Relativistic tidal properties of neutron stars

Abstract

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 $G{\ensuremath{\mu}}_{\ensuremath{\ell}}=[\mathrm{\text{length}}{]}^{2\ensuremath{\ell}+1}$ measuring the $\ensuremath{\ell}$th-order mass multipolar moment $G{M}_{{a}_{1}\dots{}{a}_{\ensuremath{\ell}}}$ induced in a star by an external $\ensuremath{\ell}$th-order gravito-electric tidal field ${G}_{{a}_{1}\dots{}{a}_{\ensuremath{\ell}}}$; (ii) a gravito-magnetic-type coefficient $G{\ensuremath{\sigma}}_{\ensuremath{\ell}}=[\mathrm{\text{length}}{]}^{2\ensuremath{\ell}+1}$ measuring the $\ensuremath{\ell}$th spin multipole moment $G{S}_{{a}_{1}\dots{}{a}_{\ensuremath{\ell}}}$ induced in a star by an external $\ensuremath{\ell}$th-order gravito-magnetic tidal field ${H}_{{a}_{1}\dots{}{a}_{\ensuremath{\ell}}}$; and (iii) a dimensionless ``shape'' Love number ${h}_{\ensuremath{\ell}}$ measuring the distortion of the shape of the surface of a star by an external $\ensuremath{\ell}$th-order gravito-electric tidal field. All the dimensionless tidal coefficients $G{\ensuremath{\mu}}_{\ensuremath{\ell}}/{R}^{2\ensuremath{\ell}+1}$, $G{\ensuremath{\sigma}}_{\ensuremath{\ell}}/{R}^{2\ensuremath{\ell}+1}$, and ${h}_{\ensuremath{\ell}}$ (where $R$ is the radius of the star) are found to have a strong sensitivity to the value of the star's ``compactness'' $c\ensuremath{\equiv}GM/({c}_{0}^{2}R)$ (where we indicate by ${c}_{0}$ the speed of light). In particular, $G{\ensuremath{\mu}}_{\ensuremath{\ell}}/{R}^{2\ensuremath{\ell}+1}\ensuremath{\sim}{k}_{\ensuremath{\ell}}$ is found to strongly decrease, as $c$ increases, down to a zero value as $c$ is formally extended to the ``black hole (BH) limit'' ${c}^{\mathrm{BH}}=1/2$. The shape Love number ${h}_{\ensuremath{\ell}}$ is also found to significantly decrease as $c$ increases, though it does not vanish in the formal limit $c\ensuremath{\rightarrow}{c}^{\mathrm{BH}}$, but is rather found to agree with the recently determined shape Love numbers of black holes. The formal vanishing of ${\ensuremath{\mu}}_{\ensuremath{\ell}}$ and ${\ensuremath{\sigma}}_{\ensuremath{\ell}}$ as $c\ensuremath{\rightarrow}{c}^{\mathrm{BH}}$ 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.