Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral Charalampos Markakis Mathematical Sciences, University of Southampton work in progress in collaboration with: John Friedman, Masaru Shibata, Niclas Moldenhauer, David Hilditch, Sebastiano Bernuzzi, Koutarou Kyutoku, Bernd Brüegmann
Introduction • Gravitational waves from neutron-star and black-hole binaries carry valuable information on their physical properties and probe physics inaccessible to the laboratory. • Although development of black-hole gravitational wave templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged. • Recent progress by groups in Kyoto (SACRA), Caltech-Cornell- CITA-AEI (SpEC), Frankfurt (Whisky), Jena (BAM), Illinois, etc. • The Valencia scheme has been a workhorse for hydro in numerical relativity… 1 a a r = ¶ - r = ( u ) ( g u ) 0 a a - g 1 b b g b = ¶ - - G = T ( gT ) T 0 b a b a ab g - g
Introduction • Gravitational waves from neutron-star and black-hole binaries carry valuable information on their physical properties and probe physics inaccessible to the laboratory. • Although development of black-hole gravitational wave templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged. • Recent progress by groups in Kyoto (SACRA), Caltech-Cornell- CITA-AEI (SpEC), Frankfurt (Whisky), Jena (BAM), Illinois, etc. • The Valencia scheme has been a workhorse for hydro in numerical relativity, but considering alternative hydrodynamic schemes can lead to further progress… • Hamiltonian methods have been used in all areas of physics but have seen little use in hydrodynamics
Introduction • Constructing a Hamiltonian requires a variational principle • Carter and Lichnerowicz have described barotropic fluid motion via classical variational principles as conformally geodesic a b dx dx dp r b ò ò d - t = = + h g d 0 h 1 ab t t r d d a 0
Introduction • Constructing a Hamiltonian requires a variational principle • Carter and Lichnerowicz have described barotropic fluid motion via classical variational principles as conformally geodesic a b dx dx dp r b ò ò d - t = = + h g d 0 h 1 ab t t r d d a 0 S • Moreover, Kelvin’s circulation theorem t d S ò a = hu dx 0 0 t a d c t implies that initially irrotational flows remain irrotational .
Introduction • Constructing a Hamiltonian requires a variational principle • Carter and Lichnerowicz have described barotropic fluid motion via classical variational principles as conformally geodesic a b dx dx dp r b ò ò d - t = = + h g d 0 h 1 ab t t r d d a 0 S • Moreover, Kelvin’s circulation theorem t d S ò a = hu dx 0 0 t a d c t implies that initially irrotational flows remain irrotational . • Applied to numerical relativity, these concepts lead to novel Hamiltonian or Hamilton-Jacobi schemes for evolving relativistic fluid flows, applicable to binary neutron star inspiral.
Carter-Lichnerowicz variational principles for barotropic flows h h • Carter’s Lagrangian: a b = - = - g u u h (on shell ) ab 2 2 ¶ a dx = = • Canonical momentum: p hu ; a = u a a a ¶ t u d 1 h a ab = - = + = p u g p p 0 • Carter’s superHamiltonian: a a b 2 h 2
Carter-Lichnerowicz variational principles for barotropic flows h h • Carter’s Lagrangian: g u a b = - = - u h ab 2 2 ¶ a dx = = • Canonical momentum: p hu ; a = u a a a ¶ u t d 1 h a ab = - = + = p u g p p 0 • Carter’s superHamiltonian: a a b 2 h 2 • Euler equation in Carter-Lichnerowicz form: dp ¶ - = - = a £ p 0 (Euler-Lagrange) a a t a u d ¶ x dp ¶ b + = - + = a u ( p p ) 0 (Hamilton) b a a b a t a d ¶ x
Constrained Hamiltonian approach a b dx dx b b ò ò d - = d a - g n n a b = h g dt h 1 dt 0 ab ab dt dt a a a - 1 a a n = a + b ( v ) fluid velocity measured by normal observers a = a v dx / dt fluid velocity measured in local coordinates n ¶ L = = = p h hu canonic al m o mentum of a fluid element a a a a ¶ v - n 2 1
Constrained Hamiltonian approach a b dx dx b b ò ò d - = d a - g n n a b = h g dt h 1 dt 0 ab ab dt dt a a - n = a a 1 a + b a ( v ) fluid velocity measured by normal observers a a = v dx / dt fluid velocity measured in local coordinates n ¶ L = = = p h a hu canonic al m o mentum of a fluid element a a ¶ a v - n 2 1 = a - = - b a + a 2 + g ab = - Constrained Hamiltonian: H p v L p h p p h u a a a b t
Constrained Hamiltonian approach a b dx dx b b ò ò d - = d a - g n n a b = h g dt h 1 dt 0 ab ab dt dt a a - n = a a 1 a + b a ( v ) fluid velocity measured by normal observers a a = v dx / dt fluid velocity measured in local coordinates n ¶ L = = = p h a hu canonic al m o mentum of a fluid element a a ¶ a v - n 2 1 = a - = - b a + a 2 + g ab = - Constrained Hamiltonian: H p v L p h p p hu a a a b t You've got the action, you've got the motio n - “ ” Dire Strai gh s t
Constrained Hamiltonian approach a b dx dx b b ò ò a b d - = d a - g n n = h g dt h 1 dt 0 ab ab dt dt a a - n = a a 1 a + b a ( v ) fluid velocity measured by normal observers a a = v dx / dt fluid velocity measured in local coordinates n ¶ L = = = p h a hu canonic al m o mentum of a fluid element a a ¶ a v 2 - n 1 = a - = - b a + a 2 + g a b = - Constrained Hamiltonian: H p v L p h p p h u a a a b t dp ¶ L - ¶ + - = Euler-Lagrange equation: a = ( £ ) p L 0 u t a a dt ¶ a x dp ¶ H + ¶ + b - + = Hamilton equation: a = p v ( p p ) H 0 t a b a a b a dt ¶ a x (independent of gravity theory)
Conservation of circulation ¶ + = ¶ + w = Eul er-Lagrange equation: ( £ ) p L ( £ ) 0 S t u a a t u ab t S w = - V orticity 2-form: p p 0 ab a b b a d d ò ò ò a = w a b = ¶ + w a b = Kelvin's theor em: p dx dx d x ( £ ) dx d x 0 u a ab t ab dt dt t t 0 • The most interesting feature of Kelvin's theorem is that, since its derivation did not depend on the metric, it is exact in time-dependent spacetimes, with gravitational waves carrying energy and angular momentum away from a system . In particular, oscillating stars and radiating binaries, if modeled as barotropic fluids with no viscosity or dissipation other than gravitational radiation exactly conserve circulation • Corollary : flows initially irrotational remain irrotational.
Irrotational hydrodynamics - = = Irrotational flow: p p 0 p S b a a b a a ¶ + b - + = Hamilton equation: p v ( p p ) H 0 t a b a a b a Hamilton-Jacobi equation: ¶ S + H = 0 t Example: In the dust limit on a Minkowsky background, on e obt ains a relativistic Burgers equation: ¶ u - u 2 + ¶ + u 2 = ¶ + + 2 = ( / 1 ) ( 1 ) 0 t S 1 ( S ) 0 t a a Obtained noncovariantly by LeFloch, Makhlofand and Okutmustur, SINUM 50, 2136 (2012) by algebraic manipu lation of the Euler equation in Minkowski and Schwarzschild charts. The fact that these are Hamilton equations and can be obtained covariantly for arbitrary spacetimes is unnoticed. Solutions to HJ equation are NOT unique. Nevertheless, 'viscosity' solutions to HJ equation are unique.
Irrotational hydrodynamics 2 ¶ + + = S 1 ( S ) 0 t Analytic 1+1 solution for homogeneously t ranslating flow: 1 u = u = - + u ( t , x ) S t x ( , ) ( t x ) x - u 2 1 Numerical soluti o n:
Irrotational hydrodynamics ¶ + + ¶ 2 = e ¶ 2 S 1 ( S ) S t x x Analytic 1+1 solution for homogeneously tra nslating flow: 1 u = u = - + u ( , t x ) S ( t x , ) ( t x ) x 2 - u 1 Numerical 'viscosity' solution:
Irrotational hydrodynamics - = = Irrotational flow: p p 0 p S b a a b a a b Hamilton equation: ¶ p + v ( p - p ) + H = 0 t a b a a b a ¶ + = Hamilton-Jacobi equation: S H 0 t For barotropic fluids, the above equation is coupled to the c ontinuity equation, resulting in a system æ ö æ ö r r u k ÷ ÷ ç ç ÷ ÷ ç ç ¶ + ¶ = 0 ÷ ÷ ç ç ÷ ÷ ç ç p d k t ÷ k H ÷ ç ç è ø è ø i i r = - r t = a g r t g = g where : g u u , det( ) ij k l = Characteristics : 0 1,2 k 2 2 - 1 k 2 2 1/2 2 2 kk 2 k 2 1/2 k l = a (1 - n c ) { n (1 - c ) c (1 - n ) [(1 - n c ) g - (1 - c )( n ) ] } - b 3,4 s s s s s Complete eigenbasis The system is strongly hyperbolic (for fini e t c ) s
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