# Roche Zone of a black hole vs Roche Zone of Neutron Star

Suppose a very unusually small back hole with the mass of about, lets say, 1.5 solar masses(hawking radiation should destroy a lot of mass let's assume). It thus would have a Roche limit smaller than larger black holes, and the Roche limit should be outside of the black hole instead of inside it, like some supermassive black holes.

Then, let's take a super massive neutron star, perhaps even with quark matter at its center. Even though this may not exist, let's say that is 3 solar masses by taking matter from a nearby red giant rather slowly. This would mean that it thus has a larger Roche limit.

Then, let's fit the two together in a binary system. Move the black hole inwards so that it is in the Roche limit of the neutron star, but its Roche limit is not affecting the neutron star.

My question is that, since a black hole can't physically be ripped apart by gravity due to it being a hole in space time itself, can it's structure be altered in any way by the gravitational pull of the neutron star? Like from a sphere to an oval or something like that? Can it separate into two smaller ones due to the gravity of the neutron star? In the system, who would win the gravity war?

Now assume that the black hole has angular momentum and change, giving it a magnetic field. Then make the neutron star a magnetar of the strongest kind, and place them in the same binary position as I described above. Would their magnetic fields have an effect on any of the bodies that are generating them? Would their magnetic fields have any effect on the other object's magnetic field?

• @uhoh ok. Anyways I read thru the criteria for tag wiki writing and I should be able to get better ones later. Feb 22 '19 at 0:07
• It's really great that you are doing this, we have a lot of undefined tags. There are a few high-fliers still remaining unfinished here as well. Thanks!
– uhoh
Feb 22 '19 at 0:28

The Roche limit is defined in classical physics, dealing with materials that behave classically. That said, of course people have tried to estimate the counterpart in relativistic conditions. It turns out that it behaves roughly like the classical case when considering a liquid body orbiting a black hole. For the neutron star case and a more massive stellar mass black hole the limit is not too far away from the ISCO.

Would a black hole be affected by tidal forces? A tidal force is after all just the effect of the radially varying curvature of spacetime, and a black hole embedded in such a curved spacetime does show distortion. Not only that, but one can apparently treat the black hole as a fluid sphere that can even dissipate! (paper, presentation2) As the plot in the second presentation shows, yes, the horizon can get distorted by a nearby massive orbiting object.

However, the laws of black hole thermodynamics do not allow the horizon area to go down. Splitting a mass 2M hole into two holes of mass M would change the area from $$4 \times (4\pi G^2 M^2/c^4)$$ to $$2\times (4\pi G^2 M^2/c^4)$$. So that is not going to happen (unless there was a process that could release a black hole's worth of entropy in the outside universe - a very tall order).

In the end, what would happen is that the deformations would dissipate orbital energy (by heating the neutron star and gravitational radiation) until the objects spiralled together, the neutron star would cross the relativistic Roche limit and shortly after the remnants would plunge inside the ISCO and be absorbed.

The magnetic field case would complicate things in various ways, but remember that even magnetar magnetic fields only have the density of $$10^4$$ times lead - compared to the neutron star density this is nothing. There would presumably be a lot of electromagnetic dissipation, jets, and complex torques, but I suspect the endgame is the same.