Tidal Love Numbers of Kerr Black Holes. (arXiv:2010.15795v1 [gr-qc])
<a href="http://arxiv.org/find/gr-qc/1/au:+Tiec_A/0/1/0/all/0/1">Alexandre Le Tiec</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Casals_M/0/1/0/all/0/1">Marc Casals</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Franzin_E/0/1/0/all/0/1">Edgardo Franzin</a>
The open question of whether a Kerr black hole can become tidally deformed or
not has profound implications for fundamental physics and gravitational-wave
astronomy. We consider a Kerr black hole embedded in a weak and slowly varying,
but otherwise arbitrary, multipolar tidal environment. By solving the static
Teukolsky equation for the gauge-invariant Weyl scalar $psi_0$, and by
reconstructing the corresponding metric perturbation in an ingoing radiation
gauge, for a general harmonic index $ell$, we compute the linear response of a
Kerr black hole to the tidal field. This linear response vanishes identically
for a Schwarzschild black hole and for an axisymmetric perturbation of a
spinning black hole. For a nonaxisymmetric perturbation of a spinning black
hole, however, the linear response does not vanish, and it contributes to the
Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an
application, we compute explicitly the rotational black hole tidal Love numbers
that couple the induced quadrupole moments to the quadrupolar tidal fields, to
linear order in the black hole spin, and we introduce the corresponding notion
of tidal Love tensor. Finally, we show that those induced quadrupole moments
are closely related to the well-known physical phenomenon of tidal torquing of
a spinning body interacting with a tidal gravitational environment.
The open question of whether a Kerr black hole can become tidally deformed or
not has profound implications for fundamental physics and gravitational-wave
astronomy. We consider a Kerr black hole embedded in a weak and slowly varying,
but otherwise arbitrary, multipolar tidal environment. By solving the static
Teukolsky equation for the gauge-invariant Weyl scalar $psi_0$, and by
reconstructing the corresponding metric perturbation in an ingoing radiation
gauge, for a general harmonic index $ell$, we compute the linear response of a
Kerr black hole to the tidal field. This linear response vanishes identically
for a Schwarzschild black hole and for an axisymmetric perturbation of a
spinning black hole. For a nonaxisymmetric perturbation of a spinning black
hole, however, the linear response does not vanish, and it contributes to the
Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an
application, we compute explicitly the rotational black hole tidal Love numbers
that couple the induced quadrupole moments to the quadrupolar tidal fields, to
linear order in the black hole spin, and we introduce the corresponding notion
of tidal Love tensor. Finally, we show that those induced quadrupole moments
are closely related to the well-known physical phenomenon of tidal torquing of
a spinning body interacting with a tidal gravitational environment.
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