# Would a neutron star-like celestial body made of bosons collapse because of missing Pauli repulsion?

Imagine the neutron star-like celestial body would be built up of bosons (assume that neutrons would be bosons). In that case there would be no Pauli repulsion (the exchange force of the exchange interaction would not be repulsive), so this would apparently lead to an immediate collapse of the neutron star. Whereas the argument for the repulsive exchange force is based on Pauli's principle, in case of bosons could not Heisenberg's uncertainty relation take up a similar role ? If $$n$$ particles are pressed in a smaller and smaller volume $$V$$ according to Heisenberg's uncertainty relation their momentum should increase like $$\Delta p = \hbar \cdot (V/n)^{-1/3}$$, this would generate an enormous pressure and may be prevent the star from collapsing ? I could be even more inquisitive and asking: Has Heisenberg's uncertainty relation ever been tested with bosons? If so, and as I've never heard about an invalidity of the uncertainty relation, can the uncertainty relation save the neutron star from collapsing ? Or is there a flaw in my reasoning ?

What volume were you thinking of? The bosons are confined within the central region of a neutron star - which is quite a big volume. So, actually I think they would form a condensate with nearly zero momentum and the uncertainty principle doesn't have a role.

Even if we consider the bosons to be confined to much smaller volumes, it isn't the case that gravity can be defeated just by increasing the pressure. In GR the RHS of the hydrostatic equilibrium equation contains a pressure term (see Tolman-Oppenheimer-Volkoff hydrostatic equilibrium). Thus increasing the pressure leads to more curvature and a higher required pressure gradient for equilibrium, and is thus ultimately self-defeating. Neutron stars collapse at finite density and pressure, well before they are small enough to be black holes or for the uncertainty principle to be an issue.

• So if the pressure, in particular pressure from Pauli repulsion, contributes to the curvature, what prevents a Neutron star (this time made of fermions) from collapsing to a black hole ? – Frederic Thomas Dec 11 '18 at 8:51
• @FredericThomas A pressure gradient prevents a neutron star from collapsing. If a neutron star becomes more massive than 2-3 solar masses then it does collapse because of the phenomenon I describe. Note that degeneracy pressure is not the main factor supporting a neutron star, it is the strong-force nucleon-nucleon repulsion between neutrons in asymmetric nuclear matter. Ideal neutron degeneracy pressure could not support neutron stars bigger than about 0.7 solar masses. – Rob Jeffries Dec 11 '18 at 9:07
• I try to recap: pure pressure would not prevent a neutron star from collapsing to a black hole since it would generate curvature, but a pressure gradient (may be a particular one) would prevent the collapse to a BH. – Frederic Thomas Dec 11 '18 at 11:26
• @FredericThomas Not really. The equation of hydrostatic equilibrium is a balance between pressure gradient on the LHS, with terms associated with gravity and pressure on the RHS. I will make an edit. – Rob Jeffries Dec 11 '18 at 16:55