Hyperbolicity of constrained Hamiltonian systems David Hilditch - PowerPoint PPT Presentation

Motivation Model Hamiltonian Application to GR Conclusions Hyperbolicity of constrained Hamiltonian systems David Hilditch Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨ at Jena, Based on arXiv:1303.4783 with R. Richter IHES Gravitation Seminar October 10, 2013 David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Outline Motivation Model Hamiltonian Application to GR Conclusions David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Why study formulations of GR? Practical side: ‘good’ PDE systems for numerical relativity. • Gravitational waves from compact objects. • Critical phenomena. • Examples: ADM, GHG, BSSNOK, Z4, FCF, EC, NOR, KST... Principle questions: physical and mathematical properties. • Does GR make sense dynamically? Gauge? • Well-posedness of I(B)VP? • Long-term existence? David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions The Harmonic formulation The vacuum field equations R ab = − 1 2 g cd ∂ c ∂ d g ab + ∂ ( a Γ b ) − g cd Γ cab Γ d + g cd g ef ( ∂ e g ca ∂ f g bd − Γ ace Γ bdf ) = 0 , with Γ a = g bc Γ abc . Kill ∂ ( a Γ b ) term? Choquet-Bruhat ’53, Friedrich & Rendall ’00, Garfinkle ’02 David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions The Harmonic formulation The vacuum field equations R ab = − 1 2 g cd ∂ c ∂ d g ab + ∂ ( a Γ b ) − g cd Γ cab Γ d + g cd g ef ( ∂ e g ca ∂ f g bd − Γ ace Γ bdf ) = 0 , with Γ a = g bc Γ abc . Kill ∂ ( a Γ b ) term? • Recognize Γ a = g ab � x b = g ab ∇ c ∇ c x b . Choquet-Bruhat ’53, Friedrich & Rendall ’00, Garfinkle ’02 David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions The Harmonic formulation The vacuum field equations R ab = − 1 2 g cd ∂ c ∂ d g ab + ∂ ( a Γ b ) − g cd Γ cab Γ d + g cd g ef ( ∂ e g ca ∂ f g bd − Γ ace Γ bdf ) = 0 , with Γ a = g bc Γ abc . Kill ∂ ( a Γ b ) term? • Recognize Γ a = g ab � x b = g ab ∇ c ∇ c x b . • Harmonic coordinates � x a = 0. Constraints C a = Γ a = 0. Choquet-Bruhat ’53, Friedrich & Rendall ’00, Garfinkle ’02 David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions The Harmonic formulation The vacuum field equations R ab = − 1 2 g cd ∂ c ∂ d g ab + ∂ ( a Γ b ) − g cd Γ cab Γ d + g cd g ef ( ∂ e g ca ∂ f g bd − Γ ace Γ bdf ) = 0 , with Γ a = g bc Γ abc . Kill ∂ ( a Γ b ) term? • Recognize Γ a = g ab � x b = g ab ∇ c ∇ c x b . • Harmonic coordinates � x a = 0. Constraints C a = Γ a = 0. • Free-evolution point of view: Constraint subsystem closed, work in expanded space. Other coordinates? Choquet-Bruhat ’53, Friedrich & Rendall ’00, Garfinkle ’02 David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Cute quotes • Friedrich and Rendall, 2000: “Ideally, one would like to exhibit a kind of ‘hyperbolic skeleton’ of the Einstein equations and a complete characterization of the freedom to fix the gauge from which all hyperbolic reductions should be derivable.” David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Cute quotes • Friedrich and Rendall, 2000: “Ideally, one would like to exhibit a kind of ‘hyperbolic skeleton’ of the Einstein equations and a complete characterization of the freedom to fix the gauge from which all hyperbolic reductions should be derivable.” • Friedrich, 1996: “To give a survey of hyperbolic reductions that is general enough to offer the optimal reduction in any given task appears a hopeless task.” David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Gauge in General Relativity We aim for a complete understanding of the local properties of different gauge choices in GR. Questions: • Can the properties of gauge choices be characterized independently of the Einstein equations? • If so can every hyperbolic gauge be coupled to the Einstein equations to form a hyperbolic PDE system? • From the (free-evolution) PDEs point of view, how does GR compare to other physical theories with gauge freedom? David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Outline Motivation Model Hamiltonian Application to GR Conclusions David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions A model constrained Hamiltonian Consider constant coefficient Hamiltonian, � ∂ i q � † � V ij � � ∂ j q F † i H = 1 � M − 1 F j p p 2 + g † q C H V ij ∂ i ∂ j q + g † p C M i M − 1 ∂ i p , • Positions and momenta ( q , p ). • F i = β i I for some shift vector β i . David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Evolution equations Variations in ( q , p ) and Hamilton’s equations give, ∂ t q = M − 1 p + F i ∂ i q − M − 1 C M† i ∂ i g p , ∂ t p = V ij ∂ i ∂ j q + F i ∂ i p − V ij C H† ∂ i ∂ j g q . • Gauge fields ( g q , g p ) not determined; their momenta are absent. David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions The constraints Variation with respect to gauge fields ( g q , g p ) reveals constraints M = C M i M − 1 ∂ i p = 0 . H = C H V ij ∂ i ∂ j q = 0 , We insist that the constraint subsystem is closed, which gives, C H V ij = ( A HM ) ( i C M j ) , M M − 1 V jk ) = ( A MH ) ( i C H V jk ) , C ( i C ( i M M − 1 V jk ) C † H = 0 . • Dirac: “The constraints are first class .” David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Gauge invariance I The equations of motion are invariant under transformation q = q − M − 1 C M† i ∂ i ψ , p = p − V ij C H† ∂ i ∂ j θ , q → ¯ p → ¯ g q → g q + ¯ g q , g p → g p + ¯ g p , with θ and ψ satisfying g q + ( A MH ) † i ∂ i ψ + β i ∂ i θ , ∂ t θ = ¯ g p + ( A HM ) † i ∂ i θ + β i ∂ i ψ . ∂ t ψ = ¯ • Dirac: “constraints generate gauge transformation”. David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Gauge invariance II Additionally require that the field strength V ij ∂ i ∂ j q is gauge invariant. • Final conditions, ( A HM ) ( i C M j ) = C H V ij , V ( ij M − 1 C M† k ) = 0 , • Can define electric and magnetic parts. Distinction if no H ’s. David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Gauge choice Still missing equation of motion for the gauge fields. Choose, ∂ t g q = ( A g q g q ) i ∂ i g q + ( A g q g p ) i ∂ i g p + ( A g q p ) p , ∂ t g p = ( A g p g q ) i ∂ i g q + ( A g p g p ) i ∂ i g p + ( A g p q ) i ∂ i q . • Assume ( A g q p ) = AC H and ( A g p q ) i = BC M i + C i C H M for some matrices A , B and C i . • This restriction can be dropped. • Structure general enough for hyperbolic gauges. David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Pure gauge subsystem Evolve, but change ID by gauge transformation q = q − M − 1 C M† i ∂ i ¯ p = p − V ij C H† ∂ i ∂ j ¯ q → ¯ ψ , p → ¯ θ . What happens? • Difference evolves according to ∂ t ¯ g q + β i ∂ i ¯ ∂ t ¯ g p + ( A HM ) † i ∂ i ¯ θ + β i ∂ i ¯ θ = ¯ θ , ψ = ¯ ψ , g p − ( A g q p ) V ij C H† ∂ i ∂ j ¯ g q = ( A g q g q ) i ∂ i ¯ g q + ( A g q g p ) i ∂ i ¯ ∂ t ¯ θ , g p = ( A g p g q ) i ∂ i ¯ g q + ( A g p g p ) i ∂ i ¯ g p − ( A g p q ) ( i M − 1 C M† j ) ∂ i ∂ j ¯ ∂ t ¯ ψ . • We call this the pure gauge subsystem . It is closed. David Hilditch Hyperbolicity of constrained Hamiltonian systems

Motivation Model Hamiltonian Application to GR Conclusions Free evolution on the expanded phase space I Define constraints (Θ , Z ). Choose: ∂ t Θ = β i ∂ i Θ + ( A Θ Z ) i ∂ i Z + ( A Θ H ) H , ∂ t Z = ( A Z Θ ) i ∂ i Θ + β i ∂ i Z + ( A Z M ) M . • Remember: can modify dynamics away from the constraint satisfying hypersurface in phase space if constraint subsystem remains closed. • If we insisted on the whole system having Poisson-bracket structure (Θ , Z ) would be canonical momenta of gauge fields. David Hilditch Hyperbolicity of constrained Hamiltonian systems