Fixed-quadrupole static tidal response of Schwarzschild black holes in a cubic Weyl effective field theory

Abstract: Static Love numbers of four-dimensional Schwarzschild black holes vanish in general relativity. We study how the fixed-quadrupole static tidal solution is modified by the parity-even cubic Weyl operator in the gravitational effective field theory. Working perturbatively in $ε{\rm e}=λ{\rm e}(Λr_s)^{-4}$, we construct the reduced quadratic radial action for static even-parity $\ell=2$ perturbations, order-reduce the higher-derivative equations, and solve the resulting boundary-value problem directly in metric variables. The order-$ε{\rm e}$ equations reduce to a first-order two-dimensional inhomogeneous system for $X_0$ and $X_K$, with $X_2$ fixed by an algebraic constraint. Horizon regularity leaves one constant, but matching to infinity shows that this freedom only renormalizes the applied tidal branch. After removing this tidal renormalization, the decaying branch is unambiguous. Calibrating the spatial sector at fixed $\ell=2$ against the associated-Legendre branches $P_2^{2}$ and $Q_2^{2}$, we obtain a fixed-quadrupole response amplitude $Δ(B/A)=-2400ε{\rm e}$. Equivalently, the scalar fixed-$\ell$ quotient gives $Δk_{2,\rm sc}^{\rm fix}=-20ε_{\rm e}$. The second number is a scalar fixed-$\ell$ conversion of the metric branch ratio, not, by itself, the analytically continued, gauge-invariant electric Love number. A comparison with canonical Teukolsky-based Love numbers requires an additional continuation in $\ell$ and a precise map of normalizations. The result should therefore be viewed as a reproducible metric-sector benchmark for the cubic Weyl EFT, complementary to gauge-invariant master-equation approaches.

• The paper demonstrates that higher-derivative cubic Weyl corrections yield a fixed quadrupole metric tidal response amplitude of -2400.
• An order-reduced perturbative EFT method is used to derive two first-order ODEs governing even-parity, ℓ=2 perturbations with proper horizon regularity.
• The analysis bridges metric-based results with gauge-invariant Love number calculations, offering insights for gravitational-wave modeling in binary inspirals.

Fixed-Quadrupole Static Tidal Response for Schwarzschild Black Holes in Cubic Weyl Effective Theory

Introduction and Context

The study addresses the modification of static tidal response amplitudes—specifically, Love numbers—of Schwarzschild black holes induced by higher-derivative terms in gravitational EFTs, focusing on the cubic (parity-even) Weyl operator. Canonical results in General Relativity (GR) show that the four-dimensional static Love numbers of the Schwarzschild solution vanish, a result with deep connections to the underlying symmetries and structure of GR. The introduction of higher-curvature terms, such as $C^3$, produces non-vanishing corrections, breaking the structural cancellation found in GR. This paper presents a direct metric computation of the fixed-$\ell=2$ static even-parity tidal response, contrasting with previous master-equation (Teukolsky-based) and worldline EFT approaches.

EFT Framework and Perturbative Methodology

The analysis is performed in a perturbative EFT framework where the gravitational action is augmented by a dimension-six, parity-even cubic Weyl contraction:

$$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$

with expansion parameter $\epsilon = \lambda_e (\Lambda r_s)^{-4}$ for $r_s = 2GM$, and $\lambda_e$ dimensionless. All calculations are truncated at $\mathcal{O}(\epsilon)$.

The background is given by the Schwarzschild solution, order-reduced to include first-order corrections to the metric components from the higher-curvature operator. Static, axisymmetric, even-parity perturbations are introduced in Regge-Wheeler gauge, retaining only $\ell=2$ components relevant to quadrupolar tides.

GR Benchmark and Love Number Normalization

For reference, the GR static $\ell=2$ perturbation is expressed through associated Legendre functions $P_2^2$ and $\ell=2$0. The regular branch at the horizon corresponds to $\ell=2$1, which grows at large radius (tidal field), while $\ell=2$2 is singular at the horizon and suppressed asymptotically (response). The canonical vanishing of Love numbers in GR is recovered since only the growing (nontidal response) branch is allowed by regularity.

The standard conventions, including large-radius asymptotics, are:

• $\ell=2$3 yields $\ell=2$4 (tidal field)
• $\ell=2$5 yields $\ell=2$6 (response)

Reduced-Action Calculation and Metric-System Structure

The quadratic action (to $\ell=2$7) is constructed, and the field equations are explicitly derived and order-reduced. Higher-derivative corrections are absorbed using the GR equations of motion, ensuring the absence of spurious (Ostrogradsky) degrees of freedom.

A crucial technical finding is that the perturbed $\ell=2$8 static system is described by two first-order ODEs for $\ell=2$9 (perturbations in $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$0 and the angular metric component), with $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$1 (from $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$2) fixed algebraically. The system does not possess nontrivial poles except at the horizon, confirming its well-posedness and compatibility with black hole boundary conditions.

Horizon Regularity and Asymptotic Matching

Near the horizon, regularity conditions fix all but one integration constant. This residual freedom is traced, via horizon-to-infinity matching, to an ambiguity in normalization of the external tidal field (the branch $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$3 at infinity). Thus, within this prescription, the only unambiguous observable is the coefficient of the decaying branch ($$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$4, i.e., the actual response).

In the asymptotic expansion, the $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$5 coefficients in $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$6 are uniquely fixed after imposing the no-tide-renormalization condition (no correction to the applied external field), yielding the decaying response amplitude.

Key Results

The main certified result is the fixed-quadrupole ($$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$7) metric response amplitude:

• $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$8

where $$S = \frac{1}{16\pi G} \int \mathrm{d}^4x\,\sqrt{-g}\left[R + \frac{\lambda_e}{\Lambda^4} C_{\mu\nu}{}^{\rho\sigma} C_{\rho\sigma}{}^{\alpha\beta} C_{\alpha\beta}{}^{\mu\nu}\right]$$9 and $\epsilon = \lambda_e (\Lambda r_s)^{-4}$0 are the amplitudes of the $\epsilon = \lambda_e (\Lambda r_s)^{-4}$1 and $\epsilon = \lambda_e (\Lambda r_s)^{-4}$2 branches, respectively, and the normalization is chosen such that $\epsilon = \lambda_e (\Lambda r_s)^{-4}$3.

An equivalent representation convenient for cross-normalization is

• $\epsilon = \lambda_e (\Lambda r_s)^{-4}$4

Here, $\epsilon = \lambda_e (\Lambda r_s)^{-4}$5 is a scalar fixed-$\epsilon = \lambda_e (\Lambda r_s)^{-4}$6 metric quotient, not analytically continued, and should not be misidentified with the gauge-invariant electric Love number $\epsilon = \lambda_e (\Lambda r_s)^{-4}$7. The computation of $\epsilon = \lambda_e (\Lambda r_s)^{-4}$8 requires analytic continuation in $\epsilon = \lambda_e (\Lambda r_s)^{-4}$9 and gauge-invariant master equation formalism, as implemented in e.g., Cano's Teukolsky-based approach [Cano, (Cano, 27 Feb 2025)].

Relation to Analytically Continued Love Numbers

The result is fundamentally a fixed-$r_s = 2GM$0 metric benchmark. The master-equation approach (Teukolsky, metric/Weyl scalars) permits a gauge-invariant definition of electric ($r_s = 2GM$1) and magnetic ($r_s = 2GM$2) Love numbers, typically involving analytic continuation in $r_s = 2GM$3. For the cubic parity-even Weyl operator, the canonical $r_s = 2GM$4 is $r_s = 2GM$5 and $r_s = 2GM$6 in the conventions of [Cano, (Cano, 27 Feb 2025)], modulo normalization factors.

The fixed-$r_s = 2GM$7 quotient computed here, $r_s = 2GM$8, differs from the canonical value because analytic continuation, normalization, and response extraction differ between frameworks. The result serves as an unambiguous metric-space diagnostic, complementing, not replacing, gauge-invariant methods.

Implications and Extensions

The formal extraction confirms that the inclusion of higher-derivative operators in gravitational EFTs generically leads to non-vanishing black hole Love numbers at fixed multipole order. This has immediate consequences for the theoretical modeling of finite-size effects in binary inspirals, relevant for gravitational-wave signal modeling. However, careful interpretation of fixed-$r_s = 2GM$9 results is essential: only analytically continued, gauge-invariant definitions are physically observable across all coordinate choices and extraction schemes.

Future research directions include:

• Extending the framework to odd-parity cubic Weyl contractions, addressing the fixed-$\lambda_e$0 magnetic response in the metric sector.
• Generalizing the action reduction and field equations to arbitrary $\lambda_e$1, with analytic continuation to extract canonical Love numbers directly.
• Matching metric response coefficients to worldline EFT Wilson coefficients, clarifying the translation between metric-sector calculations and effective action parameters.

Conclusion

This work establishes an auditable fixed-quadrupole metric tidal response calculation for Schwarzschild black holes in the presence of a cubic parity-even Weyl operator, providing a well-defined benchmark for metric-based extractions in gravitational EFTs. The main certified amplitude, $\lambda_e$2, encapsulates the decaying branch normalization at fixed multipole, supplementing gauge-invariant Love number calculations and facilitating rigorous cross-verification between analytic approaches in higher-derivative gravity.

Reference:

"Fixed-quadrupole static tidal response of Schwarzschild black holes in a cubic Weyl effective field theory" (2606.16909)