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Relativistic Astrophysics: I. Relativistic Astrophysics Fundamentals II. Specific Relativistic Astrophysics Problems David Meier Jet Propulsion Laboratory

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  • Relativistic Astrophysics:

    I. Relativistic Astrophysics Fundamentals

    II. Specific Relativistic Astrophysics ProblemsDavid MeierJet Propulsion LaboratoryCaltech

  • Introductory RemarksPurpose of these two lecturesTo introduce or review relativity and its use in astrophysicsTo 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 doPresent detailed derivations and justification for the equationsPresent detailed numerical methodsPresent results of simulations or moviesNotes on the level of the materialIf 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 equationsDO ask questions about concepts; dont worry about equation derivations

  • Lecture I: Relativistic Astrophysics Fundamentals

  • Lecture I OutlineBackground and MotivationSpecial theoryGeneral theoryPhysical situations that demand relativity be considered Astrophysical systems that demand relativity

    Evolution of Classical Relativistic Gauge FieldsEvolution of the electromagnetic fieldEvolution of the gravitational fieldEvolution of the sources

  • Background and Motivation

  • Special Theory of Relativity(Objects moving near the speed of light behave in a specific manner)Ideas introducedCosmic speed limit: v/c < 1 ; 1 < W (1 2) 1/2 < Energy/mass equivalence:EK = m c2 = W m0 c2Time dil./Fitzgerald Contr:t = t0 / W ; x = x0 / WDoppler factor: [(1 cos ) W] 1 [(1+)/(1)]1/2 2WDoppler effect & beaming: n = n0 ; Sn = 2 Sn0Space & time form a complete 4-D geometryRotations, curvilinear X-forms, and boosts are all simply coordinate changesMagnitudes of 4-vectors/tensors remain unchanged (invariant)4-velocity:U U = c24-momentum:P P = m02 c2 EK2 = p2 c2 + m02 c44-current:J J = q2 c2 + j2 Spacetime Pythagorean theorem for the invariant 4-interval:ds2 = c2 dt2 + dx2 +dy2 +dz2Spacetime (Minkowski) metric :ds2 = dxT g dx

  • General Theory of Relativity(Large amounts of mass/energy warp spacetime)Ideas introducedSpacetime 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 4Circumference 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 Mercurys orbitGravitational waves: Changes in gravitational field travel at speed of light

  • Physical Situations that Demand RelativitySpecial relativity: effects of a finite speed of light cElectrodynamics; if c, E & B would be constant & curl freeHot and/or dense matter equation of stateHeat has massRelativistic Maxwellian distribution for T > m c2 / kCreation/destruction of particles by collisions (nuclear reactions; phase changes)Radiation by plasmasSynchrotron radiation by relativistic electrons in a magnetic fieldCompton effectCompton heating of plasma by high energy photonsCompton cooling of plasma by low energy photonsBulk motion of plasmaKinetic energy has massRelativistic shock waves behave a little differently and dissipate kinetic energy to heatRelativistic charge drift velocity: beamed current and Wq

  • Physical Situations that Demand Relativity (cont.)General relativity: effects of curved spacetimeGravitational redshift and time dilation: time ticks more slowly in a strong gravitational fieldSpecial particle orbits and other radii near black holes and other relativistic stars; for a non-rotating black hole:The canonical gravitational radius:rg GM/c2Stable circular orbits (E < m0 c2): 6 rg < r < Innermost stable circular orbit (last stable orbit):rISCO = 6 rgMarginally bound orbit (E = m0 c2)rmb = 4 rgPhoton orbit:rph = 3 rgBlack hole horizon (g00 = 0):rH = rSch 2 rgBlack hole interior:r < 2 rg

  • Astrophysical Systems that Demand RelativityNumerical Astrophysics (IPAM 1) accretion disk evolution around black holesjet formation near pulsars and black holesorigin & growth of cosmic magnetic fieldsastrophysical pair plasmassupernovaeNumerical 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 Field3+1 (space + time) Formulation for the EM Field (E, B) Maxwells 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 timeInhomogeneous constraint also is propagated, if charge is conservedIf we satisfy E= 4q at t = 0, and if q/t + J = 0then E/t = c ( B ) 4 J = (4 q )/t = ( E)/t for all time

    Maxwells equations REQUIRE charge conservation for all time

    There are 6 unknowns (E, B) and six evolution equations: everything is OK

    Equation TypeEvolution EquationConstraintHomogeneousB/t = c E B = 0InhomogeneousE/t = c B 4JE= 4q

  • 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 potentialB = 0 B = A ; B/t = c E E = A/ct Maxwells 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 gaugeThere 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; /ct + A = 0, yielding four wave equations

    Equation TypeEvolution EquationConstraintInhomogeneous2A/c2t2 + 2A = ( /ct + A) 4J /c 2 (A ) /ct = 4q

    Inhomogeneous2A/c2t2 + 2A = 4J /c2/c2t2 + 2 = 4q

  • The Electromagnetic Field (cont.)NOTES:Gauge freedom arises only because we have introduced potentialsThe fields (E, B) remain invariant under a gauge transformation

    A = A + = /ct

    Gauge freedom is intimately related to the redundant constraints and to conservation lawsWe lose the redundant inhomogeneous constraint equation for the potentialsWe 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 FieldWork with 4-D geometric equations, valid in any coordinate system

    (/t, ) (, A) A(qc, J) J(E, B) Maxwells inhomogeneous equations then become (in matrix form), simply T FT = 4 J / cwith F derived from the potential A such that third derivatives cancel inT (T FT) = 0 (a vector/tensor identity)Therefore, 4-current is conserved by the EM gauge field equations T J = 0So we need to add a gauge condition, such as the Lorentz condition T A = 0because 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 A4-current source

  • Evolution ofClassical Relativistic Gauge Fields:The Gravitational Field

  • The Gravitational FieldNewtonian Gravity a time-independent scalar theoryGravitational 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 Greens function solution to Poissons equation for scalar potential 2 = 4 G m Einsteinian Gravity a time-dependent tensor theoryEinsteins inhomogeneous field equation involves the Einstein tensor G G = 8 G T / c4with 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 = 0General 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 unkns2nd order derivatives of gstress-energy-momentum tensor

  • The Gravitational Field (cont.)NOTES:The gauge in General Relativity is the coordinate systemThe 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 Einsteins field equationsApplies four coordinate conditions to determine the 4 metric potentials g00 and g0iSolves 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 solns to Einsteins equations Cartesian Minkowski Spherical-Polar Minkowski

  • The Gravitational Field (cont.)Examples of curved metrics that are solutions to Einsteins equationsSchwarzschild (non-rotating black hole)

    Kerr (rotating black hole) Kerr Schwarzschild

  • 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 / g1/2Linear velocity is: / t V = V With no forces, the equation of motion isdV / dt = V / t + V V = 0orV / t = (V V) = (V / g1/2 ) ( g / ) = 2 V cos The Coriolis pseudo-force is contained in the gradient operator , as it acts in this simple curved metricHow 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


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

  • The Field SourcesGeneral Relativistic Statistical Mechanics and Fluid TheoryGoal #1: Determine how to compute stress-energy tensor T and its evolutionStep #1a: Begin with the general relativistic Boltzmann equation for each particle species ada/d u a + a ua = a,collu = (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 { naUU + naUVa + naVaU + a }T = JaF/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]UU + [UH + HU]/c2 + p g] }T = J F/c

    NOTE: because T FT = 4 J / c, J F/c = T { [F F (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 areTFL [m+(e+p)/c2]UU + [UH + HU]/c2 + p g] TEM [ F F (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 frame4-vector variables ( U, J ): 4-velocity and 4-current measured with respect to the GLOBAL coordinates4-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 coordinates3-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 evolutionStep #2a: Weight/sum the multi-fluid equations over particle charge qa to get the charge dynamics equations T (qU + j) = 0T { [q+(eq+pq)/c2]UU + Uj + jU + 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]UU + Uj + jU + pq g

    T CT = p2 { [(1 hq)U + hJ] F/c J } / (4 )

    This is the famous Ohms 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 Ohms LawStep #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 hq)U + hJ] F/cThis is the static Ohms 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/cSimulations that use this simplified form are called resistive MHD

    Step #3c: Finally, recognize that most astrophysical plasmas are highly conductive ( 0), leaving simplyU F = 0Simulations that use this simplified form are called ideal MHDIn this case, j and q are never computed during the simulation, only after the fact with Maxwells 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 CollapseTo solve the problem of Electromagnetic Gravitational Collapse, we need to evolve both the gravitational and electromagnetic fields and their sources (matter and charge)

    PhysicsNon-Relativistic EquationsRelativistic EquationsGravity Field 2 = 4G [ = GM/r ]G = 8G T /c4Matter/t + (V) = 0(V)/t +(VV) = p + JB/ c ( e)/t + ( e V) = (p + e)VT ( U) = 0T TT = 0EM FieldB/t + c E = 0 B=0T MT = 0Charge/CurrentE = VB/c (Ohms law 1/ )J = c (B) / 4 q = E / 4U T F = 0J = c T FT / (4)

  • Lecture II: Specific Relativistic Astrophysics Problems

  • Lecture II OutlineClassification of Relativistic Astrophysics ProblemsMagnetohydrodynamics in Flat SpacetimeRelativistic Magnetohydrodynamics (RMHD)Relativistic Hydrodynamics (RHD)Force-Free Degenerate Electrodynamics (FFDE)The Grad-Schlter-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 IIAlmost exclusively, the current approach to all problems even General Relativity is to convert the equations to a 3+1 evolution problem(Even physicists sometimes cant 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 memoryEven containing a sufficiently large amount of 3-D space in memory is difficult, sometimes impossibleApproach: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 simulationFor 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. ProblemsRelativistic Numerical Astrophysics (numerical astrophysicists)

    Biggest advantage of Current N.A.: Relativity & the EM field have now been added, both very importantBiggest current drawback: Almost all simulations assume adiabatic Equation of State; NO radiation, NO cooling

    ProblemSolve G=8GT/c4 ?Curved metric (g) ?TFL0 ?Solve T FT =4J ?TEM0 ?EGRMHDXXXXXGRMHDXXXXRMHDXXXRHDXFFDEXX

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

    * A simple scalar (Klein-Gordon) field is evolved TSC= ||2 g (very limited astrophysical applications)Biggest advantage of current N.R.: Will lead to understanding of gravitational waves, a new kind of cosmic radiationBiggest current drawback: Equations are SO unstable that few have dared to add complications like matter or EM fields

    ProblemSolve G=8GT/c4 ?Curved metric (g) ?TFL0 ?Solve T FT =4J ?TEM0 ?EGRMHDXXXXXEGRHDXXXEGRScalarDXX*EGRDXX

  • 1. Magnetohydrodynamics in Flat SpacetimeNakamura & Meier (2004)

  • a. Relativistic Magnetohydrodynamics (RMHD)NotesBegin with EGRMHD equationsAssume flat, stationary Minkowski metricUse 3+1 language: e.g., 4-velocity becomes U = (W, WV)Two different formalisms used:Fully conservative scheme: Q/t = fQLends itself to high-resolution shock capture / accurate higher-order Godonov schemesEnergy is conserved explicitly: that lost by field/motion goes into heatQuasi-conservative scheme for internal energy: e/t = fe + pdV workReduces problems with negative pressures and code crashingEnergy is not conserved explicitly: numerical viscous heating leaves the grid silently!RMHDRHDMHDHDFFDEGSS Eq

  • Relativistic Magnetohydrodynamics (cont.)Conservative Equations of RMHD in 3+1 LanguageConservative evolution equations

    Conserved variables

    Post-simulation variables

  • Relativistic Magnetohydrodynamics (cont.)Important wave speeds

    Electromagnetic waves:

    Alfven waves:

    Sound waves:

    Fast magnetosonic:( B)Slow magnetosonic:(|| B)

  • Relativistic Magnetohydrodynamics (cont.)Procedure for solvingSet up initial modelCompute T and E from the primitive variables m, V, e, p, B and W Evolve D, P, E, and B forward in timeSolve 2 non-linear algebraic equations in each cell to determine primitive variables Repeat for each time stepProblems: 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 LanguageReplace total energy equation with thermal energy equation

    New evolved variable (relativistic internal energy for an adiabatic EOS):

    Many, many interesting astrophysical problems can use RMHDRelativistic jet propagation and stabilityPulsar magnetospheresSupernova explosionsRelativistic shock waves

  • b. Relativistic HydrodynamicsQuasi-Conservative Equations of RHD are simpler without the EM fieldEvolution equations and conserved variables

    Only 1 algebraic equation (for W) needs to be solved in each cell

    NOTESRHD was popular in the 1990s and early 2000s before it was realized how important the EM field was and before good MHD techniques were developedRHD is very useful for perfecting high-resolution shock capture techniques

    Astrophysical applications of RHDRelativistic astrophysics jets ONLY when the flow is super-magnetosonic and kinetic energy dominatedSupernova explosionsRelativistic shock waves

  • c. Force-Free Degenerate ElectrodynamicsRMHD 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

    NOTESThe force-free condition and Ohms 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

    VF is applied as a boundary condition only The following post-simulation variables can be computed(force-free electromagnetic field)

  • c. Force-Free Degenerate Electrodynamics (cont.)AdvantagesSimple system of equationsTreats highly-relativistic problemsDisadvantagesNot good for pulsar interiors, supernova core collapse, accretion disks or anywhere matter inertia can dominateCannot handle small-scale phenomena: current sheets, magnetic reconnectionThere is not an easy way to introduce finite resistivity or the more general charge dynamical equations

    Astrophysical applications of FFDEVery strong field problems: pulsar and black hole magnetospheresPoynting-flux-dominated jets with negligible matter inertia

  • d. The Grad-Schlter-Shafranov EquationTime-independent, axisymmetric FFDEWith /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-Schlter-Shafranov equation

    NOTESF = VF / R is the angular velocity of the magnetic field at the inner boundaryRL 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 solutionThis form of the GSS equation is sometimes called the pulsar equation

    When VF

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

  • General Relativistic MagnetohydrodynamicsThe 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 tThe 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 curvedDiagonal, stationary metricsEven the complicated Kerr metricDoes not change with time tDoes not have off-diagonal spatial elementsDivergences, curls, etc. are easy in this case

  • General Relativistic Magnetohydrodynamics (cont.)In metrics of this type, the GRMHD equations can be written as follows

    Variables of GRMHD

    Post-simulation variables

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

    The hs take the place of the 3-metric The lapse function determines the gravitational force and time dilationThe shift vector takes into account frame dragging by the rotating (or otherwise moving) black holeEvolution of the equations is very similar to RMHD, with the following differencesSpatial gradients are in curvilinear coordinatesThe (known) lapse function causes retarded evolutionThere are new terms involving gradients of , the shift vector , and other pseudo-forcesThis scheme is rather complicated, but illustrates the connection with RMHDNewer schemes, using the 4-velocity U, are simpler; and U can be >> c

    Astrophysical applications of GRMHDAccretion flows near neutron stars and black holes: detailed accretion disk simulationsJet production by relativistic accretion disksSome gamma-ray burst simulations

  • 3. Evolving General Relativistic Dynamics (EGRD)

  • Evolving General Relativistic DynamicsPreface: After all of Einsteins 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 presentHow can there be gravity with no matter?These generally are vacuum solutions outside of black hole singularitiesThere is matter; it is just outside the computational boundaries

    Method of solution:Express the general metric in 3+1 notationChoose 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 = 0G0i = 0Use 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: slicingOften called slicing because it slices up or foliates spacetime into a series of 3D spatial hypersurfaces at different time stepsThe flow of time is not necessarily perpendicular to these hypersurfaces: the time vector is actuallyt = n + where n t is the normal to the hypersurfaceSo, choosing (, ) determines the actual flow of time in the simulationGenerally, (, ) must be chosen at each time step (or nearly so) in order to sense and avoid singularitiesSome common coordinate/gauge conditions areGeodesic slicing: =1Harmonic slicing: = ( det )1/2Maximal slicing: 2 = Kij Kij where the extrinsic curvatureAvoiding physical singularities: excisionSlicing is successful in avoiding coordinate singularities (e.g., poles), but not physical singularities like at the center of black holesModern GR simulations excise (or cut out) the centers of coalescing black holes to keep the metric from becoming infiniteBoundary conditions are, of course, crucialsingularityhorizongrid

  • Evolving General Relativistic Dynamics (cont.)The 4-D Einstein Field Equations FINALLY!The Einstein curvature tensor is given byG R g R where the Ricci tensor is given by the contraction (trace) of the Riemann tensorR R R , , + and the Christoffel symbols (connection coefficients) are linear combinations of first derivatives of the metric tensor = (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 dimensionsThe 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

  • Evolving General Relativistic Dynamics (cont.)NOTEs: Sometimes the connection coefficients are used as intermediate variables instead of the extrinsic curvatureFormulations in the weak form exist for Finite Element methods, in both 3+1 and 4-DIn the coalescing black hole problem, the stability of the numerical scheme is crucial, and higher-order Godunov schemes have a lot of promise

    Astrophysical applicationsAny vacuum-dominated, strong field systemMainly single black holes and coalescing binary black holesMain issues currently are just obtaining stable, reasonable solutionsFuture 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 TogetherMiller (2001)

  • Evolving General Relativistic HydrodynamicsMethod 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)RepeatIf 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 starsTidal disruption and accretion of a neutron star by a black holeMain 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)

  • Evolving General Relativistic MagnetohydrodynamicsComments: No group appears to be doing this as yet; it sounds extraordinarily difficultHowever, 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 resultsAccurate gravitational wave generation and extraction will be much more difficult

    Astrophysical applications:Coalescence of magnetized neutron starsTidal disruption and magnetized accretion of a neutron star by a black holeRelativistic jet generation in a gamma-ray burst central engineDetailed 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 ReadingCook, G.B. 2001, Living Reviews in Relativity, Initial Data for Numerical Relativity.Font, J.A. 2003, Living Reviews in Relativity, Numerical Hydrodynamics in General Relativity.Koide, S., Shibata, K., Kudoh, T. & Meier, D.L. 2001, J. Korean Astro. Soc., 34, S215224. 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. & Mller, E. 2003, Living Reviews in Relativity, Numerical Hydrodynamics in Special Relativity.Meier, D.L. 2004, Astrophys. J., 605, 340 349. Ohms 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, The Motion of Point Particles in Curved Spacetime.Reula, O.A. 1998, Living Reviews in Relativity, 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).