I-Love-Q Relations: From Compact Stars to Black Holes

Abstract

The relations between most observables associated with a compact star, such as the mass and radius of a neutron star or a quark star, typically depend strongly on their unknown internal structure. The I-Love-Q relations (between the moment of inertia, the tidal deformability and the quadrupole moment) are however approximately insensitive to this structure. These relations become exact for stationary black holes in General Relativity as shown by the no-hair theorems. In this paper, we take the first steps toward studying how the approximate I-Love-Q relations become exact in the limit as compact stars become black holes. To do so, we consider a toy model, i.e. incompressible stars with anisotropic pressure, which allows us to model an equilibrium sequence of stars with their compactness approaching the black hole limit arbitrarily closely. We extract the I-Love-Q trio by numerically constructing such a sequence in the slow-rotation and small-tide approximations. We find that the I-Love-Q relations approach the black hole limit in a nontrivial way, with the quadrupole moment and the tidal deformability changing sign as the compactness and the amount of anisotropy are increased. Generalizing Maclaurin spheroids to anisotropic stars, we show that the multipole moments also change sign in the Newtonian limit as the amount of anisotropy is increased. We also prove analytically that the stellar moment of inertia reaches the black hole limit as the compactness reaches the black hole value in the strongly anisotropic limit. Modeling the black hole limit through a sequence of anisotropic stars, however, fails when considering other theories of gravity. We calculate the scalar dipole charge and the moment of inertia in a parity-violating modified theory and find that these quantities do not tend to their black hole counterparts as the anisotropic stellar sequence approaches the black hole limit.