General Relativistic Hydrodynamics with Viscosity

Collaborators:

Matthew D. Duez

Stuart L. Shapiro

Branson C. Stephens

Phys. Rev. D 69, 104030 (2004)

Presented by Yuk Tung Liu

14th Midwest Relativity Meeting

October 15, 2004

Motivation

• Viscosity can have significant effects on relativistic stars

- suppress gravitational-wave driven (CFS) instabilities

Motivation

• Viscosity can have significant effects on relativistic stars

- suppress gravitational-wave driven (CFS) instabilities

- drive a secular (Jacobi) bar-mode instability

Motivation

• Viscosity can have significant effects on relativistic stars

- suppress gravitational-wave driven (CFS) instabilities

- drive a secular (Jacobi) bar-mode instability

- destroy differential rotation secular evolution of

hypermassive neutron stars

Formalism

• Evolve the metric using BSSN formulation

• Gauge choices

Lapse: K-driver (approximate maximal slicing)

Shift: Gamma-driver (approximate “Gamma-freezing” condition)

Hydrodynamic Variables

Stress-energy tensor

PguuhT 0

)(3

1;)();(

uuguuau

0

1

Ph

Shear tensor:

Specific enthalpy:

Rest-mass density: 0 Pressure: PCoefficient of shear viscosity: Specific internal energy: 4-velocity: u 4-acceleration: a

-law equation of state: 0)1( P

2

Hydrodynamic Equations

Define new hydrodynamic variables:

ii uhSugeug *0/1

0*0

0*

~)(

Baryon number conservation :0)( 0 u

)/v(0)v( 0** uu iii

it

Energy equation :0 Tu

)v( **i

it ee

Navier-Stokes equation :0 kT

uuhggPgSS kk

ikikt 0,, 2

1)v

~(

~

kk ggg ,,)(2

/)1(

0 )(2

g

Viscosity Law

• Want to explore point of principle: evolve general relativistic hydrodynamics with viscosity

• Not interested in the details of viscosity in neutron stars• Assume simple viscosity of the form

=P P (P : positive constant)

• Choose P such that the viscous timescale vis= a few dynamical times (long enough for the system to be evolved quasi-statically, but short enough to make numerical treatment trackable)

• This viscosity law is consistent with a “turbulent viscosity”

Code Test – Evolution of a stable, uniformly rotating star

R/M = 4

During the entire simulation,

M / M < 0.1% ;J / J < 1.5% ;

Violation of Hamiltonian and momentum

constraints < 1%

Evolution of a Differentially Rotating Star

vis = 5.5Prot

During the entire simulation,

M / M < 0.4% ;J / J < 0.4% ;

Violation of Hamiltonian and momentum

constraints < 1%

Scaling of Secular Evolution with Viscosity Parameter

Conclusion

• We have developed a hydrodynamic code to solve the fully-relativistic Navier-Stokes equation

• Our code is able to evolve relativistic stars for dozens of rotation periods

• We studied the secular evolution of hypermassive neutron stars (next talk)

• We will use this code to study the viscosity-driven (Jacobi) bar-mode instability