Effective-Field-Theory Matching of Black-Hole Response

A worldline effective theory replaces the black hole’s short-distance region by local operators and, for absorption, retarded worldline correlations. Its coefficients are not guessed from dimensional analysis: they are determined by matching a gauge-invariant perturbative observable—typically a scattering amplitude, phase shift, or absorption probability—in the full and effective descriptions. This page gives that matching procedure for four-dimensional spin-2 response while keeping finite contact terms distinct from invariant observables.

Required background. Presymplectic Potentials and Currents supplies the variational definition of induced moments and boundary fluxes.

Helpful background. Black-Hole Geometry and Mechanics supplies the horizon scale $GM$ and regular boundary data; PDEs and Green Functions for Gravitation supplies retarded correlators and scattering solutions; Formulations of General Relativity supplies invariant curvature variables; Relativistic Two-Body Dynamics supplies the worldline and STF operator language; and Black-Hole Perturbation Theory supplies the full-theory amplitudes to be matched.

Worldline operators for conservative response

Let $u^a$ be the worldline velocity and $e^a_i$ a comoving orthonormal triad. The electric and magnetic Weyl tensors are $\mathcal E_{ij}=C_{a b c d}e^a_i u^b e^c_j u^d$ and the corresponding dual $\mathcal B_{ij}$. For an isotropic, nonspinning response, use the physical mass and current quadrupoles $Q=-\lambda_E\mathcal E$ and $J=-\lambda_B\mathcal B$ defined on External Tidal Fields. Their conservative static finite-size action is

$$S_{\mathrm{resp}} =\int d\tau \left[ \frac{\lambda_E(\mu)}4\mathcal E_{ij}\mathcal E^{ij} +\frac{\lambda_B(\mu)}3\mathcal B_{ij}\mathcal B^{ij} \right]+\cdots .$$

Here $\mu$ is the subtraction frequency scale, not the geometric mass. The functional derivatives are

$$Q_{ij}=-2\frac{\delta S_{\mathrm{resp}}}{\delta\mathcal E^{ij}}, \qquad J_{ij}=-\frac32\frac{\delta S_{\mathrm{resp}}}{\delta\mathcal B^{ij}},$$

with the derivatives taken per unit proper time. The factors follow from the physical interactions $-Q:\mathcal E/2$ and $-2J:\mathcal B/3$. For example, for a nonzero real constant $\lambda_E$, eliminate the auxiliary static quadrupole from

$$L_{\mathrm{aux}}=-\frac{Q:Q}{4\lambda_E}-\frac12Q:\mathcal E.$$

Varying $Q$ gives $Q=-\lambda_E\mathcal E$ and substitution gives $L_{\mathrm{eff}}=\lambda_E\mathcal E:\mathcal E/4$. This elementary variation prevents a factor-of-two error between a physical mass moment and a unit-coupled worldline operator. The eliminated action has a smooth $\lambda_E=0$ limit even though this particular auxiliary representation uses $1/\lambda_E$.

Operators with proper-time derivatives, such as $D\mathcal E_{ij}/d\tau$, encode analytic dynamical response. Spin permits additional contractions with the spin vector and mixing between multipoles; then the scalar coefficients are replaced by response tensors.

The declaration is: static coefficients are the zero-derivative local terms; dynamic coefficients multiply derivatives or nonlocal kernels. Time-even local terms are conservative, whereas absorption is dissipative. The theory is four-dimensional, with a spin-2 external field and either a nonspinning or spinning black-hole worldline. The applied field is the Weyl tensor in the declared comoving triad and the induced moment is its functional derivative. The full calculation uses a gauge-invariant master field or on-shell amplitude; the EFT uses a fixed operator basis and subtraction scheme at $\mu$. Response normalization is the STF relation above. Matching is done at $GM\omega\ll1$ for fixed multipole and fixed subextremal spin before a static or extremal limit is taken.

Dissipation requires retarded degrees of freedom

A real local action cannot by itself reproduce irreversible absorption. Introduce composite worldline operators $Q^E_{ij}$ and $Q^B_{ij}$ through

$$S_{\mathrm{int}} =-\int d\tau \left(Q^E_{ij}\mathcal E^{ij}+Q^B_{ij}\mathcal B^{ij}\right).$$

Their retarded two-point functions supply the imaginary response. Matching their spectral density to the long-wavelength black-hole absorption cross section fixes the dissipative coefficient without specifying microscopic horizon degrees of freedom. Goldberger and Rothstein carry out precisely this calculation for a Schwarzschild hole: their quadrupole correlator, absorption cross section, and tidal power are related in Goldberger and Rothstein 2006, Eqs. (19)–(27), pp. 11–12 (PDF).

The dictionary to physical moments is

$$Q^E_{ij}=\frac12Q_{ij},\qquad Q^B_{ij}=\frac23J_{ij}.$$

Thus if $\langle Q^E\rangle=-\alpha\mathcal E$ and $\langle Q^B\rangle=-\beta\mathcal B$, then $\alpha=\lambda_E/2$ and $\beta=2\lambda_B/3$. The source paper’s time-ordered correlator contains $|\omega|$ in its imaginary part. For positive frequency it fixes the same absorptive coefficient as the retarded response, but the retarded imaginary part extends oddly to negative frequency. These two prescriptions are not interchangeable functions on the whole real axis.

The horizon-flux orientation is inherited from the full theory: positive Schwarzschild absorption increases $M$, while Kerr modes are tested using $\dot M_H-\Omega_H\dot J_H\geq0$. A local counterterm may shift the real analytic response; it cannot change an absorption probability or this area balance.

Matching full perturbation theory to the EFT

A static quadrupolar field amplitude

The matching can be performed explicitly in the leading weak-field buffer. Use the positive Newtonian potential $U$, subtract the minimal monopole and solve only for the linear response to a fixed external STF tide. At this order the static action is

$$S_U=\frac1{8\pi G}\int dt\,d^3x\,U\nabla^2U +\frac{\lambda_E}{4}\int dt\, [\partial_i\partial_j U\,\partial_i\partial_j U]_{\mathbf x=0}.$$

Treat the last term perturbatively, evaluating its applied field at the worldline with the body’s self-field removed. Varying with compact test functions and integrating by parts twice gives

$$\nabla^2 U_{\mathrm{resp}} =2\pi G\lambda_E\mathcal E_{ij}\partial_i\partial_j\delta^3(\mathbf x).$$

Since $\nabla^2(1/r)=-4\pi\delta^3(\mathbf x)$,

$$U_{\mathrm{resp}} =-\frac{G\lambda_E}{2}\mathcal E_{ij}\partial_i\partial_j\frac1r =-\frac{3G\lambda_E}{2r^3}\mathcal E_{nn}.$$

This is the same field amplitude as the full buffer multipole expansion $3GQ_{nn}/(2r^3)$ with $Q=-\lambda_E\mathcal E$. Equivalently, in the spatial Fourier convention $\widetilde U(\mathbf k)=\int e^{-i\mathbf k\cdot\mathbf x}U(\mathbf x)d^3x$, both descriptions give

$$\widetilde U_{\mathrm{resp}}(\mathbf k) =2\pi G\lambda_E\frac{\mathcal E_{ij}k_i k_j}{|\mathbf k|^2}, \qquad \mathbf k\ne0.$$

The common local measurement is $C_{\hat0\hat r\hat0\hat r}^{\mathrm{pert}} =\mathcal E_{nn}(1+18G\lambda_E/r^5)$, at leading buffer order. This is a static field-amplitude match, not a calculation of a complete finite-frequency scattering phase. The gravitational static action and curvature-matching method are Charalambous, Dubovsky, and Ivanov 2021, Eqs. (5.16)–(5.23), pp. 36–37 (PDF).

A normalized absorptive match

For a Schwarzschild hole, a positive-frequency gravitational plane wave with unit-normalized polarization has the leading full-theory absorption cross section

$$\sigma_{\mathrm{abs},p} =\frac{4\pi}{45}(2GM)^6\omega^4.$$

Here $p$ labels either polarization, and $GM\omega\ll1$. This is an input from the ingoing gravitational scattering problem, not the scalar absorption cross section. In the unit-coupled operator normalization above, the EFT gives

$$\sigma_{\mathrm{abs},p} =\frac{\omega^3}{2m_{\mathrm{Pl}}^2}\operatorname{Im}F(\omega), \qquad m_{\mathrm{Pl}}^2=\frac1{32\pi G}.$$

Matching the two yields, for $\omega\gt0$,

$$\begin{aligned} \operatorname{Im}F &=\operatorname{Im}\alpha =\operatorname{Im}\beta =\frac{16(GM)^6}{45G}\omega,\\ \operatorname{Im}\lambda_E&=\frac{32(GM)^6}{45G}\omega,\\ \operatorname{Im}\lambda_B&=\frac{8(GM)^6}{15G}\omega. \end{aligned}$$

These factors follow from Goldberger and Rothstein 2006, Eqs. (23)–(25), p. 12 (PDF). The electric and physical-current coefficients differ, even though their unit-coupled operator correlators agree. The heating calculation checks that they give exactly incident intensity times this cross section. This is a normalized absorptive observable match; it does not determine all conservative subtraction constants from absorption alone.

Beyond the examples

A controlled calculation follows four steps.

1. Choose a full-theory observable with normalized asymptotic states, such as a partial-wave phase shift and inelasticity, and solve with the future- horizon ingoing condition.
2. Compute the same observable in the point-particle theory, including the minimal mass/spin terms, tidal operators, propagation tails, and the same infrared convention.
3. Expand both results in $GM\omega$ and match analytic real terms, logarithms, and imaginary terms separately. Never assign a long-range tail logarithm to a short-distance Love coefficient.
4. Vary gauge, field variables, and subtraction scale. Wilson coefficients may transform, but the reconstructed phase shift and absorption must stay fixed.

For Kerr, scattering-amplitude matching reveals derivative tidal operators, spin-dependent structures, dissipation numbers, and classical running (Saketh, Zhou, and Ivanov 2024, §§ 2–5, especially eqs. (2.6)–(2.23) and §§ 4.3–4.5). The static four-dimensional vacuum coefficients remain zero in the declared scheme, while some dynamic local coefficients and their beta functions do not.

Finite contact terms, running, and physical output

For positive real $\omega$ and subtraction scale $\mu\gt0$, suppose a matched partial-wave amplitude contains

$$\mathcal A_{\ell}(\omega) =\mathcal A_{\mathrm{point}}(\omega) +\omega^{2n}\left[ C_n(\mu)+\beta_n\log\left(\frac{\omega}{\mu}\right) \right] +i\,\mathcal A_{\mathrm{abs}}(\omega)+\cdots .$$

A finite change of scheme shifts $C_n$ and the definition of the logarithm’s constant part. Explicitly, $C_n\mapsto C_n+\delta C_n$ must be accompanied by subtraction of $\delta C_n$ from the remaining bracket contribution. The bracket, hence the amplitude, is identical before and after this compensation. Shifting only $C_n$ changes the amplitude. In the static example, the equivalent change is the compensated applied/response basis, which preserves both $U$ and its curvature. Scale independence fixes the running. The full amplitude, its phase shift, and its absorption probability are the physical output. The 2026 Schwarzschild MST matching makes this boundary especially concrete: a finite renormalized dynamical Love number requires both a scheme and an initial condition for the flow, whereas the bare matched result can be used directly in an observable (Kobayashi et al. 2026, §§ 3.2.2–3.2.3, eqs. (3.32)–(3.38)).

This method ends at the black-hole response coefficients. Inserting them into binary equations of motion or waveforms belongs to two-body dynamics, and replacing the horizon by a material interior belongs to compact-object physics.

Exercises

Let a matched contribution be

$$\Delta\mathcal A =\omega^6\left[ C(\mu)+\beta\log\left(\frac{\omega}{\mu}\right) \right]+i\gamma\omega^5,$$

where $\omega,\mu\gt0$ and $\beta$ and $\gamma$ are scale independent. Determine the running of $C$ required by $d\mathcal A/d\log\mu=0$, and identify what a finite scheme change can alter.

Solution

Differentiating at fixed physical frequency gives

$$0=\frac{d\Delta\mathcal A}{d\log\mu} =\omega^6\left(\frac{dC}{d\log\mu}-\beta\right),$$

so

$$\frac{dC}{d\log\mu}=\beta.$$

A finite subtraction $C\mapsto C+\delta C$ changes the quoted local dynamical coefficient; the full amplitude remains fixed only if the matching convention is changed consistently. It cannot alter the coefficient $\gamma$ of the absorptive imaginary term, nor the measured inelasticity. This check separates a scheme-dependent contact coefficient from the scheme-independent matched observable.

Return to the chapter overview for the complete static, absorptive, and effective-theory route.

References

• 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.
• 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.
• 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.