Frequency-Dependent Response and Dissipation

One retarded response function contains both tidal deformation and horizon absorption. In a nonrotating scalar response channel, its real, time-even part describes conservative response and its imaginary, time-odd part describes dissipation, with resonant poles and nonanalytic terms retaining information that a static coefficient cannot. The separation is useful only after the source, induced multipole, gauge, subtraction convention, and order of limits are fixed. This page constructs that object for linear spin-2 response of a four-dimensional black hole and does not insert it into an orbital or waveform model.

Required background. Presymplectic Potentials and Currents supplies the causal boundary pairing used to normalize response.

Helpful background. Black-Hole Geometry and Mechanics supplies regular horizons and Kerr generators; PDEs and Green Functions for Gravitation supplies retarded kernels, analyticity, and dispersion relations; Formulations of General Relativity supplies gauge-invariant perturbations; Relativistic Two-Body Dynamics supplies STF source and multipole conventions; and Black-Hole Perturbation Theory supplies the common Fourier convention.

The retarded tidal kernel

For an electric multipole of a spherical body, or a specified diagonal angular channel, define

$$Q_L(t) =-\int_{-\infty}^{t}G^E_{R,\ell}(t-t')\mathcal E_L(t')\,dt', \qquad Q_L(\omega)=-\lambda^E_\ell(\omega)\mathcal E_L(\omega).$$

Causality gives $G_R(t)=0$ for $t\lt0$. If the causal kernel is tempered (allowing contact distributions at zero) and has no exponentially growing late-time response, its Fourier–Laplace transform

$$\lambda_\ell(\omega)=\int_0^\infty e^{i\omega t}G_{R,\ell}(t)\,dt$$

is analytic for $\operatorname{Im}\omega\gt0$, with contact terms understood distributionally. Causality alone is not sufficient: a kernel growing as $e^{\Gamma t}$ initially gives convergence only for $\operatorname{Im}\omega\gt\Gamma$. Reality of the time-domain fields gives

$$\lambda_\ell(-\omega)=\lambda_\ell(\omega)^*$$

on the real axis for a nonrotating background. Consequently the real part is even and the imaginary part is odd. For the physical electric quadrupole, let $\mathcal E_{ij}(t)=\operatorname{Re}(\mathcal E^0_{ij}e^{-i\omega t})$. The interaction $-Q:\mathcal E/2$ fixes the averaged work:

$$\begin{aligned} \langle P_E\rangle &=\frac12\langle Q^{ij}\dot{\mathcal E}_{ij}\rangle\\ &=\frac{\omega}{4}\operatorname{Im}\lambda_E(\omega) \mathcal E^0_{ij}\overline{\mathcal E^{0ij}}\geq0. \end{aligned}$$

This inequality applies to a passive Schwarzschild sector with $\omega\gt0$. The factor $1/4$ combines the physical coupling’s $1/2$ with the real-monochromatic average’s $1/2$. For the physical current quadrupole $J=-\lambda_B\mathcal B$, the coupling is $-2J:\mathcal B/3$ and

$$\langle P_B\rangle =\frac23\langle J^{ij}\dot{\mathcal B}_{ij}\rangle =\frac{\omega}{3}\operatorname{Im}\lambda_B(\omega) \mathcal B^0_{ij}\overline{\mathcal B^{0ij}}.$$

These are quadrupole weights, not a formula for arbitrary $\ell$. They agree with Poisson 2004, Eq. (8.43), p. 25 (PDF). For Kerr the full tensor response must be retained; reality couples the appropriate $m$ and $-m$ sectors. Positivity applies to the horizon-generator balance and is organized by $k_H=\omega-m\Omega_H$, so asymptotic-time energy absorption can change sign in a superradiant mode.

The declaration is therefore dynamic, with conservative and dissipative pieces both retained; the spacetime dimension is four and the perturbing field has spin two. Applied and induced quantities are the normalized STF tidal and multipole moments. The bulk calculation uses a horizon-ingoing gauge-invariant master or Teukolsky field; buffer matching maps it to $\lambda_\ell$. Polynomial contact terms are defined at a subtraction scale $\mu$. We first impose both radial boundary conditions and match the overlap solution, then expand at small $GM\omega$ at fixed $(M,a,\ell,m)$; neither the far-zone nor extremal limit is taken prematurely.

Dispersion, poles, and absorption

A stable retarded black-hole response has its quasinormal poles below the real axis. Near a pole, the source can excite a large transient response, but a quasinormal pole is not a normal-mode degree of freedom with a real eigenfrequency. Along the real axis, the same ingoing condition that fixes the pole prescription fixes the absorptive spectral density. Chia obtains a low-frequency complex response by matching the ingoing Teukolsky solution to the growing and falling tidal branches (Chia 2021, §§ III–VI, especially eqs. (5)–(16)).

If one subtraction is sufficient, causality yields the schematic relation

$$\operatorname{Re}\lambda(\omega)-\operatorname{Re}\lambda(0) =\frac{2\omega^2}{\pi}\,\mathcal P \int_0^\infty \frac{\operatorname{Im}\lambda(\omega')} {\omega'(\omega'^2-\omega^2)}\,d\omega'.$$

Additional high-frequency growth requires more subtractions. Their constants are local conservative data, not predictions of the absorptive spectrum alone. This is why a statement such as “absorption determines the Love number” needs both ultraviolet behavior and a subtraction condition.

The displayed relation assumes adequate behavior at infinity after the subtraction, convergence at the lower endpoint, and the principal-value prescription at $\omega'=\omega$. One cannot insert only the low-frequency approximation to $\operatorname{Im}\lambda$ and extend it to infinity. The oscillator exercise below provides a function for which the contour and the subtraction can actually be checked.

Checking the Schwarzschild coefficient

The normalized ingoing gravitational calculation, translated to physical STF moments, gives at leading slow-driving order

$$\begin{aligned} Q_{ij}&=\frac{32(GM)^6}{45G}\dot{\mathcal E}_{ij},& J_{ij}&=\frac{8(GM)^6}{15G}\dot{\mathcal B}_{ij},\\ \operatorname{Im}\lambda_E &=\frac{32(GM)^6}{45G}\omega,& \operatorname{Im}\lambda_B &=\frac{8(GM)^6}{15G}\omega. \end{aligned}$$

The sign follows from $\dot{\mathcal E}=-i\omega\mathcal E$ and $Q=-\lambda_E\mathcal E$, not from changing the Fourier prescription. Substituting these moments into the two work formulas gives

$$\langle P_E+P_B\rangle =\frac{16(GM)^6}{45G} \left\langle\dot{\mathcal E}_{ij}\dot{\mathcal E}^{ij} +\dot{\mathcal B}_{ij}\dot{\mathcal B}^{ij}\right\rangle.$$

This is the slow-driving horizon heating result of Poisson 2004, Eqs. (8.38), (8.43)–(8.45), p. 25 (PDF). It is quadratic in the tidal amplitude and requires the averaging or settled-endpoint qualifications on the heating page. The EFT calculation independently obtains the same coefficients from the gravitational absorption cross section with its operator normalization stated. Neither comparison determines every real dynamic contact coefficient.

Local coefficients and nonlocal response

At low frequency one may organize

$$\lambda_\ell(\omega;\mu) =c_0(\mu)+c_2(\mu)\omega^2+\cdots +\lambda_{\mathrm{nonlocal}}(\omega;\mu).$$

The analytic even powers correspond to local worldline operators and can move under finite counterterm changes. Dissipation and temporal nonlocality are different classifications. An analytic retarded term such as $i\gamma\omega$ is local in a derivative expansion, but it is odd under time reversal and cannot be generated by a single-copy conservative worldline action; an open-system or doubled-field description supplies its absorptive meaning. Logarithms with a specified branch, branch cuts, and poles instead encode memory and are genuinely nonlocal in time. The scale dependence of a logarithm is canceled by running of a local coefficient, so a beta function and the complete scattering amplitude can be invariant even when the finite $c_2$ is not.

This separation has been sharpened since the original static calculations. Scattering-amplitude matching finds nonzero frequency-derivative operators, classical running, and distinct tail and short-distance logarithms (Saketh, Zhou, and Ivanov 2024, §§ 2.1–2.4 and 4.3–4.5). A later Schwarzschild MST-to-worldline calculation shows explicitly that the finite renormalized dynamical coefficients depend on the subtraction scheme and the initial condition for the flow, while a complete observable can be computed without assigning universal meaning to that finite split (Kobayashi et al. 2026, §§ 3.2.1–3.2.3). Thus zero static Love numbers do not force $c_2$, logarithmic running, absorption, or the entire nonlocal response to vanish.

Exercises

Consider

$$\lambda(\omega) =\frac{\lambda_0\Omega_0^2} {\Omega_0^2-\omega^2-i\gamma\omega} +c_2\omega^2, \qquad \lambda_0,\Omega_0,\gamma>0,$$

with real $c_2$. Check causality, the static limit, passivity, and the effect of changing $c_2$.

Solution

The poles are

$$\omega_\pm=-\frac{i\gamma}{2} \pm\sqrt{\Omega_0^2-\frac{\gamma^2}{4}},$$

Both poles lie in the lower half-plane, including the overdamped case. To establish the retarded interpretation rather than infer it from pole locations alone, set $\Omega_d^2=\Omega_0^2-\gamma^2/4$. The inverse kernel is

$$G_R(t)=\theta(t)\lambda_0\Omega_0^2 e^{-\gamma t/2}\frac{\sin(\Omega_d t)}{\Omega_d} -c_2\delta''(t).$$

Use its continuous $\Omega_d=0$ limit and the corresponding hyperbolic sine for imaginary $\Omega_d$. The kernel vanishes for negative time and its noncontact part decays. Its transform, with this causal prescription, is exactly the stated rational response plus $c_2\omega^2$. At zero frequency, $\lambda(0)=\lambda_0$. For real positive frequency,

$$\operatorname{Im}\lambda(\omega) =\frac{\lambda_0\Omega_0^2\gamma\omega} {(\Omega_0^2-\omega^2)^2+\gamma^2\omega^2}>0,$$

which makes the declared absorbed power positive. Changing $c_2$ shifts only the analytic conservative part; it leaves the poles and imaginary part unchanged. But changing $c_2$ alone changes $Q$ for a prescribed tide and is a physical change of this model. It is a scheme change only when a compensating term in the rest of the matched observable is changed too, as on the EFT page.

There is also an explicit dispersion check. Remove the contact term and write $\chi(\omega)=\lambda(\omega)-c_2\omega^2$. It decays as $1/\omega^2$ in the upper half-plane; closing the Cauchy contour there and pairing positive and negative real frequencies gives

$$\frac{2\omega^2}{\pi}\mathcal P\int_0^\infty \frac{\lambda_0\Omega_0^2\gamma} {[(\Omega_0^2-\omega'^2)^2+\gamma^2\omega'^2] (\omega'^2-\omega^2)}\,d\omega' =\lambda_0\left[ \frac{\Omega_0^2(\Omega_0^2-\omega^2)} {(\Omega_0^2-\omega^2)^2+\gamma^2\omega^2}-1\right].$$

For a direct integral verification at $\omega\gt0$, put $D(v)=(\Omega_0^2-v^2)^2+\gamma^2v^2$ and $F(v)=\lambda_0\Omega_0^2\gamma/D(v)$. The constant principal value $\mathcal P\int_0^\infty dv/(v^2-\omega^2)=0$, so subtracting $F(\omega)$ in the numerator removes the singularity. The integrand becomes

$$\frac{F(v)-F(\omega)}{v^2-\omega^2} =\frac{\lambda_0\Omega_0^2\gamma (2\Omega_0^2-\gamma^2-v^2-\omega^2)} {D(v)D(\omega)}.$$

The two remaining integrals follow from inversion $v\mapsto\Omega_0^2/v$ and the derivative of $\arctan[(v^2-\Omega_0^2)/(\gamma v)]$:

$$\int_0^\infty\frac{dv}{D(v)}=\frac{\pi}{2\gamma\Omega_0^2}, \qquad \int_0^\infty\frac{v^2\,dv}{D(v)}=\frac{\pi}{2\gamma}.$$

Substitution gives the displayed right side, $\operatorname{Re}\chi(\omega)-\chi(0)$, with $\omega=0$ obtained by continuity. For the full $\lambda$ it must be supplemented by $c_2\omega^2$: the large contour does not permit that polynomial to be discarded. This gives a concrete example of an absorptive spectrum fixing a dispersive part but not an arbitrary subtraction polynomial.

The operator interpretation and the matching of such coefficients are the subject of effective-field-theory matching.

References

• Chia, Horng Sheng. 2021. “Tidal Deformation and Dissipation of Rotating Black Holes.” Physical Review D 104, 024013. DOI.
• Kobayashi, Hajime, Shinji Mukohyama, Naritaka Oshita, Kazufumi Takahashi, and Vicharit Yingcharoenrat. 2026. “Dynamical Tidal Response of Nonrotating Black Holes: Connecting the Mano–Suzuki–Takasugi Formalism and Worldline EFT.” Physical Review D 113, 084011. 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.