General Relativistic Magnetohydrodynamics Equations

Transcription

1 General Relativistic Magnetohydrodynamics Equations Ioana Duţan IMPRS Students Workshop Berggasthof Lansegger, Austria, Aug 2006 Ioana Duţan, GRMHD Equations p.1/22

2 Introductory remarks (General) Relativistic (Magneto)hydrodynamics a (G)R (M)HD code is used to compute the flow of gas around strong field gravity sources used to study supernova collapse and formation of BH, BH-BH binaries, NS-NS binaries, pulsar wind nebulae, accretion disks, relativistic jets from AGN and microquasars, etc. Ioana Duţan, GRMHD Equations p.2/22

3 Introductory remarks Propose of this material to introduce or review some aspects of GR to introduce the 3+1 decomposition of the spacetime (in the Eulerian formulation) to provide a small derivation of the conservative systems of the hyperbolic PDE of the GRHD Notes on the level of this material if it is too easy, just treat it as review, perhaps from a different perspective if it is too fuzzy, concentrate on the concepts... Ioana Duţan, GRMHD Equations p.3/22

4 General relativity background metric element: ds 2 = g µν (x µ )dx µ dx ν for a flat spacetime of special relativity: ds 2 = dt 2 + dx 2 + dy 2 + dz 2 ds 2 < 0, interval is timelike; ds 2 > 0, interval is spacelike; ds 2 = 0, it is null 1D curve x µ (λ) in spacetime describes a series of events timelike curve (worldline) is parameterized by the proper time τ Ioana Duţan, GRMHD Equations p.4/22

5 General relativity background 4-velocity is defined as: u µ = dxµ (τ) dτ in SR, the 4-velocity components are u µ = (u 0,u 1,u 2,u 3 ) = (W,W v), where W = 1 1 v 2 imagine 2 particles with worldlines that meet at point A, having 4-velocity u µ and v µ ; their product is an invariant (it can be evaluated in an arbitrary reference frame)(fig. 1) in particular, in the Lorentz reference frame comoving with u µ you have u µ = ( 1,0,0,0), and then u µ v µ = u µv µ = v 0 = W Ioana Duţan, GRMHD Equations p.5/22

6 General relativity background Ioana Duţan, GRMHD Equations p.6/22

7 3+1 Eulerian formulation 3+1 decomposition spacetime is foliated into a set of non-intersecting spacelike hypersurfaces, parameterized by a parameter usually called time t, s.t., the evolution between these surfaces is described by two kinematic variables metric can be written in a particular way ds 2 = g µν dx µ dx ν = α 2 dt 2 + g ij (dx i + β i dt)(dx j + β j dt) which in a component form is g 00 g i0 g 0j g ij = β s β s α 2 β i β j γ ij, where γ ij = g ij i, j = 1, 2,3 γ ij = 3-metric induced on each spacelike slice Ioana Duţan, GRMHD Equations p.7/22

8 3+1 Eulerian formulation γ ik γ kj = δ j i ; gymnastics of indices β i = γ ij β j decomposition of the volume associated with the 4-metric into the volume associated with the 3-metric g = α γ, g = det(gµν ), γ = det(γ ij ) let s consider 2 close spacelike hypersuperfaces Σ(t) and Σ(t + dt) (Fig. 2) lapse function α describes the rate of advance of time along a timelike unit vector normal to the hypersurface spacelike shift vectors β i describe the motion of coordinates within a surface Ioana Duţan, GRMHD Equations p.8/22

9 3+1 Eulerian formulation 4-vector n µ is the unit normal vector to the Σ(t) n µ = ( α,0,0,0), n µ = ( 1 α, βi α ) Eulerian observers are observers having n as 4-velocity, at rest in the slice Σ(t) and moving to this slice with clocks showing proper time i.e., the basis adapted to the Eulerian observer frames is: e (µ) = {n, i } 4-vector u µ is the 4-velocity of some particle Ioana Duţan, GRMHD Equations p.9/22

10 3+1 Eulerian formulation Ioana Duţan, GRMHD Equations p.10/22

11 3+1 Eulerian formulation Let s translate the 4-velocity of a particle from the arbitrary coordinate frame (S) to the Eulerian frame (S ) S S P (t, x i ) (t, x i ) Q (t + dt, x i β i dt) (t + αdt, x i ) R (t + dt, x i + dx i ) (t + αdt, x i + β i dt + dx i ) 4-velocity = vector PR / proper time τ in the coordinate frame S: u µ = (dt,dxi ) dτ = in the Eulerian frame S : ( ) dt dτ, dxi dτ u µ = (αdt,βi dt + dx i ) dt = (αu 0,u i +β i u 0 ) = αu 0 (1, u i αu 0 + βi α ) Ioana Duţan, GRMHD Equations p.11/22

12 3+1 Eulerian formulation Lorentz factor as seen from S is: W = n µ u µ = αu 0 3-velocity of particle in Eulerian frame: v i = ui + βi αu 0 α v i = γ ij v j = γ ij u j αu 0 + βj α «= 1 αu 0 γ ij(u j + β j u 0 ) = = 1 αu 0 (γ iju j + β i u 0 ) = 1 αu 0 (g i0u 0 + g ij u j ) = u i αu 0 Normalization: 1 = g µν u µ u ν = α 2 (u 0 ) 2 + γ ij (u i + β i u 0 )(u j + β j u 0 )=» u = α 2 (u 0 ) 2 i 1 γ ij αu 0 + βi α «u j αu 0 + βj α «= α 2 (u 0 ) 2 (1 γ ij v i v j ) Lorentz factor: W = αu 0 = (1 γ ij v i v j ) 1/2 Ioana Duţan, GRMHD Equations p.12/22

13 General relativistic hydrodynamics GRHD equations consist of the local conservation laws of the matter current density and the energy-momentum (stress-energy tensor) + fluid equation of state rest mass flux (proportional to the baryon number flux) is: J µ = ρ 0 u µ ρ 0 = rest mass density (baryon number density times average rest mass of the baryons); u = fluid velocity stress-energy tensor of an ideal fluid (without non-adiabatic processes, s.a., viscosity, magnetic field, radiation) T µν = (ρ + p)u µ u ν + pg µν ρ = total mass-energy density; p = pressure Ioana Duţan, GRMHD Equations p.13/22

14 General relativistic hydrodynamics useful relations: u 0 = W α, u i W = vi βi α, u i W = v i CONSERVATIVE VARIABLES are measured by the Eulerian observers, being defined as: D = J µ n µ = ρ 0 u µ n µ = ρ 0 W, rest mass density S j = T µ νn µ ( j ) µ = αt 0 j = α(ρ + p)u 0 u j = = (ρ + p)w 2 u j W = (ρ + p)w 2 v j, momentum density E = T µν n µ n ν = α 2 T 00 = α 2 [(ρ + p)u 0 u 0 + pg 00 ] = = (ρ + p)w 2 α 2 p 1 α 2 = (ρ + p)w 2 p, energy derived conservative variable: τ = E D Ioana Duţan, GRMHD Equations p.14/22

15 General relativistic hydrodynamics T 00 = 1 α 2 T 0 i = 1 α S i T 0i = 1 α (ρ + P)W 2 (v i βi α ) + P βi α 2 T i j = S j (v i βi α ) + Pδi j differential form of baryon number conservation (continuity equation) 1st GRHD equation J µ ;µ = 1 g ( gj µ ),µ = 1 g ( gρ 0 u µ ),µ = = 1 g α γρ 0 W α «,0 = 1 g ( γd),0 + 1 g D + 1 «gρ0 W(v i βi g α ) «v i βi α,i = 0,i = Ioana Duţan, GRMHD Equations p.15/22

16 General relativistic hydrodynamics (T µ ν(e γ ) ν ) ;µ = T µ ν;µ(e γ ) ν + T µν (e γ ) ν;µ = T µν (e γ ) ν,µ Γ λ νµ (e γ) λ For γ = 0: (T µ ν(e 0 ) ν ) ;µ = (T µν n ν ) ;µ = ( αt µ0 ) ;µ = 1 g ( gαt 00 ),0 + ( gαt i0 ),i = (( 1» gα 1 γα 2 T 00 ),0 + g α (ρ + P)W2 (v i βi α ) + p βi α 2 = (( 1 «) g γe),0 +»E(v i βi g α ) + pvi,i «,i ) T µν (e 0 ) ν,µ Γ λ νµ (e 0) λ = T µ0 α,µ + αγ 0 νµ T µν = α ` T µ0 (ln α),µ + Γ 0 νµ T µν 1 g j t ( γe) + x i «ff g»e(v i βi α ) + pvi = α ` T µ0 (ln α),µ + Γ 0 νµ T µν 2nd GRHD equation Ioana Duţan, GRMHD Equations p.16/22

17 General relativistic hydrodynamics For γ = j: (T µ ν(e j ) ν ) ;µ = (T µ j) ;µ = 1 n` gt o µ g j,µ = 1 g ( gt 0 j ),0 + ( gt i j),i = (( 1 «) g γs j ),0 +»S j (v i βi g α ) + pδi j,i T µν (e j ) ν,µ Γ λ νµ(e j ) λ = T µν h g νλ (e j ) λi,µ Γλ νµg λσ (e j ) σ «= T µν g νj,µ Γ λ νµg λj 1 g j t ( γs j ) + x i g»s j (v i βi α ) + Pδi j «ff = T µν g νj,µ Γ λνµg λj 3rd, 4th, and 5th GRHD equations Ioana Duţan, GRMHD Equations p.17/22

18 General relativistic hydrodynamics GRHD equations can be written in a conservation form as { 1 g t ( γu) + } x i( gf i ) = Σ U = 2 3 D 6 4S j 7 5 F i = τ 2 D(v i βi α ) 6 4S j (v i βi α ) + pδi j τ(v i βi α ) + pvi Σ = 6 4 T µν `g νj,µ Γ λ νµ g 7 λj α `T µ0 (ln α),µ Γ 0 µν T µν 5 conservative variables fluxes source term for curved spacetime, there exist source terms, arising from the spacetime geometry for Minkowski metric Σ = 0 and g = γ = 1; strict conservation low is possible only in flat spacetime Ioana Duţan, GRMHD Equations p.18/22

19 General relativistic hydrodynamics IMPORTANT!! Recovering the primitive variables from the conservative ones, P = [ ρ 0,v j,p ] for conservative formulations, the time update of a given numerical algorithm is applied to the conservative variables after this update, the vector of the primitive variables must be re-evaluated as those are needed in the Riemann solver the relation between the 2 sets of variables is not in closed form and hence, the recovery of the primitive variables is done by using a root-finding procedure (Newton-Raphson scheme) Ioana Duţan, GRMHD Equations p.19/22

20 General relativistic MHD One have to include: evolution equations for magnetic field (Maxwell equations divergence free magnetic field and induction equation) + frozen-in condition T µν = T µν fluid + T µν elmagm U = 2 3 D S j 6 4 τ 7 5 B j F i = 2 3 D(v i βi α ) S j (v i βi α ) + pδi j 6 4 τ(v i βi α ) + pvi 7 5 ṽ i B j ṽj B i Σ = T µν `g νj,µ Γ λ νµ g λj 6 4α `T µ0 (ln α),µ Γ 0 µνt µν conservative variables fluxes source term Primitive variables: P = [ ρ 0,v j,p,b j] Ioana Duţan, GRMHD Equations p.20/22

21 Remarks GRMHD simulations of jets formation from Kerr BH Y. Mizuno, K. Nishikawa, and S. Koide Koide et al. (1998) first 3D GRMHD simulations of jets formation Ioana Duţan, GRMHD Equations p.21/22

22 References 1. Font, J. A. Numerical Hydrodynamics in General Relativity, 2. Font, J. A. General relativistic hydrodynamics and magnetohydrodynamics and their applications, Plasma Phys. Control. Fussion 47, B679-B690, Koide, S. Magnetic extraction of black hole rotational energy: Method and results of general relativistic magnetohydrodynamic simulations in Kerr space-time, Phys. Rev. D, vol. 67, Issue 10, Mizuno, Y. et al. RAISHIN: A High-Resolution Three-Dimensional General Relativistic Magnetohydrodynamics Code, astro-ph/ Ioana Duţan, GRMHD Equations p.22/22