ø    ü ÿ and à Ö

 Ø

'   "  $ ü& ÿ  

 " Ø Since Û this current is propagating along à  Ø í   "  Ü Thus the the horizon. One also have  fictitious current is conserved until real charged particles cross the horizon 4.2

4.1.3 Fictitious horizon currents for an accelerated charge à Suppose a point-like electric charge ( is moving with ð ý a constant acceleration in the -direction. Using a standard solution of this problem in the form of the Lienard-Viechert potentials, one can write the non-vanishing components of the field ,ü + ý ÿ ï ï ) * Ø ä ( Ô Ô .0/ Ö ý Ô

 3 + 2 ( ý .4/

 6 + ( Ö Ö ý .4/ Ô Ô . Ø 7)ü,+ í ý ÿ  ý :9 Ô - ï Ô Ô Ô ï Ô8 ó 2 Ø î í ð Ø 2 + Ø ñ Ô Ô ï Ô  )1 Ø í

 *51 Ø

à The only non-vanishing component of the fictitious

Ø Û is current at the surface

* Ø í

à Thus <

â

Ø

>=

Ø ä

' ä(  ý ;ü + ý ÿ ï Ô Ô Ô Ô ' * Ø í ( '  ý ü,+ ý ÿ ï Ô

à The integral of < over the total 4.3

ñ

Ô

í ó

Ô Ô

-plane is

í

(

4.1.4 Charge in homogeneous gravitational field à Flat metric in the Rindler coordinates ë+ì Ø í õ ý ë þ ë õ ë  ë  Ô Ô Ô Ô ï Ô ï Ô ï Ô Ø A$  @ ? @ of the charge (  à Field ? + õ Ø í ý Ô ï Ô ï ? @ ( ü,+ õ ý ï ï Ô Ô Ô

at rest is

ý

Ô

í kõ ý Ô Ô Ô ;ü B Ø DCòÿ ' ä à Maxwell’s equations in Rindler metric " Ø F+ Ó E el E " Ø ü)ý õˆGÿ H

+ Ø ý Iõ  @ Ó E ? @ E , and el â Ø â  J Ó Õ à Ó E are orthogonal to the horizon and < ÿ

à Let the electric charge ( be slowly taken further from the horizon and after return back.The conservation law

KML â KON N Ø < ï Û

P Ø þ

can be integrated with the result 1 Ø ýR Ø â IS ý <

4.4

Û

ï Q

ùQü ý õkÿ

1.4 1.2 1 0.8 z 0.6 0.4 0.2

–0.8

–0.8

–0.6

–0.6

–0.4

–0.2

–0.4

–0.2

0

0.2

0.4

0.2

0.4

r r

0.6

0.6

0.8

0.8

0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7

Electric field of a point charge in a homogeneous gravitational field and fictitious charge density at the horizon 4.5

4.2

Membrane interpretation

4.2.1 Maxwell’s equations in (3+1)-form à Thorne, Price and Macdonald book "Black Hole: The Membrane Paradigm"   Ø í U $WV  with Consider observers with T T à U Ø Yü X Gÿ [9 Ô being the lapse function Ò ZÕ ?  Ø Ø a b ^ ]  \ Û zero angular momentum à T

S`_

à All the objects are projected into 3D ones by the  Ø d  ï    T T projector c  à We denote: e (electric field), f (magnetic field), +Ig (electric charge density), h (electric current density), ikj (unit vector in the direction of \ ^ ), and l the norm S  d Ø ü X®ÿ [9 öø÷ p ù S`o _ ^ \ ? Õ Ô of : l m n

SA_

SAS

à Maxwell’s equations can be written as q Ø r +IgH q Ø e f Û

uU ' q s tü U ÿ|Ø h ï y z e | f v v wHe x ï { q s ütU ÿ|Ø í ' e v wfx ï y z f | 4.6

U Ø üYX

à and

ÿG [9

Ò ÕZ? à

Ø

Ô is the lapse function, }

d

å

Ø

à l

i3j , S

ä~ Ù üå ÿ í öø÷ ù o Ô ï Ù Ô Ò Ù Ô

Ø d VS SAS

is the angular velocity of rotation (in time  ) of ZAMOs

y z e m

ü

}

q ÿ e

í

ü

e

q ÿ }

à This Lie derivative describes how the vector e varies with respect to the field } . A dot denotes q differentiation with respect to  and is the three-dimensional (covariant) gradient operator in the curved “absolute” space à The first two equations have a familiar form, while that of the latter two equations is slightly unusual

U

à has appeared because the physical time differs from the global “time”  à “Lie-type” derivatives correspond to total derivatives with respect to the times of e and f , respectively, with the motion of locally non-rotating observers taken into account 4.7

4.2.2 Boundary conditions at the event horizon à Horizon is a surface of infinite gravitational redshift ü U ‚ ÿ Ø í ƒ „ ƒ â and is the surface gravity à At € à Near the horizon the metric is ë+ì Ø ü v ƒôÿ ë U Ië … ëM† Ô Ô Õ Ô Ô ï Ô ï l Ô Ô

…

à is a proper distance along the horizon from the north pole toward the equator

‡ˆŠ‹ ‰ Œ Ê Ž ‘ Œ ˆ ŒM Œ“’ ”

‡ˆ• ‰

Œ ˆ Ê Œ MŒ  “Œ – ”

— ˆ

Œ ˆ Ê D™ ˜ š Œ >Œ  ŒO›

à Macdonald and Thorne (1982) formulated the conditions at the horizon as follows: N œ  ⠄ m Ó < 1. Gauss’ law e U „ œ í %žŸ í ^  „  â _ Ô 2. Charge conservation law h V œ U   s „ f ¡ f â m ¢¤£G¦ ¥ § â 3. Ampere’s law œ U â â  â e ¡ e m ¨ 4. Ohm’s law  v Ø C â m Õ ÝÝ Ohm is the effective surface à ¨ resistance of the event horizon 4.8

4.3

Electric field of a charge near BH

à The electric field of a point-like charge ( at rest at å Ø åª© o Ø , Û near the Schwarzschild black hole was found by Linet (1976) and Lèute and Linet (1976)

' ü; ÿ|Ø í å å © ?  ( $ V ? ¨ Ô

Ø

üå í

ÿ

~ c

ü å«© í Ô ï ~ Ø üå í

~

à The electric field Ó E

¯ Ê

±³²´° ±«š È

~ ï

¨

c

ÿ í üå í ÿ ü 媩 í ÿ… í Ô ä ~ ~ ~ ÿü «å © í ÿ@í  òö¬o ~ ~ Ô Ø

Ô

öø÷ ù Ô

o

"

í U H d

E®­ ? V ­ is ± ² Ë È ¶ È ·D%¸ ¹Mº » µšË

±½¼¿¾À± Ë È Á ¾À± ² Ë È ÁË È š ·D¸%¹Mº ļ ± »kà ¶ Ë È

Ë ¾Å± ² Ë È

Á#·D¸%¹Mº ‡ Æ ‰

¾À± ² Ë [Ï È Á ¾ µšË Ï[È Î ± Á ÈÇ š ¹ÊÉÌ˺ ‡ ‰ Í »kà ¶ ° where i Î j and i Ï j are unit vectors à i Î j and i Ï j are unit å o

vectors along the directions of 4.9

and , respectively

à Electric lines of force intersect the horizon at right angles. The total flux of E across the horizon is zero (the black hole is uncharged) à The fictitious charge surface density at the horizon is ü òö opÿí ü å © í ÿ“òö¬o ï ' Ø ( ~ Ô ä ~ | â w ü ³ò¬ö okÿ < Ð  å© å© í ï ' ~ |Ô w å Ñ

å%© í

ä ~ from the horizon, the à At a distance lines of force become practically radial. The field å strength tends to (#Õ Ô . Outside a narrow region close to the horizon, the general picture is almost the same as for a charge placed at the center of the black hole 4.10

4.4

Black hole in the magnetic field

4.4.1 Killing vectors and Maxwell fields à A Killing vector in a vacuum spacetime generates a solution of Maxwell’s equations [Papapetrou (1966)]   Ø \ª   í \Ò   Ø í \Ò   ä

  Ø í \    Ø   ä Û à The definition of the Riemann tensor implies í Ø í

\Ò    ž

\Ò  ž  

\½Ó ¨ Ó  ž

à Permuting over the indices Ô , Õ , and < , adding and using the symmetries of the Riemann tensor one get Ø

\Ò    ž

\½Ó ¨ Ó ž 

à By contracting the indices Õ and < we obtain    Ø

\



¨

Ó

\ Ó

 Ø   Û , à Hence in a vacuum spacetime, ¨ Ó associated with the Killing vector \ satisfy the Maxwell’s equation 4.11

4.4.2 BH in a homogeneous gravitational field à Let us construct a solution describing a test homogeneous at infinity magnetic field in the Kerr geometry [Wald (1974)]. Let us introduce two fields

 ^ Ø í  ^ Ø í ^ ^    \ \         ä ä V_ V_ SA_ `S _  ^  ^    à At large distances V _ , while SA_  

asymptotically becomes a uniform magnetic field à For any two-dimensional surface black hole   ë  Ø < Û Ö

X

surrounding a

for both fields. Thus the magnetic monopole charge vanishes for both solutions  Ø Ø   z   z ) à One also has ( × ^ V 

_

ë  Ø í Ð   ^    < ~ Ö × V_

ë  Ø  ^     'ÒÙ ÙM~ < Ö × SA_

à The axial Killing vector \ ^ S`_ generates a stationary, axisymmetric field, which asymptotically approaches a  Ù ~ uniform magnetic field, and has electric charge M 4.12

à The timelike Killing vector \ ^ V _ generates a stationary, axisymmetric field, which vanishes at infinity í and has electric charge ä ~ à for a neutral black hole the elecromagnetic field which asymptotically approaches the homogeneous magnetic field Ú is given by the vector potential Ø  ?Ö

'

7\ ^  ï ä#Ù \ ^ V  8 Ú SA_ _ ä

à The electrostatic injection energy per a unit charge calculated along the symmetry axis is

Ü  \ ^ _ Ø í ?Ö Ú Ù V _ÝÎàÞ ß Carter (1973) proved that Û =const over the event Û Ø

horizon à A black hole immersed in a rarefied plasma will accrete charge until Û vanishes. The resulting black Ø ä Ú  Ù ~ hole charge is á The vector potential for such a black hole is Ø '  ? Ú \ ^ S`_  ä

4.13

4.5

Mechanism of the power generation

4.5.1 Potential difference à The potential difference between the (north) pole of the black hole and its equator as measured by an  P  â \ ^ ^ \ ï stationary observer V _ à SA_ is

Ø ü å æ Do Ø ÿ@í ü å æ Do Ø 
ÿ Ò Û Õä

  Ø ' d ^ ^  ä ã \ \ ï á6å ï ? Ú w V S V_ ä `S _æ d ü å„æ ço Ø ÿšØ d ü å„æ Do Ø Û VS S`S d ü å æ ço Ø  ÿ}Ø í d ü å æ ço Ø # Õ ä Ù VS S`S

Ø ' üå æ ÿ ü â í Ô ï Ù Ô á Ò Ú ä

d

á6å Û

ÿ|Ø



Û ÿ Ø }

Õ#ä á å

SAS | å æ

Ô ï Ù Ô

ÿ

à For an stationary observer co-rotating with a black

Ø â vanishes. For á å è á there is a hole Ò non-vanishing electric potential Øba a conducting surface with à Black hole horizon â Ø  v Ø C the effective surface resistance ¨ Õ ÝÝ Ohm. Ø a b A rotating conducting sphere in a magnetic field a unipolar inductor 4.14

4.5.2 Black hole magnitosphere and efficiency of the power generating process à Astrophysical black holes are surrounded by plasma. The conductivity of the plasma is so high that the electric field in the reference frame comoving with the plasma vanishes, and the magnetic lines of force are “frozen” into the plasma. The electric and magnetic fields in an arbitrary reference frame are perpendicular Ø Û to each other (degenerate fields): e f à It is usually assumed that the system (a black hole, surrounding plasma, and the electromagnetic field) are stationary and axisymmetric

P 

à In the reference frame of an observer Ö with the plasma the electric field vanishes   P  Ø Û Ö

comoving

à This property is also valid for any frame which is P  moving with respect to Ö with the velocity along the magnetic field. Let us choose special solution of the above equation which meet the symmetry property

P  â \ ^

V_

ï á6å 4.15

ü å çokÿ 

\ ^

S`_

P 

P 

can à Vector Ö is always timelike, while the vector Ø â be spacelike. In particular if á å è á it happens near the event horizon à In the force-free approximation, the rotational energy of the black hole is extracted at the rate ë ü v ÿ ü â í ÿ â ö3÷ ù Ô o é Ø í ~ Ô Ø á å á á å ? N ë X â ë  v X Ú Ô â  â à This energy is transferred along magnetic lines of force into region located far away from the black hole where the force-free condition is violated; energy is pumped into accelerated particles Ø â maximal when The power is á á Õ#ä . å à Macdonald and Thorne (1982) demonstrated that this condition is very likely to be implemented à In order of magnitude, the power of the “electric engine” is

é ê

ã 'Û

/çë erg

s æ

~ ' Ûìu~ í

Ô Ù

Ù max

Ô

Ú

Ô

' Û £ Gauss

à Sometimes this electric engine, known as Blandford (1979)– Znajek (1978) mechanism, is described in terms of electrical engineering 4.16

4.5.3 Black hole as a unipolar inductor à Equipotential curves at the horizon are the lines of o â is meridional. Hence, the constant since the field e potential difference between two equipotential lines (marked by l and 2) are

â Ø Ò ê ü ë>î

'Û

Êð

V

ÿ

Ô

e

â ë>î Ø

~

'Û ì ~ í

¢Rá

â

Ò ï á å §  v 6 ä í

Ú

' Û £ Gauss Ù

Ù max

is an element of distance along a meridian on the à black hole surface, and Ò ï is the difference between the values of ï on the equipotential 1 and 2. The approximate equality in is written for the condition ê â â á å á Õ#ä , maximal á , and the equipotentials 2 and 1 corresponding to the equatorial and polar regions, respectively

â can be written in terms of à On the other hand, Ò  â the surface current and resistance: î

â Ø â  â Ò ¨ Ò

4.17

î

à Ò is the distance along the meridian between the equipotentials 2 and 1. Substituting the expression for  â , we obtain

â Ø ñ ¨  Ò

ä

Ò

ð â

â

ß8Ò

l â

î

ß Ø

ñ Ò

ð â

î ¨ â 3ß Ò ß m  ä l â

is the total resistance between the equipotential lines 2 and 1. (If the equipotentials 2 and 1 correspond to o ê   the equator and to Õ , the integration of yields ð â ê C Ò Û Ohm) à The above formulas permit the conclusion that the rotating black hole acts as a battery with e.m.f. of order ü

'Û

:ð

V

ÿ

~ ' Ûìu~ í

Ú

' Û £ Gauss C

and internal resistance of about Û Ohm

à This mechanism (and a number of its variants) has been employed in numerous papers for the explanation of the activity of the nuclei of galaxies and quasars 4.18