# Can a rotating neutron star collapse to a black hole through a reduction in rotation?

It is well known that non-rotating neutron stars cannot grow without bound, since an increase in mass causes a proportional increase in density, and accretion beyond a critical limit would cause it to collapse to a black hole.

However, it seems that a rotating neutron star could support a supercritical mass, since the additional centrifugal force of rotation would balance its gravitational force. Of course, the star could not be arbitrarily massive since an increase in mass must be accompanied by an increase in rotation, and there is a limit to how fast a star can rotate before breaking apart.

Regardless, we know that neutron stars can rotate as fast as 716 revolutions per second, and that many have extremely strong magnetic fields produced by magnethohydrodynamic dynamos, fields which can persist due to currents in the proton-superconductor phase of matter that exist at intermediate depths within the neutron star.

Therefore, I wonder if it possible that neutron stars could, in effect, lose energy and angular momentum through the production of magnetic fields associated with rotation, and thus slow until centrifugal forces diminish enough that the star collapses to a black hole?

If this is theoretically possible, what might the signature for such an event be? On what timescale would it occur? How common could it be?

Rotation periods of the order of a millisecond can sustain a maximum mass that is only a few per cent larger than the maximum mass predicted by the TOV equations. Differential rotation can increase the effect, see this paper for some (very) extreme (and probably unphysical) examples. More details are given in the references of the paper (just take a look at the introduction).

Hypermassive neutron stars can be the remnant of neutron star mergers. Their differential rotation is killed by viscosity, and the star then collapses to a black hole. Otherways to spin-down a hypermassive neutron star id radiation emission (electromagnetic and gravitational). However, for the post-merger remnant, the most efficient mechanism is thought to be viscosity.

See Prakash, "Evolution of a Neutron Star From its Birth to Old Age," https://arxiv.org/abs/astro-ph/0012136, p. 3:

Strange matter, in the form of hyperons, a Bose condensate, or quark matter, which is suppressed to extremely large densities when neutrinos are trapped in matter, could appear at the end of the deleptonization. The appearance of strange matter leads to a decrease in the theoretical maximum mass that mat- ter is capable of supporting, leading to another possibility for black hole formation [4]. This would occur if the PNS’s mass, which must be less than the maximum mass of hot, lepton-rich matter (or else a black hole would already have formed), is greater than the maximum mass of hot, lepton-poor matter. However, if strangeness does not appear, the theoretical maximum mass instead increases during deleptonization and the appearance of a black hole would be unlikely unless accretion in this stage remains significant.

It looks to me like they're that evolution from a neutron star to a black hole can happen, but the mechanism they discuss is not the one you describe.

The fastest-rotating pulsars have frequencies of about 1000 Hz. But if a neutron star were to have an angular momentum in the relativistic domain, it would have to have a frequency on the order of $$(G/c)Mr^{-2}$$, which is more like $$10^6$$ Hz. That is, the surface rotation of a neutron star is never more than $$\sim 10^{-3}c$$, which means that the rotation is nonrelativistic. But the dynamics that control the equilibrium of a neutron star are all relativistic, so that even at formation, the kinetic energy tied up in rotation is already negligible. If the neutron star could get started on further collapse, then the kinetic energy $$\sim J^2/r^2$$ would go up and eventually become relativistic -- but it can't get started, $$r$$ is stuck at a value that is much too high, so the KE will never be relativistic.

Note, however, that on extremely long time scales like $$10^{10^{10}}$$ years or something, any object will probably undergo quantum tunneling to a black hole.

• The fastest-spinning neutron star currently known is PSR J1748-2446ad, rotating at a rate of 716 times a second (43,000 rpm), giving a linear speed at the surface on the order of 0.24 c (i.e., nearly a quarter the speed of light). Is this not relativistic? Dec 16, 2021 at 21:41
• @EssanVerne yes it is only slightly relativistic because the $\gamma$ factor is very close to 1 for that fraction of $c$. Dec 17, 2021 at 19:15