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Relativistic Astrophysics: I. Relativistic Astrophysics Fundamentals II. Specific Relativistic Astrophysics Problems

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

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Relativistic Astrophysics: I. Relativistic Astrophysics Fundamentals II. Specific Relativistic Astrophysics Problems

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  1. Relativistic Astrophysics:I. Relativistic Astrophysics FundamentalsII. Specific Relativistic Astrophysics Problems David Meier Jet Propulsion Laboratory Caltech

  2. 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

  3. Lecture I: Relativistic Astrophysics Fundamentals

  4. 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

  5. Background and Motivation

  6. 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 m0c2 • 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: UU = – c2 • 4-momentum: PP = – m02 c2 EK2 = p2 c2 + m02 c4 • 4-current: JJ = – q2 c2 + j2 • Spacetime Pythagorean theorem for the invariant 4-interval: ds2 = – c2 dt2 + dx2 +dy2 +dz2 • Spacetime (“Minkowski”) metric : ds2 = dxTg dx

  7. General Theory of Relativity(Large amounts of mass/energy warp spacetime) • Ideas introduced • Spacetime metric can be non-Minkowskian and evolving: • Time warp  gravitational force (g00) • 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

  8. 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

  9. 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 • Stable circular orbits (E < m0 c2): 6 rg < r <  • Innermost stable circular orbit (“last stable orbit”): rISCO = 6 rg • “Marginally bound” orbit (E = m0 c2) rmb = 4 rg • “Photon orbit”: rph = 3 rg • Black hole horizon (g00 = 0): rH = rSch 2 rg • Black hole interior: r < 2 rg

  10. 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

  11. Evolution of Classical Relativistic Gauge Fields:The Electromagnetic Field

  12. The Electromagnetic Field • 3+1 (space + time) Formulation for the EM Field (E, B) • Maxwell’s equations in vaccuum • 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

  13. 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 • 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

  14. 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)

  15. The Electromagnetic Field (cont.) • “Covariant” Formulation for the EM Field • Work with 4-D geometric equations, valid in any coordinate system • (/t, )  • (, A)  A • (qc, J) J • (E, B)  • Maxwell’s inhomogeneous equations then become (in matrix form), simply T  FT = 4  J / c 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. 2nd order derivatives of A 4-current source

  16. Evolution ofClassical Relativistic Gauge Fields:The Gravitational Field

  17. The Gravitational Field • Newtonian Gravity  a time-independent scalar theory • Gravitational potential and acceleration for a point mass m  =  Gm / rg =  (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 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 2nd order derivatives of g stress-energy-momentum tensor

  18. 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 spatialGij = 8  G Tij/ c4forward in time to determine the six spatialgij(t) • Examples of flat metrics that are sol’ns to Einstein’s equations Cartesian – Minkowski Spherical-Polar – Minkowski

  19. The Gravitational Field (cont.) • Examples of curved metrics that are solutions to Einstein’s equations • Schwarzschild (non-rotating black hole) • Kerr (rotating black hole) Kerr Schwarzschild

  20. 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  • 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: dU / d = U   U = 0 with becomes ^ ^ ^ ^

  21. Evolution of the Sources inClassical Relativistic Gauge Field Equations:Determining J and T

  22. 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 = qau 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  collna(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 }Twe can write the energy-momentum equations in the form we are looking for T {TFL + TEM}T = 0

  23. 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-spatial-current; 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

  24. 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 + pqg }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 + pqg 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

  25. 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 currentj 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, jand q are never computed during the simulation, only after the fact with Maxwell’s inhomogeneous equation J = c T  FT/ (4 )

  26. Summary of Nearly All of Relativistic (and non-Relativistic) Astrophysics  On a Single Slide: Ideal EGRMHD

  27. Black Hole Binary Coalescence Neutron Star Binary Coalescence The Ultimate Goal: Simulate EM Gravitational Collapse  To solve the problem of Electromagnetic Gravitational Collapse, we need to evolve both the gravitational and electromagnetic fields and their sources (matter and charge)

  28. Lecture II: Specific Relativistic Astrophysics Problems

  29. 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

  30. 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

  31. Classification of Relativistic Astrophysics Problems

  32. Classification of Rel. Astrophys. Problems • Relativistic Numerical Astrophysics (numerical astrophysicists) • 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  

  33. Classification of Rel. Astrophys. Problems (cont.) • Numerical Relativity (numerical relativists [physicists]) * 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  

  34. Quasar 1928+738 3C 273 M 87 Hummel et al. (1992) © MPIfR © ISAS, CfA 1. Magnetohydrodynamics in Flat Spacetime ©Nakamura & Meier (2004)

  35. a. Relativistic Magnetohydrodynamics (RMHD) RMHD • 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! MHD RHD FFDE HD GSS Eq

  36. Relativistic Magnetohydrodynamics (cont.) • Conservative Equations of RMHD in 3+1 Language • Conservative evolution equations • Conserved variables • Post-simulation variables

  37. Relativistic Magnetohydrodynamics (cont.) • Important wave speeds • Electromagnetic waves: • Alfven waves: • Sound waves: • Fast magnetosonic: ( B) • Slow magnetosonic: (|| B)

  38. 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 Emagdominate 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 • New evolved variable (relativistic internal energy for an adiabatic EOS): • Many, many interesting astrophysical problems can use RMHD • Relativistic jet propagation and stability • Pulsar magnetospheres • Supernova explosions • Relativistic shock waves

  39. b. Relativistic Hydrodynamics • Quasi-Conservative Equations of RHD are simpler without the EM field • Evolution equations and conserved variables • 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

  40. 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 • 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 = 0is often called the “degeneracy condition” • Manipulation of the equations gives the standard FFDE evolution equations • VF is applied as a boundary condition only • The following post-simulation variables can be computed (“force-free” electromagnetic field)

  41. 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

  42. 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 • Then, the FFDE equations can all be reduced to a single differential equation for the potential  the Grad-Schlüter-Shafranov equation • NOTES • 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

  43. 2. General Relativistic Magnetohydrodynamics (GRMHD) ©Uchida, Nakamura, & Hirose (2001) ©McKinney, & Gammie (2004)

  44. 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 • NOTES: • 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

  45. General Relativistic Magnetohydrodynamics (cont.) • In metrics of this type, the GRMHD equations can be written as follows • Variables of GRMHD • Post-simulation variables

  46. General Relativistic Magnetohydrodynamics (cont.) • NOTES: • Divergences and curls are performed in diagonal 3-space in the usual manner; e.g., • 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

  47. Black Hole Binary Coalescence 3. Evolving General Relativistic Dynamics (EGRD)

  48. 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: • Express the general metric in 3+1 notation • Choose 4 appropriate gauge conditions for the 4 quantities (,  ) • The “initial data” for the 6 independent ijare 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

  49. t n 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 +  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” • 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 singularity horizon grid

  50. 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 second-order time derivatives into one for the 3-metric and one for the extrinsic curvature: • And the constraints in this case have no explicit time derivatives so can be solved as elliptic equations at t = 0

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