Gravitational wave propagation in astrophysical plasmas

Transcription

1 Gravitational wave propagation in astrophysical plasmas An MHD approach Joachim Moortgat Department of Astrophysics Nijmegen, The Netherlands

2 Outline The MHD approximation & assumptions, Derivation of General Relativistic MHD: split & proper reference frames, Covariant Maxwell EM fields, Thermodynamics gas, Cons. laws for matter, energy & momentum, Set-up: Linearized MHD, Note on the coupling GW MHD. Wave solutions and their properties: MHD Wave equation, Alfvén, slow and fast magneto-acoustic modes, Damping of the GW. EM field Maxwell MHD GR (+MHD) EOM EFE Gas GW Evolution: ρ, µ, p EFE [end tutorial, start qualitative & quantitative results & astrophysical applications]

3 The MHD approximation

4 Ideal Magnetohydrodynamics Approximation In kinetic plasma theory, distribution function of 7 independent variables r, v, t, Velocity space effects removed (moments of Boltzmann) to reduce complexity, Assumptions to get rid of remaining 2-fluid variables and close set of eqns. In MHD model, plasma treated as continuum conducting fluids

5 Ideal Magnetohydrodynamics Approximation In kinetic plasma theory, distribution function of 7 independent variables r, v, t, Velocity space effects removed (moments of Boltzmann) to reduce complexity, Assumptions to get rid of remaining 2-fluid variables and close set of eqns. In MHD model, plasma treated as continuum conducting fluids Macroscopic variables defined as linear combination of 2-fluid vars: ρ n e m e + n i m i τ e(n e Zn i ) v 1 ρ (n em e u e + n i m i u i ) j e(n e u e Zn i u i ) p p e + p i total mass density charge density center of mass velocity current density total pressure

6 Ideal MHD Assumptions Typical length scales typical internal scales (gyro radius, collision mean free path / Debye length), Charge neutrality n e Zn i n e, Small relative velocity u e u i v, Negligible viscosity, Negligible heat flow, Neglect electron skin depth c/ω pe 0, Negligible resistivity R e / infinite conductivity vanishing electric field in rest frame.

7 3 + 1 split & proper reference frames

8 3 + 1 Space-time split Timelike observer with 4-velocity u µ perceives u µ as time axis (u µ u µ = 1). 3D hypersurfaces orthogonal to time axis are snapshots of space. Time projection operator : U µ ν u µ u ν, Space projection operator : H µ ν δ µ ν + u µ u ν.

9 Coordinate vs Non-coordinate Frames Γ µβγ = 1 2 (g µβ,γ + g µγ,β g βγ,µ ) (c µβγ + c µγβ c βγµ )

10 Coordinate vs Non-coordinate Frames Γ µβγ = 1 2 (g µβ,γ + g µγ,β g βγ,µ ) (c µβγ + c µγβ c βγµ ) Holonomic basis = coordinate, c µβγ = 0, Christoffel symbols, Twisting, turning, expansion, contraction of basis vectors by directional derivatives of metric: g βγ,µ = g βγ x µ If g βγ,µ =0, Lorentz frame, FFO.

11 Coordinate vs Non-coordinate Frames Γ µβγ = 1 2 (g µβ,γ + g µγ,β g βγ,µ )+ 1 2 (c µβγ + c µγβ c βγµ ) Holonomic basis = coordinate, c µβγ = 0, Christoffel symbols, Twisting, turning, expansion, contraction of basis vectors by directional derivatives of metric: g βγ,µ = g βγ x µ If g βγ,µ =0, Lorentz frame, FFO. Anholonomic = non-coordinate, In tidal field of GW: Basis vectors do not commute: [e α, e β ] = α e β β e α γ = cαβ e γ, Ricci rotation coefficients. [illustrationi] [illustrationii] [continue]

12 Illustration I: Proper reference frame of accelerated observer Acceleration a and rotation ω of basis, If ω = 0 Fermi-Walker (gyroscope) transport, If also a = 0 FFO, geodesic motion, parallel transport, Coordinate frame with connection coeff: Γ 0 00 = Γ 000 = 0 Γ 0 j0 = Γ 0j0 = +Γ j00 = Γ j 00 = aj Γ j k0 = Γ jk0 = ω i ǫ 0ijk

13 Illustration II: Rindler coords of FIDO near BH horizon Close to horizon, coords transf: x = 2M ( ) θ π 2, y = 2Mφ, z = 4M 1 2M r turns Schwarzschild into Rindler geometry = Minkowski: ds 2 = ( z 4M ) { [ 2 ( z dt 2 + (dx 2 + dy 2 + dz 2 ) 1 + O 4M = dt 2 + dx 2 + dy 2 + dz 2 (T = z sinh Mt 4 ) 2, ( x 4M, Z = z cosh Mt 4 ) 2 ]}, X = x, Y = y).

14 Non-coordinate frame in GW tidal field Metric tuned to TTGW: g TT µν = h + (z, t) h (z, t) 0 0 h (z, t) 1 h + (z, t) C A. Observer metric: g (µν) = η µν w. r. t. orthonormal basis vectors: e 0 = ( t, 0, 0, 0), e 1 = e 3 = ( 0, 0, 0, z), e2 = [ ] 1 h + 2 ( 0, ( 0, h 2 x, h 2 ] [ x, 1 + h + 2 ) y, 0, ) y, 0 Covariant derivatives: a T bc = e a T bc Γ d bat dc Γ d cat bd, with: Γ [01]1 = Γ [02]2 = 1 2 Γ [01]2 = Γ [20]1 = 1 2 h + t, Γ [31]1 = Γ [32]2 = 1 2 h t, Γ [32]1 = Γ [13]2 = 1 2 h + z, h z.

15 Covariant derivation of MHD

16 3 + 1 Maxwell Covariant electromagnetic field tensor split up in space & time components: F µν = (U α µu β ν + H α µh β ν)f αβ = u µ E ν E µ u ν + ǫ µνα B α F µν 1 2 ǫµναβ F αβ 3D comoving volume element ǫ αβγ ǫ αβγδ u δ and ǫ 0123 = detg = 1 B µ 1 2 ǫµαβ F αβ : magnetic field in comoving frame.

17 3 + 1 Maxwell Covariant electromagnetic field tensor split up in space & time components: F µν = (U α µu β ν + H α µh β ν)f αβ = u µ E ν E µ u ν + ǫ µνα B α F µν 1 2 ǫµναβ F αβ 3D comoving volume element ǫ αβγ ǫ αβγδ u δ and ǫ 0123 = detg = 1 B µ 1 2 ǫµαβ F αβ : magnetic field in comoving frame. E µ F µν u ν : com. electric field MHD vanishes E µ = 0 in every frame. In different frame: E = F µ0 u 0 = ǫ µ0αβ u β B α = v B Ohm s law.

18 3 + 1 Maxwell Covariant electromagnetic field tensor split up in space & time components: F µν = (U α µu β ν + H α µh β ν)f αβ = u µ E ν E µ u ν + ǫ µνα B α F µν 1 2 ǫµναβ F αβ 3D comoving volume element ǫ αβγ ǫ αβγδ u δ and ǫ 0123 = detg = 1 B µ 1 2 ǫµαβ F αβ : magnetic field in comoving frame. E µ F µν u ν : com. electric field MHD vanishes E µ = 0 in every frame. In different frame: E = F µ0 u 0 = ǫ µ0αβ u β B α = v B Ohm s law. Maxwell (current density j µ = [τ, j]): ν F µν = ǫ µνα ν B α = 4πj µ Ampère & Gauss ν F µν = 1 4 ν(b µ u ν u µ B ν ) = 0 Faraday & no monopoles

19 Thermodynamics (a) Internal energy per unit mass U as function of p and specific volume V = 1/ρ: du = pdv + dq

20 Thermodynamics (a) Internal energy per unit mass U as function of p and specific volume V = 1/ρ: du = pdv +dq No heat-flow in ideal MHD, dq = 0. (b) Adiabatic gas law: p = Kρ γ ( 4 3 γ 5 3 Γ Lorentz). (a + b) Comoving relativistic matter energy density of ideal fluid (EOS): ρ(c 2 + U) = µ = ρc 2 + p γ 1 Proper relativistic sound velocity is pressure change at constant entropy (w = µ + p is enthalpy): p µ = c 2 s = γp ad w.

21 Conservation of particles, energy & momentum Covariant energy-momentum tensor for ideal magneto-fluid: T µν = wu µ u ν + pg µν + 1 4π (F αµf να 14 g µνf αβ F αβ ) Conservation of energy & momentum: ν T µν = 0 (because ν G µν = 0). Time (µ = 0) and space µ = 1, 2, 3) projections [ with u µ = (1, v)]: (wv) t µ t + e (wv) = j E energy + e (wvv + pi) = τe + j B momentum [ ] Cons. of proper number density n = ρ m e : µ (nu µ ) = 1 ρ m e t + e (ρv) = 0.

22 Set-up Oblique GW propagation at angle θ w. r. t. B 0, Choose z-axis along GW propagation, Choose x- and y-axis such that B 0 = (B 0 x, 0, B 0 z) = B 0 (sinθ, 0, cosθ), In comoving frame E 0, v 0, τ 0, j 0 m, h 0 = 0 & Γ = 1, Equilibrium characterized by µ 0, p 0, ρ 0 0 and B 0. To study plasma properties (stability, wave modes etc.) consider plasma response to small perturbations linearized MHD equations. [NB: Metric perturbation (GW) treated on equal footing.]

23 General Relativistic Magnetohydrodynamics PART. CONS: ρ 1 t = ρ 0 v 1 ENERGY: p 1 t = γp 0 v 1 NO MONOP.: B = 0 E.O.M: w 0 v1 t = p 1 + j 1 m B 0 GAUSS E 1 = 4πτ 1 OHM: E 1 = v 1 B 0 FARADAY: E 1 + B1 t = j 1 B AMPERE: B 1 E1 t = 4πj 1 m +j 1 E

24 General Relativistic Magnetohydrodynamics PART. CONS: ρ 1 t = ρ 0 v 1 ENERGY: p 1 t = γp 0 v 1 NO MONOP.: B = 0 E.O.M: w 0 v1 t = p 1 + j 1 m B 0 GAUSS E 1 = 4πτ 1 OHM: E 1 = v 1 B 0 FARADAY: E 1 + B1 t = j 1 B AMPERE: B 1 E1 t = 4πj 1 m +j 1 E GW induced source terms: j 1 B = B0 2 t h 1 + h 1 0 and j1 E = B0 2 z h 1 h 1 + 0

25 Coupling to GW: EFE & Maxwell Einstein weak Field Eqns (GW): G µν 1 2 h µν = 8πδT µν. Poisson eq. for gravitational field with solution (in Lorenz gauge): Z δtµν (x, t ret h µν (x, t) = 4 ) d 3 x x x Not necessarily TT, but if h µν (z t) & plane TT after gauge trans. ξ(z t). Throw away all non-tt components and use h TT ij T TT ij (i, j = 1, 2).

26 Coupling to GW: EFE & Maxwell Einstein weak Field Eqns (GW): G µν 1 2 h µν = 8πδT µν. Poisson eq. for gravitational field with solution (in Lorenz gauge): Z δtµν (x, t ret h µν (x, t) = 4 ) d 3 x x x Not necessarily TT, but if h µν (z t) & plane TT after gauge trans. ξ(z t). Throw away all non-tt components and use h TT ij T TT ij (i, j = 1, 2). No coupling to ideal fluid: T TT xy = T TT yx = O[v 1 h] 2 & Txx TT = Tyy TT gauge. Back-reaction on GW through EFE: h + = 8π(δT xx δt yy ) = 4B 0 xb 1 x, h = 8π(δT xy + δt yx ) = 4B 0 xb 1 y. Cons. of stress-energy independent of GW: ν T µν EM = F µν j ν = 0. GW couples to EM field through covariant derivatives in Maxwell.

27 Schematic Overview EM fields Maxwell MHD Equation of motion Einstein field equations Gas Evolution of ρ, µ and p GR (+MHD) GW EFE

28 Summary 18 Equations (1) Cons. of number density, (1) Cons. of energy density, (1) Gauss s law, (1) No monopoles (contraint), (3) Faraday s law, (3) Ampère s law, (3) Ohm s law, (3) Cons. of momentum (or EOM), (2) Einstein field equations. EM field Maxwell MHD GR (+MHD) EOM EFE Gas GW Evolution: ρ, µ, p EFE 18 Variables (1) number density ρ m e, (1) energy density µ, (1) charge density τ, (1) pressure p, (3) current density j, (3) magnetic field B, (3) electric field E, (3) velocity v, (2) GW h + and h.

29 MHD Wave Solutions

30 The Wave mode Zoo Kinetic Thermal, unmagnetized: Transverse EM, Langmuir, Ion sound, Magnetoionic (cold plasma): Electron cyclo. maser & Low frequency: (electron / ) ion cyclotron ( Alfvén), (electron / ) ion cyclotron (Bernstein), magnetoacoustic (joins whistler at ω LH ), High frequency: Ordinary magnetoionic (o & whistler), Extraordinary magnetoionic (x & z magnetized Langmuir). warm, inhomogeneous plasma, etc... MHD Non-magnetic Sound Magnetic Alfvén Magneto-acoustic slow fast

31 Wave equation Lorentz force couples matter to EM fields; eliminate j 1 and B 1 from Maxwell. Wave equation»» 2 t 2 u2 m v 1 u 2 A t 2 (u A ) (v 1 u A ) = (u A ) 2 v 1 u A (u A ) v 1 + GW source terms q w tot 4π h (j B u A ) t (j E u A ) (u A )j B i, Defs: Total enthalpy w tot w 0 + B0 2 4π & wave vel.: u 2 A B0 2 4πw tot, u 2 m γp0 w tot + B0 2 4πw tot.

32 Wave equation Lorentz force couples matter to EM fields; eliminate j 1 and B 1 from Maxwell. Wave equation»» 2 t 2 u2 m v 1 u 2 A t 2 (u A ) (v 1 u A ) = (u A ) 2 v 1 u A (u A ) v 1 + GW source terms q w tot 4π h (j B u A ) t (j E u A ) (u A )j B i, Defs: Total enthalpy w tot w 0 + B0 2 4π & wave vel.: u 2 A B0 2 4πw tot, u 2 m γp0 w tot + B0 2 4πw tot. Solve boundary value problem algebraically with Fourier [t ω] and Laplace [z k]. Dv 1 = J 1 GW ; D is response tensor of plasma. Symmetric Hyperbolic PDE 3 3 Symmetric Matrix representation

33 Alfvén, Slow and Fast Magneto-acoustic Modes ω 2 (1 u 2 A ) k2 u 2 A 0 (ω 2 k 2 )u A u A 0 ω 2 k 2 u 2 A 0 (ω 2 k 2 )u A u A 0 ω 2 (1 u 2 A ) k2 (u 2 m u2 A ) C B vx 1 vy 1 vz 1 C A = iω2 u A k ω h + u A h u A h + u A C A Solutions for: Λ(ω, k) = (ω 2 k 2 u 2 A )(ω2 k 2 u 2 f )(ω2 k 2 u 2 s) = 0 ω k A = u A = ±k A u A = ±u A cosθ, ω k s,f = u s,f = ± 1 2 (u2 m + c 2 su 2 A ) 1 ± (1 σ) ; σ(θ) 4c 2 su 2 A (u 2 m + c 2 su 2 A )2. Inhomogeneous solution: v 1 = D 1 J 1 GW = λ ij(j 1 GW ) j Λ ; Unit polarization vectors: n Mi (k)n Mj (k) = λ ij(ω,k M ) Λ(ω,k M ).

34 Alfvén Waves Non-compressional shear wave, E component τ 1, E1 B 0 B 0 2 drift velocity along B 1 y, current density along E 1, By h Bx; 0 Equiv. to LIGO : y 1 = 1 2 h x h +y 0 ), amplitude sinθ; coherent interaction if cos θ 1... excited by h,

35 Magneto-acoustic Waves Compressional waves, gas & electromagnetic properties, perturbations in ρ, p, µ, drift velocity along GW (& wind), in PFD plasma vacuum EM wave, Bx h +Bx; 0 equiv. to x 1 = 1 2 h +x 0, coherent interaction θ; amplitude sinθ. transverse: E 1 B 1 k g;s,f,

36 Damping of the GW Most efficient interaction in PFD plasma (c s u A ), where: B 1 y(k, ω) = B0 xh (k, ω) 2 ω 2 + k 2 u 2 A ω 2 k 2 u 2 A, B 1 x(k, ω) B0 xh + (k, ω) 2 ω 2 + k 2 u 2 A ω 2 k 2 u 2. A Self-consistent dispersion relations from: h + = 4B 0 xb 1 x ω 2 k 2 = 2(B 0 x) 2 ω2 + k 2 u 2 A ω 2 k 2 u 2 A, (same for h with u A u A ) modified GW sol. (v ph,1 v gr,1 = 1) & modified plasma sol. (v ph,2 v gr,2 = u 2 A ): v 2 ph, (B0 x )2 ω 2 1+u 2 A 1 u 2 A, v 2 gr,1 1 2(B0 x )2 ω 2 1+u 2 A 1 u 2 A v 2 ph,2 u 2 A 1 (2B0 x )2 ω 2 u 2 A 1 u 2 A, v 2 gr,2 u 2 A 1 + (2B0 x )2 ω 2 u 2 A 1 u 2 A GW acts as driver (with ω = kc) when 8πG c 2 (B 0 x )2 ( ) µ 0 < ω( k)c = ω 2 c u A 1.

37 Explicit expressions & Astrophysical Applications Separate presentation...