Hamiltonian Hydrodynamics & Irrotational Binary Inspiral Charalampos M. Markakis Mathematical Sciences, University of Southampton
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 AEI-Frankfurt (Whisky), Kyoto (SACRA), Jena (BAM), Caltech-Cornell-CITA (SpEC) 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 g b = T ( gT ) T 0 b a b a ab g - g
Introduction • …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 • Constructing a Hamiltonian requires a variational principle • Carter and Lichnerowicz have described barotropic fluid motion via classical variational principles as conformally geodesic • Moreover, Kelvin’s circulation theorem 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 a b dx dx b ò • Lichnerowicz: e + = - - t dp p S h g d r ò h = 1 + = ab r r t t d d 0 a 2 h g ab • Barotropic fluid streamlines are geodesics of ¶ a dx = = a p hu ; = u • Canonical momentum: a a a ¶ u t d • Euler equation: 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 1 h a ab = - = + p u g p p (super-Hamiltonian) a a b 2 h 2
Carter-Lichnerowicz variational principles for barotropic flows a b dx dx dp b + ò r ò • Lichnerowicz: = - - t S h g d = h 1 ab t t r d d a 0 æ ö a b h dx dx h • Carter: ÷ b ç ò ÷ = ç - t S g d ÷ ç ÷ ç ab ÷ t t 2 d d 2 è ø a • Canonical momentum: a ¶ dx a = = = p hu ; u a a a t ¶ d u • Euler equation: 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 1 h a ab = - = + p u g p p (super-Hamiltonian) a a b 2 h 2
Carter-Lichnerowicz variational principles for barotropic flows a b dx dx dp b + ò r ò • Lichnerowicz: = - - t S h g d = h 1 ab t t r d d a 0 æ ö a b h dx dx h • Carter: ÷ b ç ò ÷ = ç - t S g d ÷ ç ÷ ç ab ÷ t t 2 d d 2 è ø a • Canonical momentum: a ¶ dx a = = = p hu ; u a a a t ¶ d u • Noether’s theorem: a a a a = If k is Killing, then k p conserved along streamlines: £ ( k p ) 0 a u (weak Bernoulli law)
First integrals to the Euler equation • Cartan identity: = ⋅ + ⋅ £ p k dp d k p ( ) k a a a = = t If k is Killing and the flow is irrotational ( p S ) or rigid ( u u k ), then a a a = - ⋅ = - = k p hu k constant throughout the fluid (strong Bernulli law) a For neutron-star binaries on quasicircul ar orbits, Helical Symmetry (stationarity in a a a = + W j a rotating frame) implies existence of a helical Killing vector k t . Then, is the energy of a fluid element in a rotating frame (Jacobi constant). F irst integral very useful for constructing initial data via self-consistent field methods for rotating or binary neutron stars. But the Carter-Lichnerowicz framework has not been used for evolution so far. To this end, we follow a 3+1 constrained Hamiltonian approach.
Constrained Hamiltonian approach a b dx dx b b ò ò = - - = - a - g n n a b S h g d t h 1 dt a b ab dt dt a a 2 m n 2 2 i i j j t = - = a - g + b + b d g dx dx dt ( dx dt )( dx dt ) m n ij - n a = a 1 a + b a ( v ) fluid ve locity measured by normal observers a = a v dx / d t fluid velocity measured in local coo rdinates n ¶ L p = = h = hu canonical momentum of a fluid element a a a ¶ a v - n 2 1 d p ¶ L Euler-Lagrange equ ation: a - = ¶ + ( £ ) p - L = 0 t u a a dt ¶ a x d p ¶ H b + = ¶ + - + = Hamilton equation: a p v ( p p ) H 0 t a b a a b a dt ¶ a x t a = a - = - b a + a 2 + g ab = - = - Constrained Hami ltonian: H p v L p h p p h u p a a a b t a
Hamiltonian approach (Newtonian limit) æ ö 1 f ÷ ç ò a b ÷ S = g v v - h -F dt ç ÷ ç ÷ ç ab 2 è ø i a a = v dx / dt fluid velocity ¶ L = = p v canonical momentum of a fluid element a a a ¶ v d p ¶ L - = ¶ + - = Euler-Lagrange equation: a ( £ ) p L 0 u t a a dt ¶ a x d p ¶ H + = ¶ + b - + = Hamilton equation: a p v ( p p ) H 0 t a b a a b a dt ¶ a x 1 = a - = g ab + + F Constrained Hamiltonian: H p v L p p h a a b 2
Conservation of circulation ¶ + = ¶ + w = Eul er-Lagrange equation: ( £ ) p L ( £ ) 0 u u t a a t ab S 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 a ab t u 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 p H 0 t a b a a b a t a a ¶ + = Hamilton-Jacobi equation: S H 0 t Example: In the dust limit on a Minkowsky backgrou nd, one obtains 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 algeb raic manipulation of the Euler equation in Minkowski and Schwarzschild charts. The fact that these are Hamilton equations and can be obtained covariantly for arbitrary spacetimes was missed. 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 2 ¶ + + ¶ = e ¶ 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 - u 2 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 p H 0 t a b a a b a t a a Hamilton-Jacobi equation: ¶ S + H = 0 t For barotropic fluids, the above equation is coupl ed to the continuity equation, resulting in a system æ ö æ ö k r r u ÷ ÷ ç ç ÷ ÷ ç ç ¶ + ¶ = ÷ 0 ÷ ç ç ÷ ÷ ç t ç p k d k ç ÷ H ÷ ç è ø è ø i i r = - r t = a g r t g = g where : g u u , det ( ) ij
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