Can gravity give neutrons a longer lifespan?

Question

Isolated neutrons have a lifespan of about one minute yet neutrons in a neutron star can have the lifespan of the neutron star itself and not decay into proton and electron. Is the intense gravity keeping the neutrons from decaying?

Answer 1

In neutron stars gravity does not directly stabilise the neutrons. Rather, gravity forces the particles of matter together to a very high density where it is balanced by the degeneracy pressure (fermions cannot be in the same quantum state and this prevents them from being arbitrarily close-packed). As matter get packed together the reaction $p^+ + e^- \rightarrow n^0 + \nu$ occurs turning it into neutrons. This happens because the electrons are packed so tightly that they have to reach high energies (this is due to the Heisenberg relation $\Delta x\Delta p>h$; as the position uncertainty decreases the momentum range must increase). The reverse reaction, neutron decay, creates an electron: $n^0 \rightarrow p^+ + e^- +\bar{\nu}$. But there is no space for the electron, so this is inhibited. The chemical potential is negative: you need extra energy to make the neutrons decay here.

There is another way neutrons can be stabilised by gravity and that is time dilation. Place a neutron at a very low gravitational potential and the decay rate as measured by remote observers will decline. This is a minor effect, since it scales as $1/(1-\Delta\Phi/c^2)$ for milder fields. For neutron star surfaces this might give you a lifespan 1.4 to 1.7 times longer. For black holes it scales as $1/\sqrt{1-r_S/r}$. If we put a neutron one Planck length outside a supermassice $10^9 M_\odot$ black hole we can get a factor of $4\times 10^{23}$ - suddenly that the neutron would be very long-lasting.

(Although keeping it in place would be another matter, since this is inside the smallest stable circular orbit. In fact, the acceleration will produce Unruh radiation. The extra energy of acceleration can make normally stable protons decay into neutrons, and I suspect similar effects may change the stability of neutrons.).