Relativistic Astrophysics:
I. Relativistic Astrophysics Fundamentals
II. Specific Relativistic Astrophysics
Problems
David Meier
Jet Propulsion Laboratory
Caltech
Introductory Remarks
• Purpose of these two lectures
– To introduce or review relativity and its use in astrophysics
– To provide a background in the theory of gauge fields and conservation
laws  the source of all equations solved in R.A.
– To introduce the most important R.A. problems, the equations that govern
them, and the issues surrounding them
• What these lectures will not do
– Present detailed derivations and justification for the equations
– Present detailed numerical methods
– Present results of simulations or movies
• Notes on the level of the material
– If it is too easy, just treat it as review (perhaps from a different perspective)
– If it is too difficult, concentrate on the concepts, not the equations
– DO ask questions about concepts; don’t worry about equation derivations
Lecture I:
Relativistic Astrophysics
Fundamentals
Lecture I Outline
• Background and Motivation
–
–
–
–
Special theory
General theory
Physical situations that demand relativity be considered
Astrophysical systems that demand relativity
• Evolution of Classical Relativistic Gauge Fields
– Evolution of the electromagnetic field
– Evolution of the gravitational field
– Evolution of the sources
Background and Motivation
Special Theory of Relativity
(Objects moving near the speed of light behave in a specific manner)
• Ideas introduced
–
–
–
–
–
Cosmic speed limit:
  v/c < 1 ; 1 < W  (1 – 2) –1/2 < 
Energy/mass equivalence:
EK = m c2 = W m0 c2
Time dil./Fitzgerald Contr:
t = t0 / W ; x = x0 / W
Doppler factor:
  [(1 –  cos ) W] –1  [(1+)/(1–)]1/2  2W
Doppler effect & beaming:
n =  n0 ; Sn = 2 –  Sn0
– Space & time form a complete 4-D geometry
• Rotations, curvilinear X-forms, and boosts are all simply coordinate changes
• Magnitudes of 4-vectors/tensors remain unchanged (“invariant”)
– 4-velocity:
– 4-momentum:
– 4-current:
U  U = – c2
P  P = – m02 c2
J  J = – q2 c2 + j2
 EK2 = p2 c2 + m02 c4
• Spacetime Pythagorean theorem for the invariant 4-interval:
ds2 = – c2 dt2 + dx2 +dy2 +dz2


cdt 


• Spacetime (“Minkowski”) metric :
dx   dx 
 dy 
ds2 = dxT  g  dx



dz 
g=









1
0
0
0
0
1
0
0
0
0
1
0
0
0  η

0

1
General Theory of Relativity
(Large amounts of mass/energy warp spacetime)
• Ideas introduced
– Spacetime metric can be non-Minkowskian and evolving: g =
– Time warp  gravitational force (g00)








g00
g01
g02
g03
g01
g11
g12
g13
g02
g12
g22
g23
• Light bending by stars; gravitational lensing:
• Black holes; photon orbits:
– Space warp  non-Euclidian 3-geometry (gij)
• Sum of angles in triangle  180 ; solid angles  4
• Circumference
 2 r ; surface of sphere  4 r2 ; volume/mass 
/ r3
– Time shift  frame dragging (g0i):
– Extra terms in weak gravity (Post-Newtonian expansion)
• Precession of perihelion of Mercury’s orbit
• Gravitational waves:
Changes in gravitational field travel at speed of light
g03 
g13 
g23 
g33 
Physical Situations that Demand Relativity
• Special relativity: effects of a finite speed of light c
– Electrodynamics; if c, E & B would be constant & curl free
– Hot and/or dense matter equation of state
• Heat has mass
• Relativistic Maxwellian distribution for T > m c2 / k
• Creation/destruction of particles by collisions (nuclear reactions; phase changes)
– Radiation by plasmas
• Synchrotron radiation by relativistic electrons in a magnetic field
• Compton effect
– Compton heating of plasma by high energy photons
– Compton cooling of plasma by low energy photons
– Bulk motion of plasma
• Kinetic energy has mass
• Relativistic shock waves behave a little differently and dissipate kinetic energy
to heat
– Relativistic charge drift velocity: “beamed” current and Wq
Physical Situations that Demand Relativity (cont.)
• General relativity: effects of curved spacetime
– Gravitational redshift and time dilation:
time ticks more slowly in a strong gravitational field
– Special particle orbits and other radii near black holes
and other relativistic stars; for a non-rotating black hole:
• The canonical gravitational radius:
rg  GM/c2
•
•
•
•
•
•
6 rg < r < 
rISCO = 6 rg
rmb = 4 rg
rph = 3 rg
rH = rSch  2 rg
r < 2 rg
Stable circular orbits (E < m0 c2):
Innermost stable circular orbit (“last stable orbit”):
“Marginally bound” orbit (E = m0 c2)
“Photon orbit”:
Black hole horizon (g00 = 0):
Black hole interior:
Astrophysical Systems that Demand Relativity
• Numerical Astrophysics (IPAM 1)
–
–
–
–
–
accretion disk evolution around black holes
jet formation near pulsars and black holes
origin & growth of cosmic magnetic fields
astrophysical pair plasmas
supernovae
• Numerical Astrophysics (IPAM 3)
–
–
–
–
–
–
–
–
–
–
–
–
relativistic jets
relativistic shocks
black hole astrophysics
gravitational collapse
neutron star mergers
black hole mergers
gravitational waves: timing and spectroscopy
challenges of LIGO, LISA
cosmology, cosmic background radiation polarization
gamma-ray bursters
formulations of Einstein's field equations for numerical relativity
numerical studies of higher dimensional black holes
Evolution of
Classical Relativistic Gauge Fields:
The Electromagnetic Field
The Electromagnetic Field
• 3+1 (space + time) Formulation for the EM Field (E, B)
– Maxwell’s equations in vaccuum
Equation Type
Evolution Equation
Constraint
Homogeneous
B/t =  c  E
B = 0
Inhomogeneous
E/t = c  B  4J
E= 4q
– Homogeneous constraint equation is not independent of evolution eqn.
• If we satisfy B = 0 at t = 0,
• then  B/t = c  (  E ) = 0 =  ( B)/t
for all time
– Inhomogeneous constraint also is propagated, if charge is conserved
• If we satisfy E= 4q at t = 0, and if q/t + J = 0
• then  E/t = c  (  B )  4 J =  (4 q )/t =  ( E)/t
for all time
–  Maxwell’s equations REQUIRE charge conservation for all time
– There are 6 unknowns (E, B) and six evolution equations: everything is OK
The Electromagnetic Field (cont.)
• 3+1 (space + time) Formulation for the EM Potential (, A)
– The homogeneous equations imply that the fields can be derived from a
vector and a scalar potential
• B = 0  B =   A ;
B/t =  c  E  E =   A/ct
• Maxwell’s equations then reduce to 3 evolution equations and a constraint
Equation Type
Evolution Equation
Constraint
2A/c2t2 + 2A =
Inhomogeneous  ( /ct + A)  4J /c  2   (A ) /ct = 4q
– Now, however, we have 4 unknowns (, A), but only 3 evolution equations
–  The introduction of potentials has introduced another degree of freedom:
the “gauge”
– There is an infinite number of (, A) pairs that will generate the same (E, B)
– We need another (single, but arbitrary) equation for (, A)
– Example: Lorentz gauge;  /ct + A = 0, yielding four wave equations
Inhomogeneous 2A/c2t2 + 2A = 4J /c
2/c2t2 +  2 = 4q
The Electromagnetic Field (cont.)
– NOTES:
• Gauge freedom arises only because we have introduced potentials
• The fields (E, B) remain invariant under a gauge transformation
A = A + 
 =    /ct
– Gauge freedom is intimately related to the redundant constraints and to
conservation laws
•
•
•
•
We lose the redundant inhomogeneous constraint equation for the potentials
We gain a conservation law (q/t = J)
This law says nothing about the potentials (but a lot about the sources)
So, we must construct a new, arbitrary equation  the “gauge condition” 
in order to have 4 equations for the 4 unknowns (, A)
The Electromagnetic Field (cont.)
• “Covariant” Formulation for the EM Field
– Work with 4-D geometric equations, valid in any coordinate system
•
•
•
•
(/t, )
(, A)
(qc, J)
(E, B)

 
E x E y E z 
 0

x
z B y 
 A
0
B
 E
F   y

B z 0 B x 
 E
 J


y B x
 E z

B
0



0 B x B y B z 
x
z E y 
B
0
E
M  y

B E z 0 E x 

B z E y E x 0 









– Maxwell’s inhomogeneous equations then become (in matrix form), simply
T  FT = 4  J / c
2nd order derivatives of A
4-current source
with F derived from the potential A such that third derivatives cancel in
T  (T  FT) = 0
(a vector/tensor identity)
Therefore, 4-current is conserved by the EM gauge field equations
T  J = 0
So we need to add a gauge condition, such as the Lorentz condition
T  A = 0
because one of the inhomogeneous Maxwell equations is redundant
– What happened to the homogeneous equations (T  MT = 0)?
They are already satisfied by how F is computed from A.
Evolution of
Classical Relativistic Gauge Fields:
The Gravitational Field
The Gravitational Field
• Newtonian Gravity  a time-independent scalar theory
– Gravitational potential and acceleration for a point mass m
 =  Gm / r
g =  (Gm / r2) e r= 
– Gravitational potential for a distributed mass density m at point r is
 =  G m / | r – r | d3r
the Green’s function solution to Poisson’s equation for scalar potential 
 2 = 4 G m
• Einsteinian Gravity  a time-dependent tensor theory
– Einstein’s inhomogeneous field equation involves the Einstein tensor G
G = 8  G T / c4
2nd order derivatives of g
stress-energy-momentum tensor
with G derived from metric potentials g such that third derivatives cancel in
 T  GT = 0
(a tensor identity)
Therefore, energy & momentum are conserved by the Einstein field eqns (!)
T  TT = 0
– General Relativity is a gauge theory also; we lose 4 constraints, so we still
need to specify 4 additional “gauge conditions” to get 10 eqns for 10 unkns
The Gravitational Field (cont.)
– NOTES:
• The “gauge” in General Relativity is the coordinate system
• The gauge transformation is the Lorentz transform, with vectors/tensors given by
U = L  U
g = (L 1)T  g  (L 1)
• A typical numerical scheme for integrating Einstein’s field equations
– Applies four “coordinate conditions” to determine the 4 metric potentials g00 and g0i
– Solves the following constraints at t = 0 to determine gij(t=0) and gij /t
» G00 = 8  G T00 / c4 ( “Hamiltonian” or energy constraint )
» G0i = 8  G T0i / c4 ( 3 momentum constraints )
– Integrates the six spatial Gij = 8  G Tij / c4 forward in time to determine
the six spatial gij(t)
• Examples of flat metrics that are sol’ns to Einstein’s equations
Cartesian – Minkowski
ηC 









1
0
0
0
0
1
0
0
0
0
1
0
0
0

0

1
Spherical-Polar – Minkowski
ηS P 









1
0
0
0
0
1
0
0
0
0
r2
0







r 2 sin 2  
0
0
0
The Gravitational Field (cont.)
• Examples of curved metrics that are solutions to Einstein’s
equations
– Schwarzschild (non-rotating black hole)












g 
 1  2rr
g




0
0
 2rg 
1 

r 

0
0
1
0
0
0
0
0
0
r2
0
0
r 2 sin 2












– Kerr (rotating black hole)
Kerr
Schwarzschild
1/ 2
   r 2  j 2rg2 cos 
gKerr 












    
2
2
2
 2 sin 2 
 2c2
0
0

 2  2 sin 2 
 2c2
0
0

0
2

0
0
2
0
 2 sin 2  

 2c2







2

2 
sin

2


2
0
0


 r
1/ 2


2


2
2
2
2
2
   r  j rg   j rg  sin 





   r 2  2rg r  j rg2 


2rg2rc
2

j
 r2

 r 2 1


0
2rg 
r 

The Gravitational Field (cont.)
• How does metric curvature create gravity?
– How does a curved metric change vector calculus?
• Simple example: Coriolis acceleration on a rotating sphere due to motion in 
g 
1

0

0 
sin 2  
^
• Rotational velocity is:
   / t  V = V / g1/2
^
• Linear velocity is:
 / t  V = V
• With no forces, the equation of motion is
dV / dt = V / t + V   V = 0
or
^
^
V / t = – (V   V) = – (V  / g1/2 ) (  g /  ) = – 2 V  cos 
• The Coriolis pseudo-force is contained in the gradient operator , as it acts in this
simple curved metric
– How does a time-warped metric create gravity?
• Similar process; set external forces to zero and assume radial free-fall:

W

2r

1

r

U = W  1  2 r 

 
  r 

0


0
dU / d = U   U = 0 with
 
1/ 2
g
g
1/ 2






r
ˆ
v 




rˆ
becomes  (WV ) = 

W
GM W
1/2


r2
1 2rg / r 


Evolution of the Sources in
Classical Relativistic Gauge Field Equations:
Determining J and T
The Field Sources
• General Relativistic Statistical Mechanics and Fluid Theory
– Goal #1: Determine how to compute stress-energy tensor T and its evolution
• Step #1a: Begin with the general relativistic Boltzmann equation for each
particle species ‘a’
da/d  u  a + a ua = a,coll
u = (particle 4-velocity) and
a = qa u  F / (ma c)
(particle 4-acceleration)
• Step #1b: Integrate 1st & 2nd moments of u over u to get the multi-fluid equations
T  na(U + Va) = 0
T  { naUU + naUVa + naVaU + a }T = JaF/mac  ncollna(U + Va)
• Step #1c: Weight & sum the multi-fluid equations over particle mass ma to get the
MHD equations
T  mU = 0
T  { [m+(e+p)/c2]UU + [UH + HU]/c2 + p g] }T = J  F/c
NOTE: because T  FT = 4  J / c, J  F/c =  T  { [F  F – ¼(F : F) I]/4 }T
we can write the energy-momentum equations in the form we are looking for
T  {TFL + TEM}T = 0
The Field Sources (cont.)
where the two stress-energy-momentum tensors are
TFL  [m+(e+p)/c2]UU + [UH + HU]/c2 + p g]
TEM  [ F  F – ¼ (F : F) I] / (4 )
The relativistic Boltzmann equation tells us how to compute T and how to evolve it
• Interpreting the variables: In these equations the variables are measured in a variety
of reference frames, but these turn out to be the most convenient for the user
– Thermodynamic variables (m, e, p, q, eq, pq ): mass density, internal energy, pressure,
charge density, charge -weighted energy & pressure measured in the fluid rest frame
– 4-vector variables ( U, J ): 4-velocity and 4-current measured with respect to the
GLOBAL coordinates
– 4-vectors that have only 3 independent components ( H , j, j ): heat flux and 4-spatialcurrent; U  H = 0 and U  j = 0 ; also measured w.r.t. GLOBAL coordinates
– 3-vector variables in 3+1 equations ( V, J, D, H ): 3-velocity, 3-current, electric
displacement & magnetic fields measured with respect to the MOVING metric
(keeps V < c and W real)
– Conjugate E & M 3-vector variables ( E, B ): electric field & magnetic induction
measured with respect to the GLOBAL coordinates
The Field Sources (cont.)
– Goal #2: Determine how to compute 4-current J and its evolution
• Step #2a: Weight/sum the multi-fluid equations over particle charge qa to get the
charge dynamics equations
T  (qU + j) = 0
T  { [q+(eq+pq)/c2]UU + Uj + jU + pq g }T = p2 { (U + hj)  F/c   (qU + j) } / (4 )
• Step #2b: Recognize the following current vector and tensor:
4-current vector:
J = qU + j
charge-current-pressure tensor: C = [q+(eq+pq)/c2]UU + Uj + jU + pq g
T  CT = p2 { [(1  hq)U + hJ]  F/c   J } / (4 )
This is the famous “Ohm’s law” in its most general relativistic form. It describes not
only how J is related to E (= U  F), it also shows how J evolves when V  IR
The Field Sources (cont.)
– Goal #3: Simplify Ohm’s Law
• Step #3a: Recognize that p2 / (4 ) is a very, very large coefficient; the L.H.S. is
important only for very microscopic phenomena (current sheets, reconnection, etc.)
 J = [(1  hq)U + hJ]  F/c
This is the static Ohm’s law, and simulations that use it are called “Hall MHD”
• Step #3b: Recognize that the Hall coefficient h is likely to be small, leaving an
equation for only the spatial current j
 j = U  F/c
Simulations that use this simplified form are called “resistive MHD”
• Step #3c: Finally, recognize that most astrophysical plasmas are highly conductive
(  0), leaving simply
UF =0
Simulations that use this simplified form are called “ideal MHD”
In this case, j and q are never computed during the simulation, only after the fact
with Maxwell’s inhomogeneous equation
J = c T  FT / (4 )
Summary of Nearly All of
Relativistic (and non-Relativistic) Astrophysics
 On a Single Slide: Ideal EGRMHD
The Ultimate Goal: Simulate EM Gravitational Collapse
Neutron Star Binary Coalescence
Black Hole Binary Coalescence

To solve the problem of Electromagnetic Gravitational Collapse, we need to evolve both
the gravitational and electromagnetic fields and their sources (matter and charge)
Physics
Gravity Field
Matter
EM Field
Charge/Current
Non-Relativistic Equations
 2  = 4 G 
[ = GM/r ]
/t + (V) = 0
(V)/t +(VV) = p + JB/ c 
( e)/t + ( e V) = (p + e)V
B/t + c  E = 0
B=0
E = VB/c (Ohm’s law   1/  )
J = c (B) / 4 q = E / 4
Relativistic Equations
G = 8G T /c4
T  ( U) = 0
T  TT = 0
 T  MT = 0
U T  F= 0
J = c T  FT / (4)
Lecture II:
Specific Relativistic Astrophysics
Problems
Lecture II Outline
• Classification of Relativistic Astrophysics Problems
• Magnetohydrodynamics in Flat Spacetime
–
–
–
–
Relativistic Magnetohydrodynamics (RMHD)
Relativistic Hydrodynamics (RHD)
Force-Free Degenerate Electrodynamics (FFDE)
The Grad-Schlüter-Shafranov Equation (GSS)
• MHD in a Stationary, Strong Gravitational Field (GRMHD)
• Evolving Strong Gravity with No Sources (EGRE)
• General Relativistic Hydrodynamics in a Strong
Gravitational Field (EGRHD)
• Epilogue: EGRMHD
Remarks on Lecture II
• Almost exclusively, the current approach to all problems 
even General Relativity  is to convert the equations to a
3+1 evolution problem
(Even physicists sometimes can’t think [or at least compute] in true 4-D)
– Present-day computers are simply not powerful enough to contain all of 4-D
spacetime in memory
– Even containing a sufficiently large amount of 3-D space in memory is
difficult, sometimes impossible
– Approach:
• Solve for (or simply specify) the conditions on the initial hypersurface
• Evolve these conditions forward in the chosen time coordinate, keeping only a
few hypersurfaces in memory at any moment in the simulation
– For GR this creates huge problems in trying to avoid singularities in the
spacetime (both coordinate and real)
– Employing a stable numerical scheme is also crucial
Classification of
Relativistic Astrophysics Problems
Classification of Rel. Astrophys. Problems
• Relativistic Numerical Astrophysics (numerical astrophysicists)
Problem
EGRMHD


Solve
Curved metric
G=8GT/c4 ?
(g) ?
?
Solve
EM0 ?
T
T  FT =4J ?
X
X
X
X
X
X
X
X
RMHD
X
X
X
RHD
X
X
X
GRMHD
FFDE
X
TFL0
• Biggest advantage of Current N.A.: Relativity & the EM field
have now been added, both very important
• Biggest current drawback: Almost all simulations assume
adiabatic Equation of State; NO radiation, NO cooling
Classification of Rel. Astrophys. Problems (cont.)
• Numerical Relativity (numerical relativists [physicists])
Problem


Solve
Solve
Curved metric FL
T 0 ?
TEM0 ?
4
T
T
G=8GT/c ?
(g) ?
  F =4J ?
EGRMHD
X
X
X
EGRHD
X
X
X
EGRScalarD
X
X
*
EGRD
X
X
X
X
* A simple scalar (Klein-Gordon) field is evolved TSC=    – ½ ||2 g
(very limited astrophysical applications)
• Biggest advantage of current N.R.: Will lead to understanding of
gravitational waves, a new kind of cosmic radiation
• Biggest current drawback: Equations are SO unstable that few
have dared to add complications like matter or EM fields
1. Magnetohydrodynamics in Flat Spacetime
Quasar 1928+738
3C 273
Hummel et al. (1992)
©Nakamura & Meier (2004)
M 87
© MPIfR
© ISAS, CfA
a. Relativistic Magnetohydrodynamics (RMHD)
RMHD
MHD
RHD
HD
FFDE
GSS Eq
• Notes
–
–
–
–
Begin with EGRMHD equations
Assume flat, stationary Minkowski metric
Use 3+1 language: e.g., 4-velocity becomes U = (W, WV)
Two different formalisms used:
• Fully conservative scheme: Q/t = –   fQ
– Lends itself to high-resolution shock capture / accurate higher-order Godonov schemes
– Energy is conserved explicitly: that lost by field/motion goes into heat
• Quasi-conservative scheme for internal energy: e/t = –   fe + pdV work
– Reduces problems with negative pressures and code crashing
– Energy is not conserved explicitly: numerical viscous heating leaves the grid silently!
Relativistic Magnetohydrodynamics (cont.)
• Conservative Equations of RMHD in 3+1 Language
– Conservative evolution equations
D =   DV 




t
P =  T
t
E = c2  P  DV 




t
B = c E
 B =0
t
– Conserved variables
i






i
2
2
i
1
D = W m
P =  m   e  p  / c W V  4 c  E  B 






E =W W 1 mc2  W 2  e  p   p  81  E 2  B2 





Tij =  









m  e p







/ c2 W 2 V iV j







0 E3 E 2 

E = E3 0 E1
E 2 E1 0 
E = Vc  B

– Post-simulation variables


J = 41  c  B  E 
t 




p  81  E 2  B2   ij


q = 41 
 E



 81  Ei E j  Bi B j 


Relativistic Magnetohydrodynamics (cont.)
• Important wave speeds
– Electromagnetic waves:
– Alfven waves:
– Sound waves:
VEM
VA 
cS 
=
c
B
4


 m   e 



p  B2 / 4  / c2 


p
    1 c
m   e  p  B2 / 4  / c2

– Fast magnetosonic:
( B)
– Slow magnetosonic:
(|| B)
 c
as
as
B  
p  


VF  V A2  cS2 1


VA2 
c2 


1


2
2
VS  VA  cS  1 

2





1
2
V 2  c2  
 A
S  


4cS2VA2

VA cS
VA2  cS2
for cS  VA or VA  cS
Relativistic Magnetohydrodynamics (cont.)
– Procedure for solving
• Set up initial model
• Compute T and E from the primitive variables m, V, e, p, B and W
• Evolve D, P, E, and B forward in time
• Solve 2 non-linear algebraic equations in each cell to determine primitive variables
• Repeat for each time step
– Problems: If Ekinetic or Emag dominate E, then numerical errors can cause the solution
for e & p to be negative!
• Quasi-Conservative Equations of RMHD in 3+1 Language
– Replace total energy equation with thermal energy equation
 =   V    1 V




t




– New evolved variable (relativistic internal energy for an adiabatic EOS):
 = W  p = W   1 e


• Many, many interesting astrophysical problems can use RMHD
–
–
–
–
Relativistic jet propagation and stability
Pulsar magnetospheres
Supernova explosions
Relativistic shock waves
b. Relativistic Hydrodynamics
• Quasi-Conservative Equations of RHD are simpler without the EM field
– Evolution equations and conserved variables
D =   DV 
i =     e  p  / c2 W 2 V i
D
=
W

P
m
m 





t


P =  T
E =W W 1 mc2  W 2  e  p   p


t
ij =     e  p  / c2 W 2 V iV j  p ij
E = c2  P  DV 
T

m 





t


– Only 1 algebraic equation (for W) needs to be solved in each cell
• NOTES
– RHD was popular in the 1990s and early 2000s before it was realized how important
the EM field was and before good MHD techniques were developed
– RHD is very useful for perfecting high-resolution shock capture techniques
• Astrophysical applications of RHD
– Relativistic astrophysics jets ONLY when the flow is super-magnetosonic and kinetic
energy dominated
– Supernova explosions
– Relativistic shock waves
c. Force-Free Degenerate Electrodynamics
• RMHD without any matter inertia (only charges and currents to create field)
– Start with RMHD equations with no matter inertia
q E  Jc  B = 0
(“force-free” electromagnetic field)
B = c  E
 B =0
t
E = c  B  4 J
 E = 4q
t
V
E =  cF  B
– VF < c is the “velocity of the magnetic field”; NOTE: VF is defined to be  B
• NOTES
– The force-free condition and Ohm’s law imply that E  B = 0 and B2– E2 > 0
– E  B = 0 is often called the “degeneracy condition”
– Manipulation of the equations gives the standard FFDE evolution equations
B =c  E
t
E = c  B  4 J
t
J=

c 
4 B2 
E Ε  B   B   B  E   E 
– VF is applied as a boundary condition only
– The following post-simulation variables can be computed
q = 41 
VF = c E 2B
 E
B
c. Force-Free Degenerate Electrodynamics (cont.)
– Advantages
• Simple system of equations
• Treats highly-relativistic problems
– Disadvantages
• Not good for pulsar interiors, supernova core collapse, accretion disks or anywhere matter
inertia can dominate
• Cannot handle small-scale phenomena: current sheets, magnetic reconnection
• There is not an easy way to introduce finite resistivity or the more general charge dynamical
equations
• Astrophysical applications of FFDE
– Very strong field problems: pulsar and black hole magnetospheres
– Poynting-flux-dominated jets with negligible matter inertia
d. The Grad-Schlüter-Shafranov Equation
• Time-independent, axisymmetric FFDE
– With /t = 0 and / = 0 we can define a SINGLE SCALAR POTENTIAL
 with the following properties

E =  cF
B = 1 e ˆ  2I e ˆ

R
Rc 
– Then, the FFDE equations can all be reduced to a single differential equation for the
potential  the Grad-Schlüter-Shafranov equation

• NOTES










2
L 







1  1   1 d F2  2  2 dI 2 = 0
R2 R
2c2 d  
R2c2 d
– F = VF / R is the angular velocity of the magnetic field at the inner boundary
– RL  c /F is the “radius of the light cylinder” (where VF=c if B did not bend backward)
– We MUST specify two functions of  : the rotation F() and poloidal current I()
distributions; these are NOT determined by the GSS solution
– This form of the GSS equation is sometimes called the “pulsar equation”
– When VF << c (RL/R  ), the GSS equation is identical to the force-free Tokamak
equation
– The GSS equation is relatively easy to solve numerically
– When re-derived in curved spacetime, it also describes stationary black hole
magnetospheres
2. General Relativistic Magnetohydrodynamics
(GRMHD)
©Uchida, Nakamura, & Hirose
(2001)
©McKinney, & Gammie (2004)
General Relativistic Magnetohydrodynamics
• The general 3+1 metric
– In order to discuss standard GRMHD we need to express the metric in 3+1 language;
 


ANY spacetime metric can be written in the form
  2

2


      


1
2
3 


j

2
2
2
2
2
i
i

ds =  c     dt  2icdx dt   ij dx dx
 









g



1
11
12
13




=  2 c2 dt 2   ij   icdt  dxi    jcdt  dx j 


   

 





• NOTES:
2
12
3
22
13  23 13 

– This allows us to work with a global time coordinate t
– The coefficients have the following common names
•   “lapse function”; describes how time passes at different points in spacetime
•   “shift vector”; describes how coordinates change with time
•   “3-metric”; describes how space is curved
• Diagonal, stationary metrics
– Even the complicated Kerr metric
• Does not change with time t
• Does not have off-diagonal spatial elements
– Divergences, curls, etc. are easy in this case
g 
  2
2
     

 


1



2



3

23 
1
2
 h1 
0
2
0
 h2 
0
0




0 


0 

2
h3 

3
2
 
General Relativistic Magnetohydrodynamics (cont.)
• In metrics of this type, the GRMHD equations can be written as follows
D =   D V   c 



t

 

P =  T   cP    E  Dc2    f
 P σ frame _ dragging


pseudo


t






E =   c2  P  DV    cE    Pc 2  T :σ


frame _ dragging



 
t
B = c E
t
 B =0
• Variables of GRMHD
D = W m

i

Pi =  m   e  p  / c2 W 2 V i  41 c  E  B 






E =W W 1 mc2  W 2  e  p   p  81  E 2  B2 











Tij =  m   e  p  / c2 W 2 V iV j   p  81  E 2  B2   ij  81  Ei E j  Bi B j 
















2 

1




0 E3 E
E = E3 0 E
E 2 E1 0

E = D    B
D = Vc  B
• Post-simulation variables
J = q  c  41  c  H  D 
t 



H = B    D
q = 41 
 D
General Relativistic Magnetohydrodynamics (cont.)
• NOTES:
– Divergences and curls are performed in diagonal 3-space in the usual manner; e.g.,


h
h
h

  1 2 3 V i 
 V = 1

h1h2h3 xi  hi



– The h’s take the place of the 3-metric 
– The lapse function  determines the gravitational force and time dilation
– The shift vector  takes into account “frame dragging” by the rotating (or otherwise
moving) black hole
– Evolution of the equations is very similar to RMHD, with the following differences
• Spatial gradients are in curvilinear coordinates
• The (known) lapse function  causes retarded evolution
• There are new terms involving gradients of , the shift vector  , and other pseudo-forces
– This scheme is rather complicated, but illustrates the connection with RMHD
– Newer schemes, using the 4-velocity U, are simpler; and U can be >> c
• Astrophysical applications of GRMHD
– Accretion flows near neutron stars and black holes: detailed accretion disk simulations
– Jet production by relativistic accretion disks
– Some gamma-ray burst simulations
3. Evolving General Relativistic Dynamics
(EGRD)
Black Hole Binary Coalescence
Evolving General Relativistic Dynamics
• Preface:
– After all of Einstein’s work in developing a covariant theory of gravity, numerical
relativists do their best to turn GR back into a 3+1 theory
– True 4-D simulations/models must await the advent of supercomputers perhaps a
million times more powerful than at present
– How can there be gravity with no matter?
• These generally are vacuum solutions outside of black hole singularities
• There is matter; it is just outside the computational boundaries
• Method of solution:
  2
2
     

 
1
g  

2


3

1
2

3 

 11  12  13 
 12  22  23 
 13  23  13 
– Express the general metric in 3+1 notation
– Choose 4 appropriate gauge conditions for the 4 quantities (,  )
– The “initial data” for the 6 independent ij are determined by solving the constraints
on the initial hypersurface
G00 = 0
G0i = 0
– Use the 6 non-redundant Einstein field equations to evolve the 6 ij
Gij = 0
(i j)
– Periodically check the constraints to see if they really are propagated correctly
Evolving General Relativistic Dynamics (cont.)
• Choosing the coordinate conditions: “slicing”
– Often called “slicing” because it slices up or “foliates” spacetime into a series of 3D
spatial hypersurfaces at different time steps
– The flow of time is not necessarily perpendicular to these hypersurfaces: the time
vector is actually
t
n
t =n+ 
where n  – t is the normal to the hypersurface
– So, choosing (,  ) determines the actual flow of time in the simulation
– Generally, (,  ) must be chosen at each time step (or nearly so) in order to sense
and avoid singularities
– Some common coordinate/gauge conditions are
• Geodesic slicing:  =1
• Harmonic slicing:  = ( det  )1/2
• Maximal slicing: 2 =  Kij Kij

 
where the “extrinsic curvature” Kij = 1  i  j  j i  ij 
• Avoiding physical singularities: “excision”
– Slicing is successful in avoiding coordinate singularities (e.g., poles),
but not physical singularities like at the center of black holes
– Modern GR simulations excise (or cut out) the centers of coalescing
black holes to keep the metric from becoming infinite
– Boundary conditions are, of course, crucial
2 

ct


singularity
horizon
grid
Evolving General Relativistic Dynamics (cont.)
• The 4-D Einstein Field Equations  FINALLY!
– The Einstein curvature tensor is given by
G  R – ½ g R 
where the Ricci tensor is given by the contraction (trace) of the Riemann tensor
R  R 
R  ,  – ,  +    –   
and the Christoffel symbols (connection coefficients) are linear combinations of first
derivatives of the metric tensor
 = ½ (g,  + g,  – g,  )
– As usual, Greek indices range from 0 to 3, a comma (,) denotes differentiation with
respect to a coordinate, and repeated indices indicate summation over all 4
dimensions
• The 3+1 Einstein Field Equations (one of many, many formulations)
– Standard method of numerically integrating the Field equatios is to split the secondorder time derivatives into one for the 3-metric and one for the extrinsic curvature:
 ij
= 2 Kij  i  j  j i
t
Kij
t
=
  Rij  2 Ki K j  K Kij 

 i j    Kij

Ki  j 

K j i
– And the constraints in this case have no explicit time derivatives so can be solved as
elliptic equations at t = 0
ij
2
 ij

R  K
 Kij K
= 0
 j  K   ij K  = 0
Evolving General Relativistic Dynamics (cont.)
• NOTEs:
– Sometimes the connection coefficients n are used as intermediate variables
instead of the extrinsic curvature
– Formulations in the “weak form” exist for Finite Element methods,
in both 3+1 and 4-D
– In the coalescing black hole problem, the stability of the numerical scheme is
crucial, and higher-order Godunov schemes have a lot of promise
• Astrophysical applications
–
–
–
–
Any vacuum-dominated, strong field system
Mainly single black holes and coalescing binary black holes
Main issues currently are just obtaining stable, reasonable solutions
Future issues are:
• What is the behavior and waveform of the emitted gravitational radiation?
• Can we generate a grid of waveforms to use as templates for LIGO and LISA?
• What signatures in the emitted wave have the most information about black hole mass,
binary mass ratio, initial spins, and final spin?
4. Evolving General Relativistic Hydrodynamics
(EGRHD): Putting RHD & EGRD Together
Neutron Star Binary Coalescence
©Miller (2001)
Evolving General Relativistic Hydrodynamics
• Method of solution:
– Current techniques of combining Relativistic Hydrodynamics and General
Relativistic Dynamics are fairly straightforward:
• Evolve the metric, given the fluid distribution and velocity (“lather”)
• Evolve the hydrodynamical fluid, given the metric (“rinse”)
• Repeat
– If ONLY neutron stars are involved, the computation is actually much easier that the
coalescing black hole problem (until a horizon and singularity form)!
• Astrophysical applications:
– Coalescence of non-magnetized neutron stars
– Tidal disruption and accretion of a neutron star by a black hole
– Main issues currently are
• What are the effects of the computational boundaries?
• What amount of computational resources is needed to get an accurate simulation?
• Does a black hole form immediately, or is a rapidly-spinning neutron star created?
– Future issues are:
• What is the behavior and waveform of the emitted gravitational radiation?
• Can we generate a grid of waveforms to use as templates for LIGO and LISA?
• What signatures in the emitted wave have the most information about final star mass,
binary mass ratio, initial spins, final spin, equation of state of nuclear matter?
Epilogue:
Working Toward Full Evolving General
Relativistic Hydrodynamics (EGMRHD)
Full Black Hole Formation and GammaRay Burst Simulation
Evolving General Relativistic Magnetohydrodynamics
• Comments:
– No group appears to be doing this as yet; it sounds extraordinarily difficult
– However, it would not take a huge amount of work to marry a GRMHD and
an EGRHD code (perhaps 1 or 2 full years of work)
– First order of business is to obtain astrophysical results
– Accurate gravitational wave generation and extraction will be much more
difficult
• Astrophysical applications:
–
–
–
–
Coalescence of magnetized neutron stars
Tidal disruption and magnetized accretion of a neutron star by a black hole
Relativistic jet generation in a gamma-ray burst central engine
Detailed study of “hypercritical” accretion (solar masses per second)
processes
While such interesting and complex simulations are not very
far off, we are only beginning to explore this very rich field of
relativistic astrophysics
Suggested Reading
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•
•
•
•
•
•
•
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Cook, G.B. 2001, Living Reviews in Relativity, http://relativity.livingreviews.org/Articles/lrr-2000-5/index.html.
“Initial Data for Numerical Relativity”.
Font, J.A. 2003, Living Reviews in Relativity, http://relativity.livingreviews.org/Articles/lrr-2003-4/index.html.
“Numerical Hydrodynamics in General Relativity”.
Koide, S., Shibata, K., Kudoh, T. & Meier, D.L. 2001, J. Korean Astro. Soc., 34, S215–224. “Numerical Method
for General Relativistic Magnetohydrodynamics in Kerr Spacetime”.
Komissarov, S.S. 2002, Mon. Not. Royal. Astro. Soc., 336, 759 – 766. “Time-dependent force-free, degenerate
electrodynamics”.
Komissarov, S.S. 2004, Mon. Not. Royal. Astro. Soc., 350, 427 – 448. “Electrodynamics of black hole
magnetospheres”.
Marti, J.M. & Müller, E. 2003, Living Reviews in Relativity, http://relativity.livingreviews.org/Articles/lrr-20037/index.html. “Numerical Hydrodynamics in Special Relativity”.
Meier, D.L. 2004, Astrophys. J., 605, 340 – 349. “Ohm’s Law in the Fast Lane: General Relativistic Charge
Dynamics”.
Miller, M., Gressman, P., & Suen, W.-M. 2004, Phys. Rev. D., 69, 064026. “Towards a realistic neutron star
binary inspiral: Initial data and multiple orbit evolution in full general relativity”.
Misner, C., Thorne, K.S., & Wheeler, J.A. 1973, Gravitation, (Freeman: San Francisco, CA).
NASA 2000, CAN Final Report NCCS5-153. “A Multipurpose Code for 3-D Relativistic Astrophysics and
Gravitational Wave Astronomy: Application to Coalescing Neutron Star Binaries”.
Poisson, E. 2004, Living Reviews in Relativity, http://relativity.livingreviews.org/Articles/lrr-2004-6/index.html.
“The Motion of Point Particles in Curved Spacetime” .
Reula, O.A. 1998, Living Reviews in Relativity, http://relativity.livingreviews.org/Articles/lrr-1998-3/index.html.
“Hyperbolic Methods for Einstein's Equations”.
Thorne, K.S., Price, R.H., & MacDonald, D.A. 1986, Black Holes: The Membrane Paradigm, (Yale: New Haven,
CT).