Initial Data for Magnetized Stars in General Relativity Eric Hirschmann, BYU MG12, Paris, July 2009

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.

… 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.

… 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.

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

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 Ohm’s law (MHD; connect fluid to E&M) Infinite conductivity (ideal MHD)

Analytic assumptions Symmetries: –axisymmetry –stationarity Potentially yield an axisymmetric, magnetized, relativistic neutron star with convection and differential rotation

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.

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

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

The equations (cont) Use a Green’s 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)

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

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

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 Ohm’s 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

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!

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.

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.