# Can gravity give neutrons a longer lifespan? [duplicate]

This question already has an answer here:

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?

## marked as duplicate by Rob Jeffries astrophysics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Apr 15 '18 at 6:23

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.