Tensor Wave Equation J´ erˆ ome Novak Solution of the Gravitational Wave Introduction Constrained evolution Tensor Equation Using Spectral Evolution Equation Numerical Methods Methods Vector Evolution Spherical Harmonics PDEs J´ erˆ ome Novak Time Evolution Tensor Evolution Jerome.Novak@obspm.fr Method Results Laboratoire de l’Univers et de ses Th´ eories (LUTH) Summary CNRS / Observatoire de Paris From Geometry to Numerics, November 21 st 2006
Outline Tensor Wave Equation J´ erˆ ome Novak 1 Introduction Maximally-constrained evolution scheme Introduction Constrained Evolution Equation evolution Evolution Numerical Methods Equation Numerical Methods 2 Divergence-free evolution of a vector Vector Evolution Spherical Harmonics Pure-spin vector spherical harmonics PDEs Time Evolution Differential operators in terms of new potentials Tensor Evolution New system for time evolution Method Results Summary 3 Divergence-free evolution of a symmetric tensor Method Results
Outline Tensor Wave Equation J´ erˆ ome Novak 1 Introduction Maximally-constrained evolution scheme Introduction Constrained Evolution Equation evolution Evolution Numerical Methods Equation Numerical Methods 2 Divergence-free evolution of a vector Vector Evolution Spherical Harmonics Pure-spin vector spherical harmonics PDEs Time Evolution Differential operators in terms of new potentials Tensor Evolution New system for time evolution Method Results Summary 3 Divergence-free evolution of a symmetric tensor Method Results
Outline Tensor Wave Equation J´ erˆ ome Novak 1 Introduction Maximally-constrained evolution scheme Introduction Constrained Evolution Equation evolution Evolution Numerical Methods Equation Numerical Methods 2 Divergence-free evolution of a vector Vector Evolution Spherical Harmonics Pure-spin vector spherical harmonics PDEs Time Evolution Differential operators in terms of new potentials Tensor Evolution New system for time evolution Method Results Summary 3 Divergence-free evolution of a symmetric tensor Method Results
Flat metric and Dirac gauge Following Bonazzola et al. (2004) Conformal 3+1 (a.k.a BSSN) formulation, but use of f ij (with Tensor Wave Equation ∂f ij J´ erˆ ome Novak = 0) as the asymptotic structure of γ ij , and D i the associated ∂t Introduction covariant derivative. Constrained evolution Conformal factor Ψ Evolution Equation Numerical � 1 / 12 γ ij := Ψ − 4 γ ij with Ψ := � Methods γ ˜ , so det ˜ γ ij = f f Vector Evolution Spherical Harmonics Finally, PDEs Time Evolution γ ij = f ij + h ij ˜ Tensor Evolution Method is the deviation of the 3-metric from conformal flatness. Results Summary Generalization the gauge introduced by Dirac (1959) to any type of coordinates: γ ij divergence-free condition on ˜ γ ij = D j h ij = 0 D j ˜ + Maximal slicing ( K = 0)
Flat metric and Dirac gauge Following Bonazzola et al. (2004) Conformal 3+1 (a.k.a BSSN) formulation, but use of f ij (with Tensor Wave Equation ∂f ij J´ erˆ ome Novak = 0) as the asymptotic structure of γ ij , and D i the associated ∂t Introduction covariant derivative. Constrained evolution Conformal factor Ψ Evolution Equation Numerical � 1 / 12 γ ij := Ψ − 4 γ ij with Ψ := � Methods γ ˜ , so det ˜ γ ij = f f Vector Evolution Spherical Harmonics Finally, PDEs Time Evolution γ ij = f ij + h ij ˜ Tensor Evolution Method is the deviation of the 3-metric from conformal flatness. Results Summary Generalization the gauge introduced by Dirac (1959) to any type of coordinates: γ ij divergence-free condition on ˜ γ ij = D j h ij = 0 D j ˜ + Maximal slicing ( K = 0)
Flat metric and Dirac gauge Following Bonazzola et al. (2004) Conformal 3+1 (a.k.a BSSN) formulation, but use of f ij (with Tensor Wave Equation ∂f ij J´ erˆ ome Novak = 0) as the asymptotic structure of γ ij , and D i the associated ∂t Introduction covariant derivative. Constrained evolution Conformal factor Ψ Evolution Equation Numerical � 1 / 12 γ ij := Ψ − 4 γ ij with Ψ := � Methods γ ˜ , so det ˜ γ ij = f f Vector Evolution Spherical Harmonics Finally, PDEs Time Evolution γ ij = f ij + h ij ˜ Tensor Evolution Method is the deviation of the 3-metric from conformal flatness. Results Summary Generalization the gauge introduced by Dirac (1959) to any type of coordinates: γ ij divergence-free condition on ˜ γ ij = D j h ij = 0 D j ˜ + Maximal slicing ( K = 0)
Flat metric and Dirac gauge Following Bonazzola et al. (2004) Conformal 3+1 (a.k.a BSSN) formulation, but use of f ij (with Tensor Wave Equation ∂f ij J´ erˆ ome Novak = 0) as the asymptotic structure of γ ij , and D i the associated ∂t Introduction covariant derivative. Constrained evolution Conformal factor Ψ Evolution Equation Numerical � 1 / 12 γ ij := Ψ − 4 γ ij with Ψ := � Methods γ ˜ , so det ˜ γ ij = f f Vector Evolution Spherical Harmonics Finally, PDEs Time Evolution γ ij = f ij + h ij ˜ Tensor Evolution Method is the deviation of the 3-metric from conformal flatness. Results Summary Generalization the gauge introduced by Dirac (1959) to any type of coordinates: γ ij divergence-free condition on ˜ γ ij = D j h ij = 0 D j ˜ + Maximal slicing ( K = 0)
Einstein equations Dirac gauge and maximal slicing Tensor Wave Equation Constraint Equations J´ erˆ ome Novak Introduction ∆Ψ = S Ham , Constrained evolution ∆ β i + 1 Evolution 3 D i “ D j β j ” = S Mom . Equation Numerical Methods Vector Evolution Spherical Harmonics PDEs Trace of dynamical equations Time Evolution Tensor Evolution ∆ N = S ˙ Method K Results Summary Dynamical equations ∂ 2 h ij − N 2 Ψ 4 ∆ h ij − 2 £ β ∂h ij + £ β £ β h ij = S ij Dyn ∂t 2 ∂t
Einstein equations Dirac gauge and maximal slicing Tensor Wave Equation Constraint Equations J´ erˆ ome Novak Introduction ∆Ψ = S Ham , Constrained evolution ∆ β i + 1 Evolution 3 D i “ D j β j ” = S Mom . Equation Numerical Methods Vector Evolution Spherical Harmonics PDEs Trace of dynamical equations Time Evolution Tensor Evolution ∆ N = S ˙ Method K Results Summary Dynamical equations ∂ 2 h ij − N 2 Ψ 4 ∆ h ij − 2 £ β ∂h ij + £ β £ β h ij = S ij Dyn ∂t 2 ∂t
Evolution Equation Position of the problem Tensor Wave Equation J´ erˆ ome Novak Wave-like equation for a symmetric tensor: γ ij = 1 � � 6 components - 3 Dirac gauge conditions - det ˜ Introduction Constrained ⇒ 2 degrees of freedom evolution Evolution Work with h = f ij h ij which has a given value: the condition Equation Numerical γ ij = 1 Methods � � det ˜ - non-linear condition is imposed with an iteration Vector Evolution on h ; Spherical Harmonics the evolution operator appearing is not, in general, hyperbolic PDEs Time Evolution (complex eigenvalues); with the Dirac gauge, it is (result by I. Tensor Evolution Cordero). Method Results Summary Simplified numerical problem: solve a flat wave equation for a symmetric tensor � h ij = S ij , ensure the gauge condition D j h ij = 0, has a given value of the trace.
Evolution Equation Position of the problem Tensor Wave Equation J´ erˆ ome Novak Wave-like equation for a symmetric tensor: γ ij = 1 � � 6 components - 3 Dirac gauge conditions - det ˜ Introduction Constrained ⇒ 2 degrees of freedom evolution Evolution Work with h = f ij h ij which has a given value: the condition Equation Numerical γ ij = 1 Methods � � det ˜ - non-linear condition is imposed with an iteration Vector Evolution on h ; Spherical Harmonics the evolution operator appearing is not, in general, hyperbolic PDEs Time Evolution (complex eigenvalues); with the Dirac gauge, it is (result by I. Tensor Evolution Cordero). Method Results Summary Simplified numerical problem: solve a flat wave equation for a symmetric tensor � h ij = S ij , ensure the gauge condition D j h ij = 0, has a given value of the trace.
Evolution Equation Position of the problem Tensor Wave Equation J´ erˆ ome Novak Wave-like equation for a symmetric tensor: γ ij = 1 � � 6 components - 3 Dirac gauge conditions - det ˜ Introduction Constrained ⇒ 2 degrees of freedom evolution Evolution Work with h = f ij h ij which has a given value: the condition Equation Numerical γ ij = 1 Methods � � det ˜ - non-linear condition is imposed with an iteration Vector Evolution on h ; Spherical Harmonics the evolution operator appearing is not, in general, hyperbolic PDEs Time Evolution (complex eigenvalues); with the Dirac gauge, it is (result by I. Tensor Evolution Cordero). Method Results Summary Simplified numerical problem: solve a flat wave equation for a symmetric tensor � h ij = S ij , ensure the gauge condition D j h ij = 0, has a given value of the trace.
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