COMPUTATIONS IN GENERAL RELATIVITY

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• COMPUTATIONS IN GENERAL RELATIVITY
• SEMINAR AND DISCUSSION
• 900 1100 a.m.
• Thursdays
• Spring Semester 2002
• Technology Hall N276
• Observation Data Bridle, 1994 , 1-17
• Relativity Ohanian and Ruffini, 1994 , 18-60
• Plasma Physics Stix, 1992 Black HolesThorn et
  al,1986 , 61-89
• Computational Issues Chung, 2002 , 90-92
• Neutron Star Magnetospheres Michel, 1991 ,
  93-106
• Physics of Black Hole Gravitohydromagnetics
  Punsly, 2001 , 107-186 a

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• COMPUTATIONS IN GENERAL RELATIVITY
• No. Date _____________________
  Subject_______________________________________
  
  
  
• 1 24-Jan Introduction
• 2 31-Jan Relativistic plasma physics, discussion,
  and numerical simulation
• 3 7-Feb Particle trajectories in the ergosphere,
  discussion, and numerical simulation
• 4 14-Feb Vacuum electrodynamics, discussion, and
  numerical simulation
• 21-Feb The horizon electromagnetic boundary
  condition, discussion, and numerical simulation
• 28-Feb Magnetically dominated time-stationary MHD
  jets, discussion, and numerical simulation
• 7-Mar Winds and waves in ergosphere, discussion,
  and numerical simulation
• 8 14-Mar Ergosphere driven winds, discussion, and
  numerical simulation
• 9 21-Mar Ergosphere disk dynamics, discussion,
  and numerical simulation
• 10 4-Apr Winds from event horizon
  magnetospheres, discussion, and numerical
  simulation
• 11 11-Apr Winds from event horizon
  magnetospheres, discussion, and numerical
  simulation
• 12 18-Apr Extragalactic radio sources,
  discussion, and numerical simulation
• 13 25-Apr Non-pulsed black holes, discussion, and
  numerical simulation
• 14 2-May Non-pulsed black holes, discussion, and
  numerical simulation
• Contributed seminar topics are not necessarily
  related to the main subjects. Please provide
  hard copy information of your talk. We need at
  least one volunteer at each session. Contributed
  Seminar (15-40 min.)

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Trapped In a Black Hole Speaking of black holes,
here is my story Sometime ago, I was attracted
to a black hole by supergravity and superstrings,
and sucked hopelessly deep into a singularity. I
was desperate, trying to get out. I had heard
that black holes occasionally leak out some
radiation, contrary to the story that what goes
in never gets out, not even light. To get out,
you must have the right frequency and right wave
length. Fortunately I had a pocket size
supercomputer and found the right wave length.
Thats how I was able to escape from the deadly
hell and here I am back on earth, waking up from
the bad dream. So I made up my mind to go back
and investigate the monster. The rumor is that
black holes are hanging around gobbling up
everything they can get their hands on.
According to some reliable sources, however,
thats not always so. There are some friendly
black holes. They conceal the secrets of the
Universe but most importantly, they contain the
most precious treasure called antimatter, which
disappeared after the Big Bang, but is probably
hidden in the fifth dimension. This is the gold
mine of the black hole. Whoever gets this
antimatter is to strike rich overnight, by
selling it to NASA for next generation rocket
propulsion. Did you know there are six more
dimensions hidden somewhere, making a total of
eleven dimensions? No telling how many more
treasures are hidden behind them. So it is worth
studying the black hole. This is a treasure
hunting--not on the bottom of the ocean--but
inside the black hole. Unfortunately, you have
to do so much more work, so many extra studies.
You need to have a working knowledge in plasma
physics, magnetohydrodynamics, quantum mechanics,
quantum field theory, particle physics, string
theory, shock waves, turbulence, radiation and,
most important, general relativity. This is
because all of these physics reside in the black
hole. The Large Hadron Accelerator being built in
Geneva, Switzerland, to be completed in 2005, can
hardly duplicate the mighty black hole.  
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You need to have a tremendously powerful
telescope gazing right down to the black hole,
billions of light years away, to find out whats
going on in there. If you dont have such a
telescope, then what other choices do you have?
Numerical simulations in the comfort of your
room. But, numerical simulations of the universe
are not a trivial task. Observations and
measurements in astrophysics have been in
progress for the last six centuries, but
numerical simulations began only thirty years
ago. I attended the first UK Computational
Astrophysics Conference at University of
Leicester, several months ago. Many of the
attendees were postdocs and graduate students.
One of them sent me some movies simulating the
dark matter halo and a neutron star merging into
a black hole. These were his Ph. D.
dissertation. ---movies here---- (1) Numerical
simulation of the formation and evolution of
galaxies dominated by cold dark matter, using an
N-body simulation of 30 million particles for SPH
with inviscid gas physics. A zoom into the
center of very high resolution dark matter halo
is shown, which is rotated while the zoom takes
place. (2) This is a neutron star merging into a
black hole, forming the accretion disk. No
viscous effects are taken into account.
Simulations are based on the 31 formulation with
general relativity. No turbulence, MHD, or
radiation is included in this analysis. No jet
formations were observed due to crude
approximations made in this simulation.
Curvature distortions due to angular momentum
were not taken into account. Earlier this month I
attended the 199th AAS meeting in Washington DC.
I saw many people making presentations of the
results of their observations and measurements.
Side by side they showed some pretty pictures of
someone elses numerical simulations, but they
seemed to have no idea how many simplifications
had to be made for the numerical computations to
work. What if both measurements and computations
are wrong? This is why we are here to study what
we can do to improve our skills to explore the
universe, specifically to study physics,
mathematics, and numerical simulations for the
next fourteen weeks. I hope we can stick around
together to the end, exchanging ideas, sharing
knowledge, searching for truth. So stay tuned.
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The hot, turbulent plasma that is distributed in
a region over 700,000 light years in extent in
the FRII radio galaxy, 3C 353, is a signature of
the violent world of black hole
gravitohydromagnetics. The VLA image is provided
courtesy of Alan Bridle.
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This image is an overlay of the radio (red) and
optical (blue) emission that shows jets of
magnetized relativistic plasma emerging from the
center of the elliptical galaxy in the radio
source 3C 31. This picture was generously
provided by Alan Bridle.
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Fig. 1.1. An HST image of the central disk in the
elliptical galaxy NGC 4261. Note the bright
central feature, possible accretion disk
radiation from the region of the active nucleus
shining through the dusty gaseous disk. The disk
is approximately 250 pc across and a gas
kinematical estimate of the central black hole
mass is 4.5 X 108MT. Photograph provided
courtesy of Laura Ferrarase.
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Fig. 1.2. Inserts of the large scale FR I radio
structure of NGC 4261 and the small parsec scale
VLBA jet that appears to emanate from the bright
spot in the center of the disk (which is
featured more prominently in Fig. 1.1).
Photograph provided courtesy of Laura Ferrarese.
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Fig. 1.3. The central disk of the elliptical
galaxy NGC 7052 is revealed in this HST image.
The disk is 1000 pc in diameter and the orbital
kinematics imply a central black hole mass of
3x108MT. Notice the bright central region that
shines through the disk as in NGC 4261. This is
a weak radio source and the VLA jet is misaligned
with the symmetry axis of the disk, as in NGC
4261. The photograph is provided courtesy of
Roeland van der Marel.
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Fig. 1.4. The optical jet is emanating from the
center of the inner disk in this deep HST image
of M87. The disk is 20 pc across and the orbital
motion indicates a central black hole mass of
3x109MT. The photograph is provided courtesy of
Holland Ford.
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Fig. 1.5. A 5 GHz deep VLA image of a
prototypical FR I radio source 3C 296. The jets
are very bright compared to the diffuse lob
emission. Image provided courtesy of Alan Bridle.
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Fig. 1.6. A deep VLA image of Cygnus A at 5 GHz.
The lobes are separated by 180 kpc (HO
55km/sec/Mpc,qO 0). Notice the strong hot spots
at the end of each lobe where most of the
luminosity resides. A highly collimated low
surface brightness jet extends into the eastern
lobe from a faint radio core. There are
suggestions of a counter jet in the image. The
counter jet is more pronounced in Fig. 1.10. The
VLA image was provided courtesy of Rick Perley.
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Fig. 1.7. A deep 5 GHz VLA image of the radio
loud quasar 3C 175. Notice the morphological
similarity to Cygnus A. The jet is more
pronounced relative to the lobe emission than
Cygnus A, and there is no hint of a counter jet.
This is anecdotal evidence for mildly
relativistic flows in kiloparsec scale jets.
Image provided courtesy of Alan Bridle.
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Fig. 1.8. This deep 5 GHz VLA image of the FR II
radio galaxy 3C 219 shows a strong jet and a
knot in a counter jet. It is overlaid on the
diffuse (blue) optical image of the host
elliptical galaxy. Image provided courtesy of
Alan Bridle.
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Fig. 1.10. The jet in Cygnus A is mapped from
scales on the order of 50 kpc down to less than a
light year in this series of inserts. The VLBI
maps indicate that the central engine is less
than a light year in diameter. The images are
from Krichbaum et al. 1998.
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Computations in General Relativity Our Immediate
goal is to determine what causes jets to emerge
from compact objects. To do this, we must
model the black hole physics. This research will
require (1) Combine the quantum gravity field
theory into general relativity, (2) Solve the
resulting governing equations numerically on the
computer. But the quantum field theory is
incomplete, some difficult problems unresolved.
Numerical solutions may help in revising
existing theories and redeveloping more perfect
theories by examining the numerical results.
Eventually, we may be able to help extrapolate
back to the origin of the Universe and help
predict the fate of the Universe into a distant
future.

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Past Achievements and Future Goals

• Curved spacetime geometries have been computed
  numerically using standard CFD schemes.
• Cosmological singularities have been numerically
  modeled using the Mixmaster or Gowdy Cosmology
  Model.
• Quantum gravity equations have been numerically
  solved.
• The above three problems are involved in the
  black hole physics that may hold the secrets of
  the universe.
• Can we numerically simulate any and all of the
  black hole physics ? This is what we wish to
  explore.
• (6) Can we someday model the entire
  universe with all the physics taken into account?
  This is what we wish to explore.

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Computations in General Relativity for Black
Holes with Singularities
Particle Physics (Nachtmann, 1990), Quantum
Mechanics (Hecht, 2000), Quantum Field Theory
(Kaku, 1993), Plasma Physics and MHD (in
Punsly, 2001), String Theory, Vol 12
(Polchenski, 1998) Celestial Mechanics (Taft,
1985) Gravitation and Spacetime (Ohanian
Ruffini, 1994) Physics of Black Holes (Novikov
Frolov, 1989) Cosmology and Particle
Astrophysics (Bergström Goobar, 1999)
Turbulence (Lesieur, 1997) Shock Waves and
Radiation (in Miharas Miharas, 1984) Flat and
Curved Spacetimes (Ellis and Williams, 1988)
Relativity and Scientific Computing (Hehl,
Puntigam Ruder, 1996) Computational Fluid
Dynamics (Chung, 2002)
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Why do we need the theory of relativity in
Particle Physics?
Consider a typical reaction
PPgtPP??-

• The kinetic energy of the original protons is
  converted into rest mass of new particles
• The greater the energy of the protons, the
  larger the number and mass of the particles that
  can be produced
• Collisions between fast, relativistic particles
  are important in this process

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Why do we need the theory of relativity in
quantum gravity, supersymmetry, supergravity and
superstrings?

• Equivalence Principle The laws of physics in a
  gravitational field are identical to those in a
  local accelerating frame --Einstein
• Construct a theory that is invariant under
  general coordinate transformations, that is, a
  theory in which one can choose coordinates such
  that the gravitational field vanishes locally
• Construct tensors of arbitrary rank or indices
  and their covariant derivatives

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ROLE PLAYED BY QUANTUM EFFECTS IN BLACK HOLE
PHYSICS

• Quantum effects significant for black holes of
  mass smaller than solar mass with a spacetime
  singularity
• Virtual particles are constantly created,
  interact with one another, and are annihilated in
  a vacuum
• In an external field, some virtual particles may
  acquire sufficient energy for becoming real
• The result is the effect of quantum creation of
  particles from vacuum by an external field

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Special Relativity in Flat Space
ds2gaßdxadxß a,ß(0,1,2,3) gaßmetric
tensor (Minkowski tensor) Spacelike metric ds2
-(cdt)2 dx2 dy2 dz2
 

gaß
Timelike metric ds2 (cdt)2 dx2 dy2 dz2
gaß
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GENERAL RELATIVITY IN CURVED SPACETIME
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Curvilinear coordinates ga Covariant
tangent vectors (undeformed) ga Contravariant
tangent vectors (undeformed)
Initial curvature
Deformed curvature due to angular momentum and
torsional deformations
Ga Covariant tangent vectors (deformed) Ga
Contravariant tangent vectors (deformed)
xa Reference Cartesian coordinates
ga . gß dß
a
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Covariant metric tensor (undeformed)
Covariant metric tensor (deformed)
Contravariant metric tensor (undeformed)
Contravariant metric tensor (deformed)
Squared line segments (undeformed )
Squared line segments (deformed)
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General Relativity
Riemann curvature tensor in general relativity

Ricci tensor Einstein
equation Torsionally strained line segments
  
gaßa Curvature of tangent vectors (vector
curvature)
scalar curvature
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• THE SCHWARZSCHILD SOLUTION
•  
•  
• This is applicable to the gravitational collapse
  of any
• nonrotating, electrically neutral star.
• For rotating black holes in a magnetic field we
  require
• Kerr-Neumann spacetime, Boyer-Lindquist
  coordinates
• (Reissner-Nordström Solution).

   
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THE KERR BLACK HOLE SOLUTION
? Boyer-Lindquist Coordinates   ?
The available extractible energy is the reducible
energy (Mred)max 0.29M
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BLACK HOLES AND GRAVITATIONAL COLLAPSE
Large relativistic effects are found in the
gravitational field in the neighborhood of
an extremely compact mass, .
For , the
gravitational fields are so strong that
nothing can escape from this grip.
   
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Null Surface
t/rs
Worldlines by integrating
r/rs
rltrs Spacelike
rgtrx Timelike
Fig. 8.3 The forward light cones near and inside
a black hole. As r -gt , the light cone
assumes its usual shape and direction, that is,
dr/dt /- 1. The curve AB, BC is the worldline
of an ingoing light signal.
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Timelike
Spacelike
 
Interior region
Exterior region
(No singularity at r rs)
Fig. 8.5 The maximal Schwarzschild spacetime in
Kruskal coordinates.
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HORIZONS AND SINGULARITIES IN THE ROTATING BLACK
HOLES
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THE MAXIMAL KERR GEOMETRY
The worldline begins outside of the black hole,
crosses the horizons r r and r r- , and
approaches near r 0, moving in the outward
direction and approaching the surface r r- from
the inside. The surface r r- belongs to a
white hole rather than a black hole. It is one
way out rather than one way in. This implies an
infinite sequence of universes.
Fig 8.14 A possible worldline for a particle
moving along the axis of a black hole.
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NAKED SINGULARITY

• A singularity not surrounded by a horizon is
  called a naked
• singularity .
• A rotating hole of large spin
  has such a naked
• singularity.
• Although naked singularity does exist
  mathematically, as
• solutions of the Einstein equations, we do not
  know whether
• they are ever found in the real world.
• Penroses Cosmic censorship conjecture points
  out a
• singularity hidden within a horizon. This
  remains unproved.
• Shapiro and Teukolsky (1991) assert that the
  gravitational
• collapse of an elongated mass distribution
  generates a naked
• singularity. This is unproven.

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SUMMARY OF HORIZONS AND SINGULARITIES IN THE
ROTATING BLACK HOLES

• a lt GM There are two horizons, r , and r-
• a GM There is only one horizon
• a gt GM There is no horizon. If there exists a
  horizon
• there may be a naked singularity, but this has
• not been proved

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EXTRACTION OF ENERGY FROM THE ROTATING BLACK HOLE
(Penrose, 1969 Christodoulou and Ruffini, 1970,
1971)
Positive and Negative Event Horizons
The Event Horizon Surface Area
The Black Hole Mass
for Schwarzchild Black Hole
The available extractible energy is the
reducible mass
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BLACK HOLE THERMODYNAMICS THE HAWKING PROCESS

• The horizon, or one-way membrane, of a black
  hole acts as
• a perfect absorber or an ideal heat sink.
• Hawking demonstrated that when the black hole
  forms,
• the quantum fields settle into a state that
  involves a steady
• outward emission of radiation from the
  horizon toward
• infinity. The energy spectrum of this
  radiation is thermal,
• with a temperature

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Fig. 1-1 Characteristic electron density and
temperature of plasmas. The unshaded area
represents classical kinetic plasmas.
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MAXWELLS EQUATIONS
Schwarzschild (S0, Q0) Kerr (S?0,
Q0) Reissner-Nordström (S0, Q?0) Kerr-Newman
(S?0, Q?0)
Plasma Generator
Plasma Accelerator
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THE EQUATIONS OF PERFECT MHD PLASMAS
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• THE THEORY OF RELATIVITY IN PLASMA PHYSICS AND
  MAGNETOHYDRODYNAMICS
• The origins of the theory of relativity lie in
  electromagnetism
• The experimental basis for the invariance of
  electric charge, the covariance of
  electrodynamics, the explicit transformation
  properties of electric and magnetic fields
• Relativistic equations of motion for spin
• Relativistic kinematic and dynamic equations
• Alfvén velocity
• Alfvén waves

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RELATIVISTIC PLASMA PHYSICS
Black hole gravitohydromagnetics (GHM) is
essentially plasma physics in the
magnetosphere near a black hole Information
can be transmitted from one region to another
only by means of the modes of propagation
allowed by the plasma state in the ergosphere
In this region (ergosphere) the velocity is near
the speed of light and thus the relativistic
structure of plasma waves dominates the black
hole physics
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RELATIVISTIC ELECTROMAGNETICS
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RELATIONSHIPS AMONG ALFVÉN WAVE, FAST WAVE AND
SLOW WAVE  Momentum Second Order
Maxwell Fourier Space Perturbed
equations   Alfvén Velocity  
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SLOW WAVES, FAST WAVES, AND ALFVÉN WAVES
Momentum
Maxwell
Alfvén Velocity
Slow Wave
s
Fast Wave
Fig. 2.1. The Friedrichs or phase polar diagram
of the three plasma wave phase velocities. The
polar diagram plots the velocities in the case
the . Note that UFgt
UIgtUSL.
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THE 31 SPLIT OF SCHWARZSCHILD SPACETIME
For fiducial observers (FIDO) or zero angular
momentum observers (ZAMO)
Gravitational acceleration (general relativity)
The motion of freely falling observer (FFO)
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BOUNDARY CONDITIONS ON THE HORIZON
All components of E and B tangential to the
horizon blow up as a ? 0
Because the FIDOs near the horizon ( a ? 0 ) are
moving outward at nearly the speed of light, the
tangential components of all magnetic fields
appear to be ingoing electromagnetic waves.
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THE 31 SPLIT OF KERR SPACETIME, RAPIDLY ROTATING
HOLES
Fig. 24. The shapes of the horizons of three
black holes, all with the same irreducible mass
Mirr , as depicted by embedding diagrams. Each
diagram should be rotated about its vertical
axis. The hole in ( a ) is nonrotating and
spherical that in ( b ) is rotating sufficiently
fast that its poles are completely centrifugally
flattened that in ( c ) is rotating so fast that
its polar regions (dashed) have acquired negative
Gaussian curvature and consequently are embedded
in a Minkowski space Eq. (3.76b). (Figure
adapted from Smarr 1973a.)
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BLACK HOLE MAGNETOSPHERE, INNER REGION
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BLACK HOLE MAGNETOSPHERE, STRETCHED HORIZON
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Total Magnetic Flux ?
1
Constant ?
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Governing Equations for Plasma Wave Propagation
0
Momentum
(1) (2)
Maxwell
0
(3) (4) (5) (6)
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(7) (8) (9) (10) (11) (12)
Neglecting the diffusion term (second
derivative), we have
Summary To derive the plasma wave propagation
equations, we require the continuity equation,
momentum equations (5) and the Maxwell equation
(9) as follows
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Derivation of Alfvén Velocity
To derive the Alfvén velocity, we consider an
incompressible flow Perturbed magnetic
field Assume the plasma is of infinite extent
and the only spatial variations are
in the x3 direction (
from (13)). Substitute (14)
into (12), Substitute (14) into (11),
The x3 component of (11) becomes The
time derivative of (15) with (16) The
time derivative of (16) with (15) Note
that (18) and (19) satisfy where the
Alfvén velocity is identified as
(13) (14) (15) (16) (17) (18) (19) (20) (21)
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Electromagnetic Waves in a Compressible
Conducting Fluid
Electromagnetic governing equation
(22) (23) (24) (25) (26) (27) (28) (29) (30)
Linearization
Linearized equations
Perturbations
Orientation of coordinate system for propagation
of hydromagnetic waves
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Perturbed equations Substituting (34) into
(32) Substituting (35) into (36) where A is
the Alfvén velocity.
(31) (32) (33) (34) (35)
(36) (37)
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(38) (39)
Note that is uncoupled from and and
those transverse disturbances in the x1 direction
propagate with a phase velocity.
x3
Alfvén Wave
A
A
B0
ai
?
ai
?
Slow Wave

x2
Fast Wave
x1
Phase polar diagram
Wave normal surface
Orientation of velocity
Alfvén Wave Parameters
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B0
B0
B0
uA
ua
u
Disturbance propagates as longitudinal
acoustic wave with a phase velocity a
For propagation across the magnetic field only
the disturbance propagates as a longitudinal
magnetosonic wave with a phase velocity
Disturbance propagates as transverse Alfvén
wave with a phase velocity A
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For arbitrary directions of propagation, the
equations of and can be written as
(40) (41a) (41b)
This dispersion equation is quadratic in u2. In
general, two modes can propagate modified
hydromagnetic wave and the modified acoustic wave
with reference to the wave normal surfaces shown
below.
A
a
uA
uA
ua
a
A
ua
uA
Modified hydromagnetic wave, Agta
Modified acoustic wave, Alta
Wave normal surfaces for uA and ua.
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• Summary for Plasma Waves
• The oblique Alfvén wave (intermediate wave) can
  propagate field
• aligned currents and carries a charge density
  having a significant
• electrostatic polarization.
• Magneto-acoustic waves (fast waves,
  compressional Alfvén waves)
• carry no current along the magnetic field
  direction nor charge density
• having no electrostatic polarization.
• For high frequency waves, the Alfvén mode is
  the only wave that can
• carry field aligned currents and have an
  electrostatic polarization.
• Acoustic waves (slow waves) carry no current
  nor charge density.

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Poyntings Theorem
Faraday Ampere Poynting Vector, The
Poynting vector is normal to both electric and
magnetic intensity vectors. It represents the
energy per unit time which crosses a unit area.
(1) (2) (3) (4) (5)
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    Derivation of Electromagnetic Stress and
Energy  (1) (2)   Multiply (1)
by H and (2) by E (dot product) and subtract (2)
from (1), leading to the electromagnetic
energy   (3)   Integrate (3) over the
volume and apply the Green-Gauss
theorem   (4)   where G is the Poynting
vector  (5)   which is normal to both E
and H.   Take the cross products of (1) with D
and (2) with B and add   (6) with
(7) where and are the electric
and magnetic stress tensors respectively, given
by
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Electric Stress Tensor   (8)   Magnetic
Stress Tensor   (9)   The integral
form of (6) may be written as     If the fields
do not vary with time, we have   (10)   or
using the Greer-Gauss theorem,  (11)
 or (12)
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Maxwell and Momentum Equations for the Derivation
of MHD Boundary Conditions
Maxwell Equations
(1) (2)
(3) (4)
Combining the Faraday and Amperes equations
together with (1) and (4),
(5) (6)
Magnetic Reynolds Number
Momentum Equation
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SHOCK WAVE DISCONTINUITIES
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WHY CONSERVATION FORM?
Solutions of this equation in nonconservation
form will not resolve discontinuities resulting
from shock waves
The solution of this equation will be smooth,
from which the discontinuous primitive variables
can be extracted
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Physics of Turbulence
Turbulence caused by convection wave
instabilities (turbulence models, large eddy
simulations, direct numerical simulations)

Free boundary layer turbulence due to difference
in magnitudes of velocity between two streamlines
Wall boundary layer turbulence due to high
velocities interacting with shock waves and
microscale eddies in the secondary boundary layer
Turbulence caused by plasma wave instabilities
(a-disk model, direct numerical simulations, etc.)

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The basic black hole magnetosphere model in
Punsly 2001 is based on F. Curtis Michel 1991
Fig. 1.15. The self-excited Faraday disk. A disk
rotating through a magnetic field produces an EMF
and drives current through a stationary shunt,
here replaced by a solenoidal coil which in turn
provides the magnetic field in the first place.
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Rotating Disk Modeled by Faraday Disk
Fig. 1.16. The axisymmetric self-excited Faraday
disk. By restyling the solenoid and the sliding
contact at the disk edge, the self-exciting
system can be made manifestly axisymmetric, in
contradistinction to the theorem against such
dynamos. It is necessary to spiral the wires
returning the current to the periphery, but this
spiraling is not a violation of axisymmetry.
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Pederson field lines and Hall field lines
JP Pederson current (produces toroidal magnetic
field lines) Jh Hall current (produces
poloidal magnetic field lines)
(a)
(b)
To define the Hall current, we invoke the
generalized Ohms law Where Jih is the Hall
current vE electron-ion collision
frequency Ok electron cyclotron frequency s
conductivity In general, the direction of Hall
current deviates from Pederson current due to
stray magnetic field lines. If OK ltlt vE, then the
Hall current is negligible.
Fig. 1.17. A planetary self-exciting Faraday
sphere. (a) Cross-sectional view showing the
rotating core with uniform magnetization, which
induces an electric field that causes the current
to flow in the surrounding shell. (b) Current
flow in the shell showing that the Pederson
currents and magnetic field lines crossing the
shell conspire to drive the Hall currents in one
azimuthal sense, which can thereby provide the
source of the magnetization in the first place.
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Black Hole Magnetospherical Coordinates
Magnetospherical Coordinates
Standard Spherical Coordinates
r radial ? meridional f tangential
(circumferential) P poloidal T
toroidal Poloidal magnetic flux
is in the plane of meridional and
radial flux vectors
97
Goldreich-Julian field lines and charge density
(b) Goldreich-Julian case
(a) Vacuum case
Fig. 2.1. Magnetic and electric field lines about
an aligned rotator. Solid lines are the dipole
magnetic field lines, while the dotted lines are
the electrostatic field lines, (a) for the vacuum
case and (b) for the Goldreich-Julian case. From
F.C. Michel, 1982, Rev. Mod. Phys., 54, 1 (figure
4).
98
Magnetic field lines as equipotential
Fig. 2.2. Magnetic field lines as equipotentials.
The field lines can be labeled, given axial
symmetry, by the total magnetic flux ( f ) that
would be enclosed by rotating the field line
about the axis. The flux f between two field
lines of potential difference is
therefore geometrically in a fixed ratio along
each field line. From F.C. Michel, 1982, Rev.
Mod. Phys., 54, 1 (figure 5).
99
Magnetosphere defined by light cylinder
Fig. 2.3. Hypothetical aligned rotator magnetic
fields. Dashed vertical line locates the
light-cylinder. The field line f0 is the last
open field line. Shaded region contains the
closed field lines. From F.C. Michel, 1974b, Ap.
J., 187, 585 (figure 1).
100
Electron energy due to pair production
Fig. 2.5. Space-charge flow limitation by pair
production. Electrons gain energy (?-1)mc2 as
they accelerate to height h (solid curve). At
height h? the curvature radiation first produces
photons, which are energetic enough to be
converted into pairs at h2. Because only a very
small downward flux of positrons can be
tolerated, the accelerating field must
essentially vanish quite close to h2 in order
that positrons produced at and above h2 are not
returned. Consequently the electron energy never
reaches the value ?0 it would have attained if
acceleration all the way to h0 had been possible,
as illustrated. For h0 gt h2, pair production
becomes unimportant in limiting ?, and for h0 lt
h?, there is little or no pair production at all.
From F.C. Michel, 1982, Rev. Mod. Phys., 54, 1
(figure 8).
Height of one polar cap radius Lorentz factor
101
Physics of pair production
Fig. 2.6. Nature of a pair production discharge.
All the potential drop must appear in the gap
(h) otherwise the system would be flooded by
downward-accelerated electrons. Above the gap the
energetic primaries would continue to radiate and
produce yet more pairs, resulting in a relatively
dense pair plasma. The gap width would be
maintained so that pair production within the gap
would be kept just as threshold (otherwise the
average number of particles there would
exponentiate). Pair production takes place at
points 1, 3, 5, and gamma radiation at points
2, 4, 6, . Because the process requires curved
field lines, it automatically marches toward
the least-curved field line, suggesting that it
would either extinguish itself or exhibit a
relaxation type of oscillation (note event 6 is
numbered inconsistently). From A.F. Cheng and
M.A. Ruderman, 1977b, Ap. J., 214, 598 (figure 1).
102
Goldreich-Julian Magnetosphere
103
Simple disk/neutron star interaction model
104
Modified disk/neutron star interaction model
Fig. 6.2. Modified interaction if magnetic field
lines are ejected. If finite conductivity is
insufficient to limit the current flow, J x B
forces should eject the magnetic field lines. The
steady state solution might then be as shown,
where only a restricted magnetic flux crossed the
disk. Because , the flux
reduction also reduces the available
electromotive force. This would return us to the
geometry of figure 2.3, with the disk acting the
role of a neutral sheet in the equatorial plane.
The field structure is identical to that proposed
by Roberts and Sturrock (1973), which gives a
deceleration parameter n 7/3. From F.C. Michel,
1983b, Ap. J., 266, 188 (figure 1) 1982, Rev.
Mod. Phys., 54, 1 (figure 22b).
105
Accretion onto a Magnetized Neutron Star
Fig. 10.1. Confined magnetosphere. Pressure of
in-falling (or resident) plasma produces a
characteristically shaped magnetosphere.
106
Magnetic Paddle Wheel
107
Spiral Fluid Motion
108
EVIDENCE OF A BLACK HOLE CENTRAL ENGINE IN RADIO
LOUD AGN WITH A STRONG MAGNETIC FIELD
The central engine of a radio loud AGN
supplies far more power to the radio lobes
than is indicated directly from the radio
luminosity (Cygnus A, 3C-405). Rapidly
rotating supermassive black holes are the most
viable known power sources for explaining all
of the properties of the radio loud AGN
population. The quasar in the host galaxy is
now commonly believed to be the result of
large viscous losses in an accretion flow that
scales with the accretion rate.
109
EXTRACTING ENERGY FROM A BLACK HOLE
Supermassive black holes are believed to be
located in the central engines of
extragalactic radio sources. Accretion flows
yield the quasar emission and the physical
state of the central black hole is determinant
for the existence of bipolar radio jets.
Yet, the signature of a black hole is the manner
in which it sucks mass-energy inescapably
toward the event horizon. Thus, how can
energy be extracted from the central black
holes of radio loud AGN?
110
BLACK HOLE GRAVITOHYDROMAGNETICS Relativistic
Plasma Physics

• Coupling of the gravitational field and a
  large-scale magnetospheric plasma governs the
  accretion disk and black hole physics
• Complete understanding of the Kerr spacetime on
  higher dimension Riemann geometry (higher order
  derivatives of the slope of the tangent vectors
  are necessary to demystify the black hole
  physics)

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Kerr-Newman Spacetime (A Black Hole Has No
Hair)
Rotating or charged black holes are alive
and some of their energy is extractable
Christodoulou and Ruffini, 1971. A black
hole can have M (mass), a (angular momentum
per unit mass), and Q (charge), and nothing
else (no hair).
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WHAT IS IMPORTANT IN BLACK HOLE RELATIVISTIC
PLASMA PHYSICS?
The Alfvén wave has an electrostatic
polarization and can propagate field aligned
currents in a black hole driven plasma wind
The fast wave has no electrostatic polarization
and can not propagate field aligned currents
A plasma-filled waveguide terminated by a
unipolar inductor illustrates the role of a
unipolar inductor in a relativistic pulsar MHD
wind theory A Faraday wheel disconnected from
the end of a plasma- filled waveguide is used
to elucidate the electrodynamic properties of
the event horizon in the context of MHD winds
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How does the theory of relativity interact with
hydrodynamics?

• Turbulence MHD Turbulence
• Convective Turbulence
• Shock Waves Radiative
• Electromagnetic
• Convective
• Radiation Reflection
• Absorption
• Emission
• Scattering
• Refraction
• Defraction

119
PARTICLE TRAJECTORIES IN THE ERGOSPHERE

• Boyer-Lindquist coordinates are invaluable in
  general relativity
• They are not very useful for understanding the
  nature of the
• physical interaction since they are neither
  orthonormal nor
• orthogonal, because of spacetime curvature
• The four velocities of the distant observers,
  is spacelike
• within the ergosphere.
• Compute special relativistic physics in local
  orthonormal frames
• Using the foliation of spacetime then yields
  physics in the global
• Boyer-Lindquist coordinates

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MAXWELLS EQUATIONS
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VACUUM ELECTRODYAMICS

• In vacuum electrodynamics, an axisymmetric
  electromagnetic field can
• not extract energy from a rotating black hole.
• The event horizon is an asymptotic infinity for
  accreting charge neutral
• electromagnetic sources no hair theorem.
• The net electromagnetic field has a component
  due to the Kerr-Newman
• black hole that results from charge accretion.
• The horizon boundary condition is where the
  spacetime near the horizon
• has no relevance electrodynamically in any
  global plasma flow. It is a
• sink for inflowing plasma.

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Fig 4.7 The magnetic field of an axisymmetric
current loop with a radius of r 1.5r , a
0.43, centered about a black hole with a/M 0.9
Fig 4.8 The magnetic field of an axisymmetric
current loop with a radius of r 1.5r , a
0.29, centered about a black hole with a/M
0.995
132
Fig 4.9 The magnetic field of an axisymmetric
current loop with a radius of r 1.05r , a
0.124, centered about a black hole with a/M 0.9
Fig 4.10 The magnetic field of an axisymmetric
current loop with a radius of r 1.001r , a
0.017, centered about a black hole with a/M 0.9
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Derivation of MHD Boundary Conditions
The conservation of the MHD governing equations
(7) (8) (9)
Integrate (7) over the volume
Verify that (8) represents the correct
conservation form. To do this, perform the
indicated differentiation and obtain the
primitive variable equations (5) and (6). The
surface integral in (9) provides the Neumann
boundary conditions to be imposed on the event
horizon as well as on other boundaries.
135
Derivation of MHD Boundary Conditions Continued
Non Relativistic
Extract the electromagnetic components, momentum
Extract the electromagnetic components, Maxwell
Relativistic
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Nonrelativistic Electromagnetic Terms
Momentum
Maxwell
Blandford-Jnajek Relativistic Horizon Boundary
Condition
(Force-free and frozen-in)
137
Punslys Relativistic Horizon Boundary Conditions
Lapse Function
Electromagnetically induced equilibrium
Plasma boundary conditions, stationary frames
Plasma boundary conditions, ZAMO frames
138
Summary of Differences in Boundary Condition
Treatments
139
Gravitohydromagnetics Event Horizon Boundary
Conditions
Blanford-Znajek 1977-2002

• Assumes the force-free limit J x B 0, the
  frozen-in condition Ev x B 0, free to impose a
  magnetically dominated solution everywhere. This
  expedience circumvents the necessity to introduce
  black hole gravitohydromagnetics. As such, the
  solution is very amenable to usage, thus
  receiving a great deal of popularity in the past.
• Assumes that, near the event horizon, there is a
  unipolar induction in which a spark gap forms as
  in Faraday wheel, thus ignoring the horizon
  boundary conditions arising from the
  electromagnetically induced equilibrium. There,
  field aligned poloidal currents emanate from the
  event horizon and there is no significant source
  of poloidal current anywhere within the wind zone.

Punsly 1996-2001

• The intermediate (Alfvén) mode propagates
  information about the global charge and current
  density because they involve non-zero
  perturbations to the current density. The
  force-free and frozen-in conditions are not
  assumed.
• Assumes an asymptotic infinity condition to the
  paired wind system at the event horizon which is
  inertially dominated. Plasma interactions must be
  taken into account for the event horizon
  boundary conditions.

140
Blandford, R.D. and Znajek, R.L. 1977.
Electromagnetic extraction of energy from Kerr
black holes. Mon. Not. R. Astro. Soc. 179,
433-456. SUMMARY
When a rotating black hole is threaded by
magnetic field lines supported by external
currents flowing in an equatorial disc, an
electric potential difference will be induced.
If the field strength is large enough, the vacuum
is unstable to a cascade production of
electron-positron pairs and a surrounding
force-free magnetosphere will be established.
Under these circumstances it is demonstrated that
energy and angular momentum will be extracted
electromagnetically. These ideas are
incorporated into a model of active galactic
nuclei containing a massive black hole surrounded
by a magnetized accretion disc. In this model
relativistic electrons can be accelerated at
large distances from the hole and therefore will
not incur serious losses, which is a defect of
some existing models.
141
PERFECT MHD WINDS AND WAVES IN THE ERGOSPHERE
142
INGOING PERFECT MHD WIND FRONT
143
ALFVÉN WAVES, FAST AND SLOW WAVES, HIGH AND LOW
FREQUENCY WAVES
Maxwells Equation (curls of Ampere and
Faraday equations)
Fast Waves Fast waves carry no charge
density The fast wave does not propagate as
Alfvén Waves
The Alfvén wave has an electrostatic
polarization and can propagate field aligned
currents. The fast wave has no electrostatic
polarization and can not propagate field aligned
currents. For high frequency waves, the Alfvén
mode is the only wave that carries field aligned
currents and has an electrostatic polarization.
144
ERGOSPHERE DRIVEN WINDS

• Analogy to the physics of the Faraday Wheel

A unipolar inductor drives current because the
rotationally induced EMF is unbalanced by the
electrostatic force in a Faraday wheel. This
rotational inertia is converted to Poynting flux
through Fµ?J? forces.

• The causal structure of the dynamo

The ergosphere dynamo occupies a region of
spacetime upstream of the fast critical force of
the ingoing wind. The outer boundary surface of
the dynamo region resides upstream or
coincidental with the inner Alfvén critical
surface. This establishes a causal relationship
through Alfvén waves between the dynamo and the
plasma source and outgoing wind.

• The dragging of inertial frames

Frame dragging pulls plasma across the poloidal
magnetic field relative to the black hole
rotation.
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THE TORSIONAL TUG OF WAR

• A large scale magnetic flux threads the
  ergosphere and also extends to large
• distances from the black hole. There is a
  tenuous plasma frozen onto the
• magnetic field lines in gyro-orbits both in
  the ergosphere and far from the hole
• (see next page).
• The plasma rotates with an angular velocity
  such that
• There must exist a toroidal magnetic field, BT,
  so that the plasma can slide
• azimuthally with respect to the corotating
  frame of the magnetic field and
• remain frozen-in globally.
• The existence of BT, as plasma is introduced
  globally on vacuum axisymmetric
• poloidal field lines is a result of a
  torsional tug of war between plasma at r ??
• and plasma in the ergosphere. The plasma at
  r ?? sends torsional Alfvén
• waves inward, telling plasma and the field
  near the black hole not to rotate so
• fast, thus consequently twisted by a
  torsional tug of war.

146
ERGOSPHERIC DRIVEN WINDS
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Fig. 8.5. The accretion history of an
axisymmetric magnetized plasma ring. A poloidal
magnetic field is supported by the azimuthal
current in the ring. ( a ) The ring is nearing
the hole with poloidal component, uP, of the four
velocity. The shaded region indicates where most
of the accreting plasma is located. The time
sequence shows the reconnection process. ( c) A
circular ring (seen in cross section) of X-type
reconnection sites is about to form. (e) The
large-scale flux is buoyant and moves outward,
completely decoupled from the plasma ring. The
accreting plasma ring becomes a circular set of
O-points (seen in cross section) at which
magnetic loops are destroyed before the plasma
reaches the horizon.
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Fig. 8.6. The global energetics of an
ergospheric disk dynamo in an azimuthal flux
tube. A thin disk (shaded) is bounded from above
by a slow switch-off shock front at which the
energy flux, ke, is transformed from mechanical
form to Poynting flux radiated from the disk
surface. Plasma is dragged azimuthally across the
poloidal magnetic field lines in the shock by the
dragging of inertial frames. This is the force
driving the cross-field dynamo current in the
shock front. The enhanced current flow in the
shock layer is indicated by the thick arrowed
line. Plasma is resistively heated by u F J
dissipation in the shock. A relativistically hot
flow exits the shock and settles into the disk on
negative energy trajectories (plasma is rotating
at nearly the negative speed of light in the ZAMO
frames.) The strong headlight effect associated
with this ultrarelativistic motion beams the
synchrotron and annihilation radiation from the
plasma onto negative energy trajectories. These
superradiant photons spin down the black hole.
The influx of these negative energy photons can
be considered the outflow of energy flux, ke,
that allows the black hole to power the dynamo.
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Fig. 11.3. The field line angular velocity,
, as a function of latitude, , on the plasma
horizon
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• SUMMARY
• Black Hole Gravitohydromagnetics
• Introduction
• The struggle between electromagnetic and
  gravitational forces, known as torsional tug of
  war between ergospheric plasma and the distant
  asymptotic plasma leads to extraction of the
  rotational inertia of the black hole. These
  interactions result in a powerful pair of
  magnetized particle beams or jets that are
  ejected at nearly the speed of light, transfering
  energy fluxes exceeding 10-E47 erg/s.
• Black hole no hair theorem. A stationary
  black hole resulting from a collapse of neutral
  matter interacting gravitationally (equivalent to
  spin-2 massless bosan field) is described by a
  metric having only three free parameters mass
  (m), angular momentum (a), and electric charge
  (Q) Doshkevich 1965 Novikov, 1969, eliminating
  the magnetic field Ginzberg1964.
• Relativistic plasma physics
• Attach a Faraday wheel or unipolar inductor
  to the end of a semiinfinite plasma-filled
  waveguid. Solve the relativistic MHD equations.
• The Alfvén wave has an electrostatic
  polarization and can propagate field aligned
  currents.
• The Fast wave has no electrostatic
  polarization and can not propagate field aligned
  currents.
• (3) Particle trajectories in the Ergosphere
• Consider a particle rotating with an angular
  velocity as viewd from asymptotic infinity at
  event horizon in ZAMO frame. All particles must

185

• corotate with the horizon as seen globally,
  accompanied by frame dragging due to timelike
  coordinates changing to spacelike cooordinates
  inside the ergosphere.
• In this process the absorption of negative
  energy matter leads to the rotational energy
  being extracted by the black hole, governed by
  the second law of black hole thermodynamics.
• (4) Vacuum Electrodynamics
• No hair theorem dictates that the event
  horizon is the asymptotic infinity, not the
  unipolar inductor.
• The horizon acts as a sink for inflowing
  plasam.
• (5,6,7) Magnetically dominated time-stationally
  perfect MHD winds in the ergosphere
• Plasma must exist in a black hole
  magnetosphere if an active energy extraction is
  to be possible.
• The magnetosphere is open circuited by the
  Goldreich-Julian current charge of axisymmetric
  magnetospheres The vacuum electric field
  switches sign in the magnetic flux tubes across
  the pair production region (ingoing if negative,
  outgoing if positive)
• The spacetime near the horizon is nothing
  more than an asymptotic infinity to the paired
  wind system.
• The coupling of the gravitational field to
  the plasma in the ergosphere through the dragging
  of inertial frams and torsional tug of war
  between the ergospheric plasama and distant
  asymptotic plasma

186
is responsible for driving the global poloidal
current system, resulting in a jet. (8,9)
Ergospheric disk dynamos and winds from event
horizon The GHM interaction of flux tube
model threading the equatorial plane of the
ergosphere, not the horizen, demonstrates how the
rotational energy of the hole powers the outgoing
wind. The ergospheric dynamos on flux tubes
threading the event horizon can be obtained in
terms of an MHD asymptotic infinity. (10, 11)
Extragalactic radio sources and non-pulsed black
holes The black hole GHM describes radio
loud AGNs. The Kerr-Newman black hole and
its magnetosphere can power a magnetically
dominated plasma winds, resulting in a jet
fomation with particle creation.