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Hamiltonian Hydrodynamics & Irrotational Binary Inspiral

Charalampos

• M. Markakis

Mathematical Sciences, University

• f

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 ( ) ( ) 1 ( ) u g u g T gT T g

a a a a b b g b b a b a ab g

r r  = ¶

• =
• 

= ¶

• 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

• r Hamilton-Jacobi

schemes for evolving relativistic fluid flows, applicable to binary neutron star inspiral.

Carter-Lichnerowicz variational principles for barotropic flows

• Lichnerowicz:
• Barotropic fluid streamlines are geodesics
• f
• Canonical momentum:
• Euler equation:

b a

dx dx S h g d d d

a b ab

t t t = -

• ò

£ (Euler-Lagrange)

u

dp p d x

a a a a

t ¶

• =
• 

= ¶    ; p hu u

a a a

¶ = = ¶  dx u d

a a

t = ( ) (Hamilton) dp u p p d x

b a b a a b a a

t ¶ + = 

• 

+  = ¶    1 (super-Hamiltonian) 2 2 h p u g p p h

a ab a a b

=

• =

+   

1 dp p h

r

e r r + = + =

ò

2

h gab

Carter-Lichnerowicz variational principles for barotropic flows

• Lichnerowicz:
• Carter:
• Canonical momentum:
• Euler equation:

2 2

b a

h dx dx h S g d d d

a b ab

t t t æ ö ÷ ç ÷ =

• ç

÷ ç ÷ ÷ ç è ø

ò

 

 b a

dx dx S h g d d d

a b ab

t t t = -

• ò

£ (Euler-Lagrange)

u

dp p d x

a a a a

t ¶

• =
• 

= ¶    ; p hu u

a a a

¶ = = ¶  dx u d

a a

t = ( ) (Hamilton) dp u p p d x

b a b a a b a a

t ¶ + = 

• 

+  = ¶    1 (super-Hamiltonian) 2 2 h p u g p p h

a ab a a b

=

• =

+   

1 dp h

r

r = + ò

Carter-Lichnerowicz variational principles for barotropic flows

• Lichnerowicz:
• Carter:
• Canonical momentum:
• Noether’s

theorem: 2 2

b a

h dx dx h S g d d d

a b ab

t t t æ ö ÷ ç ÷ =

• ç

÷ ç ÷ ÷ ç è ø

ò

 

 b a

dx dx S h g d d d

a b ab

t t t = -

• ò

If is Killing, then conserved along streamlines: £ ( ) (weak Bernoulli law)

u

k k p k p

a a a a a =

; p hu u

a a a

¶ = = ¶  dx u d

a a

t =

1 dp h

r

r = + ò

First integrals to the Euler equation

• Cartan

identity:

£ ( ) If is Killing and the flow is irrotational ( ) or rigid ( ), then constant throughout the fluid (strong Bernulli law) For neutron-star binaries on quasicircul

k t

p k dp d k p k p S u u k k p hu k

a a a a a a a

= ⋅ + ⋅ =  = = - ⋅ = - =  ar orbits, Helical Symmetry (stationarity in a rotating frame) implies existence of a helical Killing vector . Then, is the energy of a fluid element in a rotating frame (Jacobi constant). F k t

a a a

j = + W  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

1 2 2 2

1 ( ) fluid ve ( )( locity measured by normal observers / fluid velocity measured in local coo )

b b a b ab a a a a a a a i i j j ij

dx dx S h g d d g dx dx dt dx dt t h dt dt dt v v dx dt t dx d

m m b n n a b a

t a g b a g a b b

• = -
• = -
• n n

= - =

• +

n = + = +

ò ò

2

rdinates canonical momentum of a fluid element Euler-Lagrange equ 1 ( £ ) ation: Hamilton equation: ( ) Constrained Hami

a a a a a t a a a t a a a b a a a b b

v d dt x d L p h hu p L p L p H p v H dt x p p

u

n n ¶ ¶ = = = ¶

• = ¶ +
• 
• ¶

= = ¶ +   +  ¶ +

• =

¶

2

ltonian:

a ab a a a a b t

H p p h p p v p h L u t a

a

b a g + + = - =

• = -

= -

Hamiltonian approach (Newtonian limit)

1 2 / fluid velocity canonical momentum of a fluid element Euler-Lagrange equation: Hamilton equation: ( £ )

a a a a t a f a b ab i a a a a

S v v h dt v dx dt v v d dt x L p p L p L

u

g ¶ = = ¶ = ¶ +

• æ

ö ÷ ç ÷ =

• F

ç ÷ ç ÷ ç è ø = ¶

• ¶

 =

ò

( ) 1 Constrained Hamiltonian: 2

b b a a ab a t a a a b a a b a

p H p d v H dt x H v L p p h p p p g = ¶ +   ¶ +

• =

¶ =

• =

 + + + F

Conservation of circulation

( £ ) ( £ )

• rticity 2-form:

Eul ( £ er-Lagrange equation: V Kelvin's theor ) em:

t t

t a a t ab ab b a a a b a b a ab t ab a b

p L p p d d p dx dx x dx x dt dt d d

u u u

w w w w ¶ + =   ¶ + = =   = 

• =

 ¶ + =

ò ò ò 

 

• 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.

S

t

S

Irrotational hydrodynamics

Irrotational flow: Hamilton equation: ( ) Hamilton-Jacobi equation: Example: In the dust limit on a Minkowsky backgrou

a a b a a t a a a b b b b a t a a t

p p p p p p p S v H H S H    ¶ +   = - ¶

• =

 =

• 

= +  = ¶ +

2 2 2

( / 1 ) ( 1 ) 1 ( nd, one obtains a relativistic Burgers equation: Obtained noncovariantly by LeFloch, Makhlofand and Okutmustur, SINUM 50, 2136 (2012) by algeb )

t a a tS

S u u u ¶

• + ¶

+ = =  ¶ + +  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 2

Analytic 1+1 solution for homogeneously 1 ( ) t ranslating flow: 1 ) ( , ) ( ) Numerical soluti (

• ,

1 n:

t x

S x S t x t x S t u u u u = =  = ¶

• +

+ + 

Irrotational hydrodynamics

2 2 2

Analytic 1+1 solution for homogeneously tra 1 ( ) nslating flow: 1 ) ( ( , 1 , ) ( ) Numerical 'viscosity' solution:

t x x x

S x S S S t t x t x u u u u e = =  =

• +

¶ + + ¶ ¶

Irrotational hydrodynamics

Irrotational flow: Hamilton equation: ( ) Hamilton-Jacobi equation: For barotropic fluids, the above equation is coupl

a a b a a t a a a b b b b a t a a t

p p p p p p p S v H H S H    ¶ +   = - ¶

• =

 =

• 

= +  = ¶ + ed to the continuity equation, resulting in a system , where : det ) (

k k t k i i t t ij

p H g u u r r u d r r a g r g g æ ö æ ö ÷ ÷ ç ç ÷ ÷ ç ç ¶ + ¶ = ÷ ÷ ç ç ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø =

• =

=

  

Irrotational hydrodynamics

Irrotational flow: Hamilton equation: ( ) Hamilton-Jacobi equation: For barotropic fluids, the above equation is coupl

a a b a a t a a a b b b b a t a a t

p p p p p p p S v H H S H    ¶ +   = - ¶

• =

 =

• 

= +  = ¶ +

1,2

0 (Hamilton) where : det( ) Characteris ed to the continuity equation, resulting in tics : a sy stem ,

k t t k t k ij i i k

g u u p H r r u r r a g r g g d l æ ö æ ö ÷ ÷ ç ç ÷ ÷ ç ç ¶ + ¶ = =

• =

= ÷ ÷ ç ç ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø =

   2 2 1 2 2 1/2 2 2 2 2 1/2 3,4 s s s s s s

Complete eigenbasis The system is stro (1 ) { (1 ) (1 ) [( ngly hyperbolic (for finite 1 ) (1 )( ) ] } ( ) ( ) )

k k kk k k

c c c c c c g u gT g T

a a b g b b a ab g

l a n n n n g n b r

• =
• 
• ì

ï¶

• =

ï ï í ï¶

• 

= G ï î

1,2 2 2 1 2 2 1/2 2 2 2 2 1/2 3,4 s s s s s

(Valencia), Characteristics : (1 ) { (1 ) (1 ) [(1 ) (1 ) ( ) } ]

k k k k k kk k k

T hu u pg c c c c c

b b b a a a

r l an b l a n n n n g n b

• =

+ ï =

• =
• 

Recovery of primitives from conservative variables

To evolve the Hamiltonian system numerically, we need to reconstruct the fluxes given the conservative variables at each time step. To do so, we need to recover the primitive variables { , } given {

i

h u r

1 2

, } { , } 1) Recover from and by solving the algebraic equation ( ) (which follows from 1) 2) Recover from by dividing by . Since 1 there is no

t i i i t ij i j ij i j i i

p u hu h p h h u u u p p h u p h h a g r r r r a g g g

• =

= = + + ³

 

division by 0.

Conclusions

Notable features:

• Recovery of ui

requires dividing pi =hui by specific enthalpy h which is 1 at the surface. Thus, no division by 0 is involved when recovering primitive from conservative variables (unlike Valencia)

• At the surface where pressure vanishes, cs

also vanishes, eigenbasis not complete, system becomes weakly hyperbolic (like Valencia). Nevertheless, for HJ equations written as conservation laws, there exist relaxation schemes that restore strong hyperbolicity [S. Jin and Z. Xin, SINUM 35, 2385 (1998).]

k k t k i i

p H r r u d æ ö æ ö ÷ ÷ ç ç ÷ ÷ ç ç ¶ + ¶ = ÷ ÷ ç ç ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø

 

1( ) ( 1, 1)

t i i t i i i t i

p w p H w a p H w a e e ì ï¶ +  = ï ï ï ¶ +  =  í ï¶ +  =

• >

ï ï ï î 

Conclusions

Notable features:

• Recovery of ui

requires dividing pi =hui by specific enthalpy h which is 1 at the surface. Thus, no division by 0 is involved when recovering primitive from conservative variables (unlike Valencia)

• At the surface where pressure vanishes, cs

also vanishes, eigenbasis not complete, system becomes weakly hyperbolic (like Valencia). Nevertheless, for HJ equations written as conservation laws, there exist relaxation schemes that restore strong hyperbolicity [S. Jin and Z. Xin, SINUM 35, 2385 (1998).]

• Thus, the artificial atmopshere

required in the Valencia formulation can in principle be avoided in the Hamiltonian formulation. Work in progress…

• Avoiding the atmosphere would save unnecessary numerical operations (mainly for

reconstruction) outside the star, which can lead to higher accuracy at lower computational cost.

• Scheme may be combined with symplectic

integration and constraint damping methods that preserve symplecic structure and circulation to further increase accuracy.

• SPH schemes based on the Lagrangian
• r Hamiltonian formulation possible

Reference C.M. Markakis, arXiv:1410.7777

k k t k i i

p H r r u d æ ö æ ö ÷ ÷ ç ç ÷ ÷ ç ç ¶ + ¶ = ÷ ÷ ç ç ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø

 