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 G_=length^2+1 measuring the th-order mass multipolar moment GM₀₁{a_} induced in a star by an external th-order gravito-electric tidal field G₀₁{a_}; (ii) a gravito-magnetic-type coefficient G_=length^2+1 measuring the th spin multipole moment GS₀₁{a_} induced in a star by an external th-order gravito-magnetic tidal field H₀₁{a_}; 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 G_/R^2+1, G_/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/ (c₀^2R) (where we indicate by c₀ the speed of light). In particular, G_/R^2+1k_ is found to strongly decrease, as c increases, down to a zero value as c is formally extended to the ``black hole (BH) limit'' c^BH=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 cc^BH, but is rather found to agree with the recently determined shape Love numbers of black holes. The formal vanishing of _ and _ as cc^BH 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.