Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral - - PowerPoint PPT Presentation
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,
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
carry valuable information on their physical properties and probe physics inaccessible to the laboratory.
templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged.
CITA-AEI (SpEC), Frankfurt (Whisky), Jena (BAM), Illinois, etc.
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 = ¶
= ¶
=
carry valuable information on their physical properties and probe physics inaccessible to the laboratory.
templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged.
CITA-AEI (SpEC), Frankfurt (Whisky), Jena (BAM), Illinois, etc.
numerical relativity, but considering alternative hydrodynamic schemes can lead to further progress…
have seen little use in hydrodynamics
via classical variational principles as conformally geodesic
1
b a
dx dx dp h g d h d d
a b r ab
d t t t r
= +
via classical variational principles as conformally geodesic
implies that initially irrotational flows remain irrotational.
1
b a
dx dx dp h g d h d d
a b r ab
d t t t r
= +
c
d hu dx d
t
a a
t =
S
t
S
via classical variational principles as conformally geodesic
implies that initially irrotational flows remain irrotational.
Hamiltonian or Hamilton-Jacobi schemes for evolving relativistic fluid flows, applicable to binary neutron star inspiral.
1
b a
dx dx dp h g d h d d
a b r ab
d t t t r
= +
c
d hu dx d
t
a a
t =
S
t
S
(on shell 2 ) 2 h h h g u u
a b ab
= - =
; p hu u
a a a
¶ = = ¶ dx u d
a a
t = 1 2 2 h p u g p p h
a ab a a b
= = =
2 2 u h h h g ua
b ab
=
£ (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 2 2 h p u g p p h
a ab a a b
= = =
1 2
1 ( ) fluid velocity measured by normal observers / fluid velocity measured in local coordinates canonic
al m
b b a b ab a a a a a a a a a a a
dx dx h g dt h dt dt dt v v dx dt v L p h hu
a b ab
d d a g a b
= n = + = n n ¶ = = = ¶
mentum of a fluid element
1 2
1 ( ) fluid velocity measured by normal observers / fluid velocity measured in local coordinates canonic
al m
b b a b ab a a a a a a a a a a a
dx dx h g dt h dt dt dt v v dx dt v L p h hu
a b ab
d d a g a b
= n = + = n n ¶ = = = ¶
2
mentum of a fluid element Constrained Hamiltonian:
a ab a a a b t a
p p h p p h v L u H b a g =
= - = +
1 2
1 ( ) fluid velocity measured by normal observers / fluid velocity measured in local coordinates canonic
al m
b b a b ab a a a a a a a a a a a
dx dx h g dt h dt dt dt v v dx dt v L p h hu
a b ab
d d a g a b
= n = + = n n ¶ = = = ¶
2
“ ” Dire Strai mentum of a fluid element Constrained Hamiltonian: t gh s
a ab a a a a b t
You've got the action, you've got the motio p p H v L h p hu n p b a g + + = -
1 2
1 ( ) fluid velocity measured by normal observers / fluid velocity measured in local coordinates canonic
al m
b b a b ab a a a a a a a a a a a
dx dx h g dt h dt dt dt v v dx dt v L p h hu
a b ab
d d a g a b
= n = + = n n ¶ = = = ¶
2
mentum of a fluid element Constrained Hamiltonian: Euler-Lagrange equation: Hamilton equation: = ( £ ) =
a a a t b a a a b a a a a b a t a a t
dp L p L dt x dp H p v dt x p p h p p h H v L u
u
b a g ¶
= ¶ ¶ + ¶ + + = - =
¶
( ) (independent of gravity theory)
b a a b a
p p H
+ =
( £ ) ( £ )
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 ¶ + = ¶ + = = =
¶ + =
depend on the metric, it is exact in time-dependent spacetimes, with gravitational waves carrying energy and angular momentum away from a
barotropic fluids with no viscosity or dissipation other than gravitational radiation exactly conserve circulation
S
t
S
Irrotational flow: Hamilton equation: Hamilton-Jacobi equation: Example: In the dust limit on a Minkowsky background, on ( ) e obt
a a t a a t b b a b a a a b b
p p S p p H p S p H v + ¶ + ¶
= = =
=
2 2 2
ains a relativistic Burgers equation: Obtained noncovariantly by LeFloch, Makhlofand and ( / 1 ) ( 1 ) 1 ( Okutmustur, SINUM 50, 2136 (2012) by algebraic manipu )
t a a tS
S u u u ¶
+ = ¶ + + = 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.
2 2
t x
2 2 2
t x x x
Irrotational flow: Hamilton equation: Hamilton-Jacobi equation: For barotropic fluids, the above equation is coupled to the c ( )
a a b a a t a a t a b b b a b
S p p p p p S H p H v + ¶ + ¶
= + =
1,2
where : det( ) Characteristics :
,
k k t k i i t t ij k
p H g u u r r u d r r a g r g g 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
(1 ) { (1 ) (1 ) [(1 ) (1 )( ) Complete eigenbasis The system is strongly hyperbolic (for fini e } t ] )
k k kk k k
c c c c c c l a n n n n g n b
=
Notable features:
recovered via dividing pi=hui by specific enthalpy h which is 1 at the surface (no division by zero)
becomes weakly hyperbolic instability on surface
surface: sound speed in a realistic NS crust (outer 1 km) cs ~ 0.05
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
2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5
2 S
(km) R
S ~ 0.05
c
Credit: Christian Krüger
Notable features:
ui is recovered via dividing pi=hui by specific enthalpy h which is 1 at the surface (no division by zero)
becomes weakly hyperbolic instability on surface
surface: sound speed in a realistic NS crust (outer 1 km) cs ~ 0.05. Then, extrapolating the EOS to the exterior (h<1) allows one to evolve smooth fields and obtain pointwise convergence on the surface, which is unattainable with an artificial atmosphere.
that preserve symplecic structure and circulation
Reference
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