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Tidal effects on compact objects provide profound theoretical insights into the structure of the field equations, and are wonderful probes of the equation of state of matter, the nature of black holes and of the underlying theory of gravity. The natural framework for understanding tides is a perturbative scheme. Here, we point out ambiguities in determining tidal response functions within such a framework, which may lead to bias in constraining physical parameters with gravitational-wave observations if the computed quantities are not properly linked to observables. We propose a Canonical Tidal Response Function (CTRF) definition to compare values of tides in theories beyond vacuum General Relativity in a unified manner. As an example, we provide black hole tidal response functions, including both conservative and dissipative pieces, in various theories of gravity. Tidal dissipation Love numbers for black holes are derived here for the first time in most of the non-Einsteinian theories considered in this paper., Comment: 24 pages

The response of astrophysical compact objects to external tidal fields carries valuable information on the nature of these objects, on the equation of state of matter, and on the underlying gravitational theory. In this work, we highlight subtleties in describing relativistic dynamical tidal responses that arise from ambiguities in the decomposition of a perturbed metric into external tidal and induced response pieces. Observables are unambiguous. However in practice, differences arising from implicit assumptions in the definition of tidal deformabilities may lead to a bias in constraining nuclear physics or gravitational theories, if not properly tied to observational data. We propose calibration of a tidal response function for any compact objects in vacuum General Relativity. Within this framework, the dynamical tidal Love numbers of a Schwarzschild black hole in both even and odd sectors vanish at any multipole order. The calibration allows one to define dynamical tidal deformabilities of relativistic stars, such as neutron stars, as the difference from the BH values (zero) under the unified definition in a simple manner. As a straightforward extension of our framework, we compute the next-to-leading dissipative tidal response of Schwarzschild black holes for the first time., Comment: 22 pages

Stationary, asymptotically flat, black hole solutions of the vacuum field equations of General Relativity belong to the Kerr family. But how does one approach this state, dynamically? Linearized fluctuations decay at late times, at fixed spatial position, as a Price power law for generic initial conditions. However, little attention was paid to forced and nonlinear spacetimes, where matter and nonlinearities play a role. We uncover a new, source-driven tail governing waves generated by pointlike matter and nonlinearities, which can dominate over Price's decay., Comment: 13 pages, to appear in Physical Review D Letters

A set of tidal Love numbers quantifies tidal deformation of compact objects and is a detectable imprint in gravitational waves from inspiralling binary systems. The measurement of black hole Love numbers allows to test strong-field gravity. In this paper, we present a parametrized formalism to compute the Love numbers of static and spherically symmetric black hole backgrounds, connecting the underlying equations of a given theory with detectable quantities in gravitational-wave observations in a theory-agnostic way. With this formalism, we compute the Love numbers in several systems. We further classify black hole Love numbers according to whether they vanish, are nonzero, or are ``running'' (scale-dependent), in theories or backgrounds that deviate perturbatively from the GR values. The construction relies on static linear perturbations and scattering theory. Our analytic and numerical results are in excellent agreement. As a side result, we show how to use Chandrasekhar's relations to relate basis of even parity to odd parity., Comment: 25 pages, 4 figures, v2: minor corrections, accepted for publication in PRD

The tidal response of compact objects in an inspiraling binary system is measured by a set of tidal Love and dissipation numbers imprinted in the gravitational waveforms. While a four-dimensional black hole in vacuum within General Relativity has vanishing Love numbers, a black hole in alternative theories of gravity can acquire non-vanishing Love numbers. The dissipation numbers may quantify Planckian corrections at the horizon scale. These properties will allow a test of classical theories of gravity in the strong-field regime with gravitational-wave observation. Since black holes are not in the exact vacuum environment in astrophysical situations, the following question arises: can the environment affect the tidal response? In this paper, we investigate the stability of the tidal response of a Schwarzschild black hole for frequency-dependent tidal-field perturbations against a small modification of the background. Our analysis relies on the scattering theory, which overcomes difficulties in computing the relativistic tidal Love numbers. The tidal Love and dissipation numbers can be understood from the property of sufficiently low-frequency scattering waves. We show that the tidal Love numbers are sensitive to the property of the modification. Therefore, we need careful consideration of the environment around the black hole in assessing the deviation of the underlying theory of gravity from General Relativity with the Love numbers. The modification has less impact on the dissipation numbers, indicating that quantifying the existence of the event horizon with them is not spoiled. We also demonstrate that in a composite system, i.e., a compact object with environmental effects, the Love and dissipation numbers are approximately determined by the sum of the numbers of each component., Comment: 44 pages, 19 figures, v2: minor corrections, v3: minor corrections, accepted for publication in PRD

We study hairy black holes in Einstein-Maxwell-complex scalar theory in four-dimensional asymptotically global anti-de Sitter (AdS) spacetime when the Robin boundary conditions are imposed on the scalar field. This setup is dual to the double trace deformation of strongly interacting field theory on $R \times S^2$ by charged scalar operators. We identify the instability of the Reissner-Nordstr\"{o}m-AdS (RNAdS) black holes under the Robin boundary conditions and construct backreacted geometries branching at the onset of the instability. Also considering associated horizonless geometries called boson stars, we obtain phase diagrams with fairly rich structure in the grand canonical ensemble depending on the boundary condition parameter or the deformation parameter, where phase transition occurs between thermal AdS, RNAdS, charged boson stars, and hairy black holes., Comment: 30 pages, 10 figures v2: version accepted for publication in JHEP

We study the underlying structure of the vanishing of the Love numbers of both Schwarzschild and Kerr black holes in terms of spacetime conformal symmetry in a unified manner for the static spin-$s$ fields. The perturbations can be reduced with the harmonic decomposition to a set of infinite static scalar fields in a two-dimensional anti-de Sitter spacetime~$({\rm AdS}_2)$. In the reduced system, each scalar field is paired with another, implying that all multipole modes of the perturbation can be regarded as symmetric partners, which can be understood from the property of the supersymmetry algebra. The generator of the supersymmetric structure is constructed from a closed conformal Killing vector field of ${\rm AdS}_2$. The associated conserved quantity allows one to show no static response, i.e., vanishing of the Love and dissipation numbers. We also discuss the vanishing Love numbers of the Kerr black hole with the nonzero dissipation numbers for the non-axisymmetric perturbations in terms of a radial constant found in a parallel manner as the axisymmetric field case even though the interpretation for the structure is controversial. The symmetric structure corresponds to the ``ladder'' symmetry in Hui et al. [JCAP 01, no.01, 032 (2022)] although the geometrical origin is different. Our ladder operator includes the generators of hidden symmetries in previous works., Comment: 41 pages, v2: added discussion on the result of the Kerr black hole in terms of the Teukolsky equation, v3: minor corrections, accepted for publication in PRD

We study quasinormal modes (QNMs) of massive Klein-Gordon fields in static Ba\~{n}ados-Teitelboim-Zanelli (BTZ) black holes in terms of ladder operators constructed from spacetime conformal symmetries. Because the BTZ spacetime is locally isometric to the three-dimensional anti-de Sitter spacetime, ladder operators, which map a solution of the massive Klein-Gordon equation into that with different mass squared, can be constructed from spacetime conformal symmetries. In this paper, we apply the ladder operators to the QNMs of the Klein-Gordon equations in the BTZ spacetime. We demonstrate that the ladder operators can change indices of QNM overtones, and all overtone modes can be generated from a fundamental mode when we impose the Dirichlet or Neumann boundary condition at infinity. We also discuss the case with the Robin boundary condition., Comment: 32 pages, 5 figures, v2: minor revisions, v3: minor revisions, accepted for publication in Phys. Rev. D

We study late-time behaviors of massive scalar fields in general static and spherically symmetric extremal black hole spacetimes in arbitrary dimensions. We show the existence of conserved quantities on the extremal black hole horizons for specific mass squared and multipole modes of the scalar fields. Those quantities on the horizon are called the Aretakis constants and are constructed from the higher-order derivatives of the fields. Focusing on the region near the horizon at late times, where is well approximated by the near-horizon geometry, we show that the leading behaviors of the fields are described by power-law tails. The late-time power-law tails lead to the Atetakis instability: blowups of the transverse derivatives of the fields on the horizon. We further argue that the Aretakis constants and instability correspond to respectively constants and blowups of components of covariant derivatives of the fields at the late time in the parallelly propagated null geodesic frame along the horizon. We finally discuss the relation between the Aretakis constants and ladder operators constructed from the approximate spacetime conformal symmetry near the extremal black hole horizons., Comment: 32 pages; v2: minor revisions, accepted for publication in Phys. Rev. D

We discuss dynamics of massive Klein-Gordon fields in two-dimensional Anti-de Sitter spacetimes ($AdS_2$), in particular conserved quantities and non-modal instability on the future Poincar\'e horizon called, respectively, the Aretakis constants and the Aretakis instability. We find out the geometrical meaning of the Aretakis constants and instability in a parallel-transported frame along a null geodesic, i.e., some components of the higher-order covariant derivatives of the field in the parallel-transported frame are constant or unbounded at the late time, respectively. Because $AdS_2$ is maximally symmetric, any null hypersurfaces have the same geometrical properties. Thus, if we prepare parallel-transported frames along any null hypersurfaces, we can show that the same instability emerges not only on the future Poincar\'e horizon but also on any null hypersurfaces. This implies that the Aretakis instability in $AdS_2$ is the result of singular behaviors of the higher-order covariant derivatives of the fields on the whole $AdS$ infinity, rather than a blow-up on a specific null hypersurface. It is also discussed that the Aretakis constants and instability are related to the conformal Killing tensors. We further explicitly demonstrate that the Aretakis constants can be derived from ladder operators constructed from the spacetime conformal symmetry., Comment: 24 pages, 1 figure, v2: minor revisions, v3: minor revisions, accepted for publication in Phys. Rev. D

We analytically and numerically study quasinormal frequencies (QNFs) of neutral and charged scalar fields in the charged anti-de Sitter (AdS) black holes and discuss the stability of the black holes in terms of the QNFs. We focus on the range of the mass squared $\mu^2$ of the scalar fields for which the Robin boundary condition parametrised by $\zeta$ applies at the conformal infinity. We find that if the black hole of radius $r_{+}$ and charge $Q$ is much smaller than the AdS length $\ell$, the instability of the charged scalar field can be understood in terms of superradiance in the reflective boundary condition. Noting that the s-wave normal frequency in the AdS spacetime is a decreasing function of $\zeta$, we find that if $|eQ|\ell/r_{+}$ is greater than $(3+\sqrt{9+4\mu^2\ell^2})/2$, where $e$ is the charge of the scalar field, the black hole is superradiantly unstable irrespectively of $\zeta$. On the other hand, if $|eQ|\ell/r_{+}$ is equal to or smaller than this critical value, the stability crucially depends on $\zeta$ and there appears a purely oscillating mode at the onset of the instability. We argue that as a result of the superradiant instability, the scalar field gains charge from the black hole and energy from its ambient electric field, while the black hole gives charge to the scalar field and gains energy from the scalar field but decreases its asymptotic mass parameter., Comment: 52 pages, 27 figures, v2: changes of the title and abstract, revision of introduction, conclusion, and errors in some results, addition of figures associated with numerical results for other mass squared cases and an analytic investigation, v3: minor revisions, published in CQG