Tidal Response, Love Numbers, Absorption, and Heating

A black hole responds to an external tide in two qualitatively different ways. Its conservative field can contain an induced multipole, measured by a Love coefficient, while its horizon absorbs energy and angular momentum, measured by the imaginary part of a retarded response. The first effect vanishes for the standard static four-dimensional vacuum black-hole problem; the second does not. This chapter builds the applied/induced split, proves the scoped static result, derives horizon heating, and matches both sectors to a worldline effective theory.

Helpful background. Black-Hole Geometry and Mechanics supplies regular Schwarzschild and Kerr horizons; PDEs and Green Functions for Gravitation supplies radial boundary-value problems and retarded kernels; Formulations of General Relativity supplies gauge-invariant perturbations; and Relativistic Two-Body Dynamics supplies STF tidal and multipole notation.

The tidal response problem

Throughout the chapter, $G$ is explicit and Fourier modes vary as $e^{-i\omega t+im\phi}$. In a buffer region $GM\ll r\ll\mathcal R$, the growing radial solution is matched to an applied electric or magnetic STF tidal moment, and the falling solution is matched to an induced mass or current multipole. Our electric normalization is

$$Q_L(\omega)=-\lambda^E_\ell(\omega)\mathcal E_L(\omega),$$

with an analogous magnetic relation after the current-moment convention is fixed. A scalar coefficient describes a spherical body or a specified diagonal angular sector; Kerr generally requires a response map. The response is static only after a declared $\omega\to0$ limit; otherwise it is dynamic. Time-even real local terms are conservative in a nonrotating scalar sector, while the absorptive spectral part is dissipative. For rotating tensor response, distinguish the symmetric conservative map from an antisymmetric part on real STF components. The baseline problem is linear spin-2 response in four spacetime dimensions on a Schwarzschild or Kerr vacuum background.

Every calculation states its gauge or gauge-invariant master field, its buffer matching map, its response normalization, and its subtraction scheme. Horizon flux is oriented inward. At leading quadratic perturbative order, positive Schwarzschild absorption increases $M$ in the averaged or settled-state physical-process interpretation. For Kerr the nonnegative averaged combination is

$$\langle\dot M_H\rangle-\Omega_H\langle\dot J_H\rangle =\frac{\kappa}{8\pi G}\langle\dot A_H\rangle,$$

not necessarily $\langle\dot M_H\rangle$ by itself. Arbitrary finite horizon cuts also have an endpoint-expansion term; a positive physical shear flux alone does not define their instantaneous mass evolution. The near-zone, low-frequency, asymptotic, and extremal limits are never interchanged without a new argument. These conventions prevent the static conservative zero from being generalized to absorption or to a dynamic coefficient.

Route through the six pages

The pages are ordered so that each new coefficient has already been defined before it is calculated.

Step	Page	Main task
1	External Tidal Fields and Induced Multipoles	Recover the Newtonian STF quadrupole and verify one curvature measurement in two coordinates.
2	Relativistic Love Numbers and Gauge Invariance	Carry that measurement through response, action and compensated contact-basis descriptions.
3	Static Love Numbers of Schwarzschild and Kerr in Four-Dimensional Vacuum GR	Exclude singular Schwarzschild invariant branches; derive the Kerr Gamma connection and integer-multipole conservative zero.
4	Horizon Absorption and Tidal Heating	Match normalized electric shear, physical tidal work and plane-wave absorption; retain finite-cut qualifications.
5	Frequency-Dependent Response and Dissipation	Check a retarded kernel, an explicit subtracted dispersion relation and the low-frequency physical response.
6	Effective-Field-Theory Matching of Black-Hole Response	Reproduce the static field amplitude and normalized absorption, then distinguish running/contact terms from the complete observable.

The static Schwarzschild construction uses horizon-regular light-cone gauge and residual-gauge invariant combinations (Binnington and Poisson 2009, Eqs. (3.9)–(3.22), pp. 9–11 (PDF)). The dissipative route uses the horizon shear and first law (Poisson 2004, §§ IV.B–IV.C and VIII.E–VIII.G, pp. 11–12 and 24–25 (PDF)). The effective description then matches the same absorption to a retarded worldline correlator (Goldberger and Rothstein 2006, § II, eqs. (19)–(27), pp. 11–12 (PDF)).

The physical quadrupole dictionary

Throughout the worked quadrupolar examples, $Q_{ij}$ is the physical STF mass moment and $J_{ij}=S_{ij}$ is Poisson’s physical current moment. The following normalizations belong together:

Quantity	Electric sector	Magnetic sector
Induced moment	$Q=-\lambda_E\mathcal E$	$J=-\lambda_B\mathcal B$
Physical interaction	$-Q:\mathcal E/2$	$-2J:\mathcal B/3$
Conservative local term	$\lambda_E\mathcal E:\mathcal E/4$	$\lambda_B\mathcal B:\mathcal B/3$
Unit-coupled EFT operator	$Q^E=Q/2$	$Q^B=2J/3$
Real-monochromatic averaged power	$\omega\operatorname{Im}\lambda_E\,\mathcal E^0:\overline{\mathcal E^0}/4$	$\omega\operatorname{Im}\lambda_B\,\mathcal B^0:\overline{\mathcal B^0}/3$

The action row is conservative and local; absorption needs retarded degrees of freedom. For a slowly driven Schwarzschild quadrupole, $\operatorname{Im}\lambda_E=32(GM)^6\omega/(45G)$ and $\operatorname{Im}\lambda_B=8(GM)^6\omega/(15G)$ at leading order. Their different weights yield equal electric and magnetic coefficients in the heating formula. Confusing a unit-coupled operator with a physical mass moment produces a factor-of-two error.

What the static zero does and does not say

Under asymptotic flatness, four-dimensional vacuum Einstein dynamics, linear stationary spin-2 tides, future-horizon regularity, and a declared matched response scheme, the conservative Schwarzschild and Kerr static Love coefficients vanish in the declared prescription. For Schwarzschild the proof rejects the response branch through its singular gauge-invariant metric combinations. For Kerr the calculation instead retains the full ingoing hypergeometric solution, separates its applied and response branches at noninteger radial separation parameter, and takes the physical integer limit. The resulting response is purely imaginary and $m$-odd, so its symmetric conservative part vanishes; its entire response branch need not vanish. The electric dimensionless conversion is

$$r_s^{2\ell+1}\operatorname{Re}\mathcal K^{(2)}_{\ell m} =2R_0^{2\ell+1}k^{E,\mathrm{Love}}_{\ell m}, \qquad r_s=2GM.$$

The full complex BP-normalized response is denoted $\widehat k$ on the static page; it is distinct from the conservative $k^{E,\mathrm{Love}}$. This prescription and the conversion are derived there, following Charalambous, Dubovsky, and Ivanov 2021, Eqs. (2.12)–(2.14) and (6.1)–(6.5), pp. 7–8 and 39 (PDF). A Kerr mode stationary at infinity can still have $\omega-m\Omega_H\neq0$ at the horizon, so an imaginary dissipative response may survive. Zero asymptotic-frequency work does not by itself imply zero torque or zero area growth.

The conclusion does not cover matter, stellar interiors, higher dimensions, charged or hairy solutions, a cosmological constant, modified gravity, nonlinear tides, or every dynamic local coefficient. Nor does it remove scheme dependence from the finite analytic part of a frequency expansion. Recent amplitude and MST-to-worldline calculations find running dynamical operators and emphasize the renormalization condition needed to quote their finite parts (Saketh, Zhou, and Ivanov 2024, §§ 4.3–4.5 and 6).

Orbital implementation remains with Relativistic Two-Body Dynamics, material response remains with Relativistic Matter and Compact Objects, and waveform inference lies downstream. Here the endpoint is a black-hole response coefficient or flux with its assumptions and normalization attached.

Chapter review

A four-dimensional Schwarzschild calculation reports three facts:

1. the regular static $\ell=2$ solution has no matched falling branch;
2. at small nonzero frequency, $\operatorname{Im}\lambda_2(\omega)=\gamma\omega+O(\omega^3)$ with $\gamma>0$; and
3. a real coefficient multiplying $\omega^2$ shifts after a finite subtraction.

Classify each fact as static or dynamic, conservative or dissipative, and state which parts can be called invariant physical output.

Solution

Fact 1 is static and conservative. With the horizon condition, gauge- invariant matching, and response convention fixed, it states the four-dimensional vacuum black-hole Love number $k^E_2=0$.

Fact 2 is dynamic and dissipative. For a real quadrupolar drive $\mathcal E(t)=\operatorname{Re}(\mathcal E^0e^{-i\omega t})$, $\gamma>0$ gives positive averaged power

$$\langle P_E\rangle =\frac{\omega}{4}\operatorname{Im}\lambda_E\, \mathcal E^0:\overline{\mathcal E^0}.$$

Its match to the horizon flux or absorption probability is physical.

Fact 3 is dynamic and conservative but local. The finite coefficient alone is scheme dependent only when the rest of the matched description changes consistently. An uncompensated change changes the physical response. Its running, together with the compensating logarithm, and the complete scattering observable are invariant after consistent matching. None of the three facts implies anything universal about a material star, higher-dimensional black hole, modified-gravity solution, or an unnormalized bulk-master coefficient.

References

• Binnington, Taylor, and Eric Poisson. 2009. “Relativistic Theory of Tidal Love Numbers.” Physical Review D 80, 084018. DOI.
• Charalambous, Panagiotis, Sergei Dubovsky, and Mikhail M. Ivanov. 2021. “On the Vanishing of Love Numbers for Kerr Black Holes.” Journal of High Energy Physics 05, 038. DOI.
• Goldberger, Walter D., and Ira Z. Rothstein. 2006. “Dissipative Effects in the Worldline Approach to Black Hole Dynamics.” Physical Review D 73, 104030. DOI.
• Poisson, Eric. 2004. “Absorption of Mass and Angular Momentum by a Black Hole: Time-Domain Formalisms for Gravitational Perturbations, and the Small-Hole/Slow-Motion Approximation.” Physical Review D 70, 084044. DOI.
• Saketh, M. V. S., Zihan Zhou, and Mikhail M. Ivanov. 2024. “Dynamical Tidal Response of Kerr Black Holes from Scattering Amplitudes.” Physical Review D 109, 064058. DOI.