Tidal deformations of a spinning compact object

TL;DR

The paper develops a perturbative framework to compute tidal deformabilities (tidal Love numbers) of slowly spinning compact objects in general relativity, extending static results to axisymmetric tidal fields and to ${\cal O}({\chi}^2)$. It derives an analytic exterior solution for the tidally distorted spacetime, revealing spin-induced couplings between electric and magnetic tides and introducing a family of spin-dependent Love numbers, while showing that Kerr BH Love numbers vanish to second order in the spin. Regularity at the horizon and a careful separation of the tidal field from the response yield a vanishing set of Love numbers for a Kerr BH, generalizing the Schwarzschild result and supporting the no-hair paradigm under tidal perturbations. The work also provides a tidally-deformed Kerr BH metric with horizon and geodesic properties, discusses implications for gravitational-wave templates and strong-field tests of general relativity, and outlines a path to compute spinning neutron-star Love numbers via interior solutions and matching in future work.

Abstract

The deformability of a compact object induced by a perturbing tidal field is encoded in the tidal Love numbers, which depend sensibly on the object's internal structure. These numbers are known only for static, spherically-symmetric objects. As a first step to compute the tidal Love numbers of a spinning compact star, here we extend powerful perturbative techniques to compute the exterior geometry of a spinning object distorted by an axisymmetric tidal field to second order in the angular momentum. The spin of the object introduces couplings between electric and magnetic deformations and new classes of induced Love numbers emerge. For example, a spinning object immersed in a quadrupolar, electric tidal field can acquire some induced mass, spin, quadrupole, octupole and hexadecapole moments to second order in the spin. The deformations are encoded in a set of inhomogeneous differential equations which, remarkably, can be solved analytically in vacuum. We discuss certain subtleties in defining the multipole moments of the central object, which are due to the difficulty in separating the tidal field from the linear response of the object in the solution. By extending the standard procedure to identify the linear response in the static case, we prove analytically that the Love numbers of a Kerr black hole remain zero to second order in the spin. As a by-product, we provide the explicit form for a slowly-rotating, tidally-deformed Kerr black hole to quadratic order in the spin, and discuss its geodesic and geometrical properties.

Figures (1)

• Figure 1: The curvature of the intrinsic geometry of a tidally-deformed spinning BH for different values of the spin $\chi$ and of the intensity of the tidal field $\alpha$. We used extreme values of $\chi$ and $\alpha$ in order to magnify the effect of the deformations. The coordinates $(X,Y)$ are related to $(r,\vartheta)$ through $r=\sqrt{X^2+Y^2}$, $Y/X=\tan^{-1}\vartheta$.