Relativistic tidal properties of neutron stars

We study the various linear responses of neutron stars to external relativistic tidal fields. We focus on three different tidal responses, associated to three different tidal coefficients: (i) a gravito-electric-type coefficient Gμ_=[length]^2+1 measuring the th-order mass multipolar moment GM_a₁a_ induced in a star by an external th-order gravito-electric tidal field G_a₁a_; (ii) a gravito-magnetic-type coefficient Gσ_=[length]^2+1 measuring the th spin multipole moment GS_a₁a_ induced in a star by an external th-order gravito-magnetic tidal field H_a₁a_; and (iii) a dimensionless ``shape'' Love number h_ measuring the distortion of the shape of the surface of a star by an external th-order gravito-electric tidal field. All the dimensionless tidal coefficients Gμ_/R^2+1, Gσ_/R^2+1, and h_ (where R is the radius of the star) are found to have a strong sensitivity to the value of the star's ``compactness'' c≡GM/(c₀²R) (where we indicate by c₀ the speed of light). In particular, Gμ_/R^2+1~k_ is found to strongly decrease, as c increases, down to a zero value as c is formally extended to the ``black hole (BH) limit'' cBH=1/2. The shape Love number h_ is also found to significantly decrease as c increases, though it does not vanish in the formal limit c→cBH, but is rather found to agree with the recently determined shape Love numbers of black holes. The formal vanishing of μ_ and σ_ as c→cBH is a consequence of the no-hair properties of black holes. This vanishing suggests, but in no way proves, that the effective action describing the gravitational interactions of black holes may not need to be augmented by nonminimal worldline couplings.