Black holes are said to have no hair because all of their multipole moments can be expressed in terms of just their mass , charge and spin angular momentum .
The recent discovery of approximately equation-of-state-independent relations among certain multipole moments in neutron stars suggests that they are also approximately bald .
We here explore the yet unknown origin for this universality .
First , we investigate which region of the neutron star ’ s interior and of the equation of state is most responsible for the universality .
We find that the universal relation between the moment of inertia and the quadrupole moment is dominated by the star ’ s outer-core , a shell of width ( 50–95 ) % of the total radius , which corresponds to the density range ( 10 ^ { 14 } – 10 ^ { 15 } ) g/cm ^ { 3 } .
In this range , realistic neutron star equations of state are not sufficiently similar to each other to explain the universality observed .
Second , we study the impact on the universality of approximating stellar isodensity contours as self-similar ellipsoids .
An analytical calculation in the non-relativistic limit reveals that the shape of the ellipsoids per se does not affect the universal relations much , but relaxing the self-similarity assumption can completely destroy it .
Third , we investigate the eccentricity profiles of rotating relativistic stars and find that the stellar eccentricity is roughly constant , with variations of roughly ( 20–30 ) % in the region that matters to the universal relations .
Fourth , we repeat the above analysis for differentially-rotating , non-compact , regular stars and find that this time the eccentricity is not constant , with variations that easily exceed 100 % , and moreover , universality is lost .
These findings suggest that universality arises as an emergent approximate symmetry : as one flows in the stellar-structure phase space from non-compact star region to the relativistic star region , the eccentricity variation inside stars decreases , leading to approximate self-similarity in their isodensity contours , which then leads to the universal behavior observed in their exterior multipole moments .