Assume there is a neutron star with a magnetic field and no net electrical
charge. What happens to the magnetic field if it gains some mass and
collapses into a black hole?
Astrophysics – What Happens if a Magnetar Collapses into a Black Hole?
astrophysicsblack-holesmagnetic fieldsneutron-stars
Related Solutions
The solution for a black hole with non-zero spin and non-zero charge is completed with the vector potential associated with the electromagnetic field. If both spin and charge are nonzero, the vector potential will have $A_{i}$ that depend on the spatial coordinates, and thus, the black hole will have a nonzero magnetic field.
In spheriodial coordinates, the vector potential comes out to:
$\begin{equation} A_{a} = -\frac{er}{r^{2}+a^{2}\cos^{2}\theta}dt_{a} + \frac{era\sin^{2}\theta}{r^{2}+a^{2}\cos^{2}\theta}d\phi_{a} \end{equation}$
Taking the curl of the spatial part (thus, assuming that we're going to calculate 'the magnetic field observed by someone who is not moving relative to $r,\theta,\phi$, for instance), we find the relevant two components of the Maxwell tensor:
$\begin{align} F_{r\phi}&=\frac{(a^{2}\cos^{2}\theta - r^{2})ea\sin^{2}\theta}{(r^{2}+a^{2}\cos^{2}\theta)^{2}}\\ F_{\theta\phi}&=\frac{(r^{2}+a^{2})era\sin(2\theta)}{(r^{2}+a^{2}\cos^{2}\theta)^{2}} \end{align}$
We know that $F_{r\phi}$ is proportional to $B_{\theta}$, while $F_{\theta\phi}$ is proportional to $B_{r}$ (the exact factors require calculating the determinant of the metric tensor, and I don't think calculating these terms exactly is the point of this excersise). The angular dependence of this field, though, should make it clear that the field is different than that of a true dipole. The question about the field lines crossing the horizon is a trickier one, since these coordinates are singular on the horizon, and that calculation would have to be carried out in a coordinate system that is non-singular there. But I would expect there to be a normal component of the magnetic field to the horizon, since there is nothing in these $F_{ab}$ terms that is singular on the horizon.
There is a test of the collapse of a magnetized neutron star (case 3) in this preprint: http://arxiv.org/abs/arXiv:1208.3487.
Here is some of the text and figures from that paper that describe how the magnetic field is expelled:
- Magnetized Collapse to a Black Hole
Our final and most comprehensive test is represented by the collapse to a BH of a magnetized nonrotating star. This is more than a purely numerical test as it simulates a process that is expected to take place in astrophysically realistic conditions, such as those accompanying the merger of a binary system of magnetized neutron stars [26, 27], or of an accreting magnetized neutron star. The interest in this process lays in that the collapse will not only be a strong source of gravitational waves, but also of electromagnetic radiation, that could be potentially detectable (either directly or as processed signal). The magnetized plasma and electromagnetic fields that surround the star, in fact, will react dynamically to the rapidly changing and strong gravitational fields of the collapsing star and respond by emitting electromagnetic radiation. Of course, no gravitational waves can be emitted in the case considered here of a nonrotating star, but we can nevertheless explore with unprecedented accuracy the electromagnetic emission and assess, in particular, the efficiency of the process and thus estimate how much of the available binding energy is actually radiated in electromagnetic waves. Our setup also allows us to investigate the dynamics of the electromagnetic fields once a BH is formed and hence to assess the validity of the no-hair theorem, which predicts the exponential decay of any electromagnetic field in terms of Quasi Normal Mode (QNM) emission from the BH.
...
As the collapse proceeds, the restmass density in the center and the curvature of the spacetime increase until an appararent horizon is found at t = 0.57 ms and is marked with a thin red line in Fig. 12. As the stellar matter is accreted onto the BH (the rest-mass outside the horizon $M_{b, out} = 0$ is zero by $t \geq 0.62$ ms), the external magnetic field which was anchored on the stellar surface becomes disconnected, forming closed magneticfield loopswhich carry away the electromagnetic energy in the form of dipolar radiation. This process, which has been described through a simplified non-relativistic analytical model in Ref. [52], predicts the presence of regions where |E| > |B| as the toroidal electric field propagates outwards as a wave. This process can be observed very clearly in Fig. 13, which displays the same three bottom panels of Fig. 12 on a smaller scale of only 15 km to highlight the dynamics near the horizon. In particular, it is now very clear that a closed set of magnetic field lines is built just outside the horizon at t = 1.0 ms, that is radiated away. Note also that our choice of gauges (which are the same used in [61]) allows us to model without problems also the solution inside the apparent horizon. While the left panel of Fig. 13 shows thatmost of the rest-mass is dissipated away already by t = 0.65 ms (see discussion in [62] about why this happens), some of the matter remains on the grid near the singularity, anchoring there the magnetic field which slowly evolves as shown in the middle and right panels.
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Best Answer
This paper by Lyutikov and McKinney describes this precise scenario. I'll provide a brief summary of their results here.
One of the crucial underlying assumptions made when deriving the no-hair theorem is that the region outside the event horizon is a vacuum. However, a rapidly spinning, magnetized neutron star can induce vacuum breakdown, leading to the existence of self-generated currents in the magnetosphere. That is, an initial spinning dipole field with no external currents evolves into a magnetized black hole supported by currents outside the event horizon. In particular, rotation-induced poloidal currents cause magnetic field lines which originate near the poles of magnetar to "open up" to infinity.
The presence of a conductive plasma in the magnetosphere freezes magnetic field lines into the surface. As the magnetar collapses, the closed field lines are absorbed into the event horizon. However, the open lines, which connect the magnetar surface to infinity, persist and end up connecting the event horizon to infinity. The resulting field relaxes to a split monopole configuration which looks something like this:
Over time, the equatorial current sheet dissipates and the field lines which are anchored to the event horizon are able to "slide off" and the magnetic field decays. However, this process relies on the finite resistivity of the plasma in the magnetosphere, and is slow when compared to the timescale one would naively expect from the no-hair theorem. As a result, magnetized black holes can persist for quite a while even in the absence of externally supplied accretion material.