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Tetrad Approaches to Numerical Relativity Tetrad Approaches to Numerical Relativity Tetrad Approaches to Numerical Relativity James Bardeen Luisa Buchman KITP July 1, 2003 References Buchman and Bardeen, gr-qc/0301072 Estabrook, Robinson, Wahlquist, gr-qc/9703072 van Putten and Eardley, gr-qc/9505023 Choquet-Bruhat and York, gr-qc/0202014 Nester, J. Math. Phys. 33, 910 (1992) James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 1 Tetrad Approaches to Numerical Relativity Advantages of tetrad versus metric • The tetrad metric, and therefore the raising and lowering of indices, is trivial. There are many fewer terms in the equations, with a more consistent structure. • Fewer connection coefficients (24 versus 40 in 4D, 9 versus 18 in 3D). • Variables are more closely related to physical/geometric quantities. • Tetrad gauge conditions (the evolution of the acceleration and angular velocity of the tetrad frames) can be formulated in a coordinate-independent way. • Most of the variables are coordinate scalars, so Lie derivatives in time derivative operators do not involve derivatives of the shift vector. Potential disadvantages of tetrad formalism • Need gauge conditions to evolve both tetrad frames and coordinates. • Frames may not have any fixed orientation relative to the constant time hypersurfaces on which the numerical evolution is defined, so the “spatial” tetrad directional derivatives may contain coordinate time derivatives and also may not coincide with spatial coordinate axes. • Commutation of tetrad directional derivatives is non-trivial. • There are more tetrad vector components to evolve than coordinate metric components (9 versus 6 in 3D). • May be harder to develop asymptotically stationary gauge conditions. James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 2 Tetrad Approaches to Numerical Relativity Threading approaches (1 + 3) Tetrad congruence world lines not forced to be orthogonal to constant-time hypersurfaces. We consider three types of tetrad evolution gauges consistent with symmetric hyperbolicity of a “Christoffel” system. • Fixed acceleration and angular velocity of tetrad frames. • “Lorentz” gauge of van Putten and Eardley. • “Nester” gauge of Estabrook, et al. Hypersurface-orthogonal frames (3 + 1) Tetrad congruence worldlines are forced to be orthogonal to the constant-time hypersurfaces. A dynamic equation for the lapse evolves the hypersurfaces and the tetrad acceleration. I consider two basic types of evolution systems: • An “Einstein-Christoffel” system, based on first-order equations connecting the connection coefficients. • An “Einstein-Bianchi” system (Choquet-Bruhat and York) which uses the Bianchi identities to evolve the Riemann tensor. Both systems are simplified to use the minimum number of variables, while maintaining symmetric hyperbolicity. James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 3 Tetrad Approaches to Numerical Relativity Reduction to practice · Relate tetrad variables and directional derivatives to coordinates and coordinate derivatives required for numerical calculation. Derive eigenvectors of characteristic matrix for arbitrary directions of propagation. 1D test applications: · colliding plane waves, with general polarization · spherical symmetry (Schwarzschild geometry) Notation α,β,γ,... space-time tetrad indices (0-3) λ,µ,ν,... space-time coordinate indices (0-3) a,b,c,... spatial tetrad indices (1-3) i,j,k,... spatial coordinate indices (1-3) λµ λ <λ η Tetrad vectors α , α β =αβ µ ∂ Tetrad directional derivatives D =λ α α ∂xµ µν =ηαβλµλν Coordinate metric g α β James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 4 Tetrad Approaches to Numerical Relativity Coordinates Introduce a tetrad lapse and shift such that 1 ∂ k ∂ D = −β . 0 α k ∂t ∂x Project the spatial tetrad vectors into the hypersurface along the tetrad world lines to define a truly spatial directional derivative ∂ operator ∂ =ek such that D = A D +∂ . a a ∂xk a a 0 a Aa is the 3-velocity of the tetrad worldline relative to the hypersurface normal. Necessary conditions for non-singular coordinates are α > < µν = µ ν 1) 0 and 2) AaAa 1 . The inverse spatial metric h ea ea should be positive-definite. Connection coefficients Γ =λ ∇ λ =−Γ αβγ α γ β βαγ = 1λ λ λ + 1λ λ λ − 1λ λ λ β α, γ γ α, β α β, γ 2 2 2 Space-time split in E-W dyadic notation Acceleration 3-vector a =Γ =λ λ ,λ b b00 0 b 0 Angular velocity 3-vector ω =−1ε Γ b 2 abc bc0 "Extrinsic curvature" K =Γ ab b0a Spatial connection dyadic N = 1ε Γ ab 2 bcd cda James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 5 Tetrad Approaches to Numerical Relativity Twist vector, vanishes iff tetrad is hypersurface orthogonal, antisymmetric part of K : ab Ω ≡ 1ε K . a 2 abc bc Antisymmetric part of N , only part which transforms ab non-trivially under conformal rescalings of spatial metric: n ≡ 1ε N . a 2 abc bc Evolution of the (in general) 9 K and 9 N are determined ab ab by the Einstein equations and ordering identities, while the evolution of the 3 and the 3 ω is determined by gauge ab b conditions. Einstein equations = Γ − Γ Riemann tensor Rαβγδ Dγ αβδ Dδ αβγ +.... Initial value equations G =R +R +R =D 2n +.... 00 2323 3131 1212 b b G =R +R = −D K −D K +D K +K +.... 01 0212 0313 2 12 3 13 1 22 33 Evolution equations R −δ G = −R +R −δ R +R +R ba ba 00 0b0a cbca ba 2323 3131 1212 =D K −D a −ε D N +.... 0 ab a b acd c db Note that form of the the evolution equations is not symmetric, even though the Ricci tensor is symmetric. James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 6 Tetrad Approaches to Numerical Relativity Identities for evolution and constraints Riemann identities = Ordering identities Rαβγδ Rγδαβ : D N +D ω +ε D K +...=0 , 0 ab a b acd c db − − + + + = D N D N D N N ... 0 . 2 12 3 13 1 22 33 Cyclic identity: + + = Ω + = R R R D 2 ... 0 . 0123 0231 0312 b b Also require gauge evolution equations for a and ω . b b Simplest (but not suitable in general): a and ω are b b fixed functions of the coordinates (fixed gauge). Gauge Evolution Lorentz gauge (van Putten and Eardley) ω ω =λµω =Γ Connection 1-forms µαβ, with γαβ γ µαβ αβγ , ∇µω = satisfy µαβ 0. − = − − D a D K a 2n K TrK a , 0 b c cb c c cb b ω + =− − − ω D D N a 2n N (TrK) . 0 b c cb c c cb b Nester gauge (Estabrook, et al), based on propagating tetrad frames with a "teleparallel" (zero curvature, non-zero torsion) connection, − =−ω − Ω + ω D a D K 2 N (TrN) , 0 b c cb c c cb b ω + =−ω − Ω − D D N 2 K (TrN)a . 0 b c cb c c cb b James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 7 Tetrad Approaches to Numerical Relativity Equations for the tetrad vectors, derived from the commutators of the tetrad vectors expressed in terms of the connection coefficients. Evolution: = −∂ α − +ε ω D A a (ln )K A , 0 a a a ac abc b c ∂β k k + m =− +ε ω k D ea ea Kac ec . 0 ∂xm abc b Constraints: ∂ek ε m b = k − k +ε +ε ω k ea N e (TrN)ec Aa K e , cab ∂xm dc d cab bd bdf f d ∂ m Aa ε e =2Ω + A N −(TrN)A +ε A K +ε ω A −a . cab a ∂ m c d dc c cab a bd bfd f d b x Initial Value Problem ¡ = Assume tetrad worldlines are orthogonal to the t=0 hypersurface, 0 , so £ = ¢ = Aa 0, a a. 1. Start with a spatial metric h and an extrinsic cu ij rvature Kij, obtained by standard methods in the coordinate framework. Construct orthonormal k = m n triad fields ea in the initial hypersurface. Then Kab ea ebKmn , and the k Nab are determined from the commutators of the ea. 2. The 3D Nester teleparallel gauge conditions, TrN =0, ∇×n=0, can be used to fix the initial spatial orthonormal triad. Under a conformal ¤ =φ4 k =φ−2 ¤ k transformation of the coordinate metric, hij hij, ea ea , −2 ¤ −2 n =φ n −D lnφ , the symmetric part of N scales as φ , so the a a a ab Nester conditions are conformally invariant. A "bare" 3-geometry has = = na 0 and TrN 0. James Bardeen & Luisa Buchman, U Washington (KITP Gravitation Program 7/01/03) 8 Tetrad Approaches to Numerical Relativity Pseudo-hyperbolic and hyperbolic systems If one pretends that the Da are purely spatial directional derivatives, the evolution equations in the fixed, Lorentz, and Nester gauges have a very simple symmetric hyperbolic structure, with all propagation at light speed and variables coupled in pairs to form eigenvectors along each tetrad direction. However, the Aa will not stay zero, so the true hyperbolic structure includes the time derivatives hidden in the Da. A system of equations of the form D q+Cf D q=S(q) , 0 f where the Cf are characteristic matrices, is really (I+Cf A )D q+Cf ∂ q=TD q+Cf ∂ q=S(q) . f 0 f 0 f The true characteristic matrices are Cf =T−1Cf . Also, the nominal constraint equations are not the true constraint equations. Fortunately, the T matrix has a simple structure, and its inverse can be found explicitly. It is block diagonal in groups of 8 variables labeled by the index , ( ,,,,,,,ω ). The true hyperbolic c N1ccN2 N3cac K1cK2cK3cc system is still symmetric hyperbolic, with T as the symmetrizing < matrix, as long as AaAa 1.
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