Initial Data for Magnetized Stars in General Relativity Eric
Hirschmann, BYU MG12, Paris, July 2009 Slide 2 Some background
Interested in the evolution of binary systems of compact objects,
e.g. NS-NS, NS-BH, BH- BH. sig recent progress on BH-BH more work
needed on non-vacuum case Potentially sig. additional physics
neutrinos, nuclear EOS, B, plasmas, rad hydro From NR side, been
slow to incorporate many of these effects. Slide 3 introduction We
have begun a program to incorporate additional physics into
non-vacuum binaries. Evidence that presence of (strong) magnetic
fields can affect gravitational wave signal. Slide 4 and motivation
Drawback in our current simulations is use of seeded fields small
magnetic field added below truncation error in constraints grows
via MRI Works, but ad hoc Can it be improved? Also of interest to
consider single, magnetized stars. GR models of magnetars Goal:
Evolve/simulate more realistic models of magnetized NS singly and
in binaries. Slide 5 Modeling magnetized stars Equilibrium (fluid)
NS configurations in GR Much work done Includes stability studies
and nonlinear evolution Much less done on magnetized NS in GR
Bocquet et al (rigid rotation, poloidal) Cardall et al (static,
poloidal) Kiuchi et al (rigid rotation, toroidal; axisymmetric
evolution) Little or nothing (?) on convective motions in GR Slide
6 Analytic assumptions Our matter is a perfect fluid + E&M rest
mass density internal energy density pressure fluid velocity
Maxwell Ideal gas equation of state: Polytropic relation
Relativistic Ohms law (MHD; connect fluid to E&M) Infinite
conductivity (ideal MHD) Slide 7 Analytic assumptions Symmetries:
axisymmetry stationarity Potentially yield an axisymmetric,
magnetized, relativistic neutron star with convection and
differential rotation Slide 8 Another assumption: hypersurface
orthogonality? In general, must solve for most of metric. Hard
simplify? Assume that 2D surfaces orthogonal to Killing vectors
exist and can be extended (HO) Requires circularity flow fields
follow lines of latitude only no convection either poloidal or
toroidal B fields, but not both Pros: Calc spacetime means 4 metric
components Cons: Strong (mathematical) restriction on physics; e.g.
no convection, simple B fields, etc. Slide 9 Our approach: drop
circularity Solve the full stationary, axisymmetric problem Assume
same things as before (ideal MHD, polytropes, 2 KVs) Perform a
Kaluza-Klein like decomposition Slide 10 The equations 3+2
Poisson-type eqns for the scalars 4 pairs of first order eqns for
(conformal) 2-metric and 3 2-forms are independent of analytic:
solvable via quadrature numerical: treat as ODEs / 1D integrations
Slide 11 The equations (cont) Use a Greens function approach for
elliptic eqns: Nonlinear integral equations, but iterate to
convergence. Work in radial (compactified) and angular coords
Allows exact BCs at infinity Based on self-consistent field
approach of Hachisu, et al (1988) Slide 12 The equations (cont) Eqn
of structure / eqn of (magneto-) hydrostatic equilibrium e.g., in
absence of B-fields with B-fields, includes internal currents with
ansatz, can solve exactly relates the physical, free data to the
star and thermo props from it, we get the surface of the star Ideal
MHD condition (Lorentz) (axisymmetry, stationary, MHD) constrains
allowable physics in principle, all present with one constrained
Slide 13 Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19
Slide 20 Slide 21 Slide 22 Slide 23 Slide 24 Slide 25 Slide 26
Slide 27 Slide 28 Slide 29 Slide 30 Slide 31 Slide 32 Slide 33
Slide 34 Slide 35 Slide 36 Slide 37 Slide 38 Summary and future We
can construct magnetized neutron stars with general magnetic field
topology. Add differential rotation Add convection Evolve these
singly Stability, growth and final mag of B GW emission and in
binaries Other directions: More realistic equations of state; QED
limit Accretion disks Slide 39 Slide 40 Analytic assumptions Our
matter (stress tensor) is a perfect fluid + EM Rest mass density,
internal energy density, pressure, fluid velocity, Maxwell
Polytropic equation of state: Relativistic version of Ohms law to
connect the fluid to the EM (MHD with displacement current: RMHD)
Infinite conductivity (ideal MHD) Symmetries: axisymmetry and
stationarity Potentially yield an axisymmetric, magnetized,
relativistic neutron star with convection and differential rotation
Slide 41 Related work Equilibrium, axisymmetric NS in GR have been
studied since 1970s, but Rigid rotation, no convection Seldom
include B fields (special) Hypersurface orthogonality (HO) Few
stability studies One (formal) study relaxing HO yuck! Slide 42
Another assumption: hypersurface orthogonality? In general (for
axisymmetric, stationary), must solve for most components of metric
in two variables. This is hard can one simplify? Assume that 2D
surfaces perpendicular to the symmetries exist and can be extended
Turns out to require circularity flow fields follow lines of
latitude only no convection either poloidal or toroidal B fields,
but not both Pros: Calc of metric means only 4 components Cons:
Strong (mathematical) restriction on physics; e.g. no convection,
simple B fields, etc. Slide 43 Where it stands We have a formalism
that will allow the computation of axisymmetric, equilibrium NS
with magnetic fields (both poloidal and toroidal), convection and
differential rotation. But we must still debug our code.
