A new scheme for matching general relativistic ideal magnetohydrodynamics to its force-free limit

We present a new computational method for smoothly matching general relativistic ideal magnetohydrodynamics (MHD) to its force-free limit. The method is based on a flux-conservative formalism for MHD and its force-free limit and a vector potential formulation for the induction equation to maintain the zero divergence constraint for the magnetic field. The force-free formulation we adopt evolves the magnetic field and the Poynting vector, instead of the magnetic and electric fields. We show that our force-free code passes a robust suite of tests, performed both in one-dimensional flat spacetime and in three-dimensional black hole spacetimes. We also demonstrate that our matching technique successfully reproduces the aligned rotator force-free solution. Our new techniques are suitable for studying electromagnetic effects and predicting electromagnetic signals arising in many different curved spacetime scenarios. For example, we can treat spinning neutron stars, either in isolation or in compact binaries, that have MHD interiors and force-free magnetospheres.

Figure 1

Results from one-dimensional force-free tests: dashed lines indicate initial data, solid lines the analytic solution (at the indicated time) and crosses the numerical solution, except for the force-free breakdown test (bottom), where the solid line indicates the numerical solution.

Figure 2

Left: Split monopole initial poloidal magnetic fields lines in blue (black in grey scale). Black shaded area designates the BH interior. Right: Same as in left, but at $\mathrm{t}=5\mathrm{M}$.

Figure 3

Poloidal magnetic field lines for the Schwarzchild electrovacuum Wald solution. Poloidal magnetic fields lines at $t=0\mathrm{M}$ in blue (black in grey scale), and at $t=5\mathrm{M}$ in red (grey in grey scale). Black shaded area designates the BH interior. The lines at $t=5\mathrm{M}$ are overlapping with those at $t=0\mathrm{M}$.

Figure 4

Left: Magnetospheric Wald initial poloidal magnetic fields lines in blue (black in grey scale). Black shaded area designates the BH interior. Right: Same as in left, but at $t=126\mathrm{M}$.

Figure 5

Left: Far field poloidal magnetic field lines in blue (black in grey scale). Yellow (grey in grey scale) shaded area designates the stellar interior. Right: Same as in left, but zooming in on the near zone and showing the interior magnetic field, and how it smoothly joins to the exterior one. The time $t=6\pi /\Omega$, at which point the field has reached a stationary state.

Figure 6

Angular frequency of the magnetic field lines ${\Omega }_F$ normalized by the spin angular frequency of the star $\Omega$ vs the polar angle $\theta$ on the x-z plane for $r=0.3R_{\mathrm{LC}}$ (solid lines) and $r=0.7R_{\mathrm{LC}}$ (dashed lines). The value of this ratio should be unity. Two resolutions are shown here: low resolution (top) and high resolution (bottom). It is clear that the magnetosphere within the light cylinder is in near corotation with the star even for the low-resolution case.

Figure 7

Convergence test for the time independence of the vector potential $A_i$. The plot shows the $L_2$ norms of the difference at $t=5\mathrm{M}$ between the numerical and analytic solutions in the volume contained inside a coordinate sphere of radius $r=90\mathrm{M}$. The plot demonstrates that the error norms converge to zero as $\sim \Delta x^2$, i.e., the order of convergence of our code is 2. Note that $L_2(A_x)\simeq 3.52\times {10}^4$, so that $L_2(\Delta A_x)/L_2(A_x)\sim {10}^{-6}$ even for the lowest-resolution run.