Nonlinear Tidal Deformation

Tidal response leaves the linear, adiabatic regime when the applied curvature is no longer small relative to the object’s self-gravity, when the driving time approaches an internal mode time, or when the induced displacement is no longer small compared with the object. These are distinct routes to nonlinearity. For a black hole the response must be described through horizon geometry and absorption, not by importing a material surface or a coordinate bulge.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon geometry; Black-Hole Perturbation Theory supplies linear tidal response; Formulations of General Relativity supplies nonlinear variables; Relativistic Two-Body Dynamics supplies orbital scales; Computational Relativity supplies strong-field validation.

Three expansion parameters

Let body 1 have mass $M_1$, characteristic radius $R_1$, and internal mode frequency $\omega_*$. A companion of mass $M_2$ passes at instantaneous separation $r$. Its leading electric tidal curvature has magnitude

$$\mathcal E\sim\frac{GM_2}{r^3}.$$

Compare it with the body’s self-gravity curvature $GM_1/R_1^3$:

$$\epsilon_{\rm tide} =\frac{\mathcal E R_1^3}{GM_1} \sim\frac{M_2}{M_1} \left(\frac{R_1}{r}\right)^3.$$

Linear deformation requires $\epsilon_{\rm tide}\ll1$ and an induced displacement $\epsilon_\xi=\xi/R_1\ll1$. Adiabatic response additionally requires

$$\epsilon_{\rm ad}=\frac{\Omega_{\rm drive}}{\omega_*}\ll1.$$

A response may be small but nonadiabatic near resonance, or large but slowly varying near quasi-equilibrium disruption. The three parameters should therefore be reported separately.

For a material body, define the electric tide in its local rest frame by $\mathcal E_{ij}=C_{i0j0}$ with the site’s curvature convention. A mass quadrupole can admit a low-frequency expansion such as

$$Q_{ij} =-\lambda\,\mathcal E_{ij} -\lambda^{(2)}\mathcal E_{k\langle i}\mathcal E_{j\rangle}{}^k +\text{derivative and magnetic-tide terms}.$$

Angle brackets select the symmetric trace-free part: remove the trace so that $Q_{ij}$ describes quadrupolar shape rather than a change of monopole mass. With $c=1$ and explicit $G$, $[\mathcal E]=L^{-2}$ and $[Q]=ML^2$, so $[\lambda]=ML^4$ and $[\lambda^{(2)}]=ML^6$. The minus sign is tied to the displayed electric-tide convention. The coefficients depend on equation of state, compactness, field content, and matching prescription. Their detailed material calculation belongs with compact-object structure; this chapter uses them as declared inputs and follows the spacetime consequences. Covariant tidal moments and their geometric action are developed in Poisson 2005, Eqs. (1)–(12), pp. 1–4 and Poisson and Vlasov 2010, §§II–V.

Black-hole deformation is quasi-local

A black hole has no material surface. For a nonrotating black hole in four-dimensional vacuum general relativity, the static conservative quadrupolar Love numbers vanish in the standard asymptotically flat matching convention, but a dynamical horizon can still shear, absorb flux, and develop time-dependent intrinsic multipoles. “No static Love number” does not mean “no tidal dynamics.”

Useful nonlinear diagnostics include the horizon area $A_H$, quasi-local angular momentum $J_H$, intrinsic scalar curvature of horizon sections, and geometrically defined multipoles. Their changes should be related to horizon flux rather than a coordinate embedding radius. A finite-amplitude encounter can produce large local distortion even when the remnant’s late-time conservative response is Kerr-like.

The zero-response statement here is specifically the four-dimensional, asymptotically flat, nonrotating vacuum result in the conventional static matching definition Binnington and Poisson 2009, §I.C and §III, pp. 4–8. Rotating, dynamical, higher-dimensional, nonvacuum, or alternative subtraction problems require their own response definition.

For a neutron star or other extended body, $R_1$ and $\omega_*$ are physical material scales. For a black hole, use $R_H=\sqrt{A_H/4\pi}$ only as an areal scale and replace surface displacement by invariant horizon multipoles or shear. This distinction prevents a visual coordinate shape from being treated as an observable.

Regime map and checks

The perturbative regime requires

$$\epsilon_{\rm tide}\ll1, \qquad \epsilon_\xi\ll1, \qquad \epsilon_{\rm ad}\ll1$$

unless frequency dependence is retained explicitly. The dynamical-linear regime has small amplitude but $\epsilon_{\rm ad}\sim1$. The nonlinear-deformation regime has $\epsilon_\xi$ or $\epsilon_{\rm tide}$ approaching unity. Near capture, strong-field orbital curvature and relativistic velocity add independent expansion failures.

Checks include dimensional consistency of $\mathcal E$, recovery of $r^{-3}$ behavior at large separation, agreement of invariant multipoles across gauges, and closure of energy–angular-momentum transfer. A coordinate displacement alone fails all but the first visual intuition.

Check the regime

Take equal masses and $R_1/r=0.2$, with $\Omega_{\rm drive}/\omega_*=0.9$. Estimate $\epsilon_{\rm tide}$ and classify amplitude and adiabaticity.

Solution

For $M_2/M_1=1$,

$$\epsilon_{\rm tide}\sim(0.2)^3=8\times10^{-3},$$

so the instantaneous tidal amplitude can remain perturbatively small. But $\epsilon_{\rm ad}=0.9$ is not small: the response is nonadiabatic and may be resonantly enhanced. Small tidal curvature does not justify an adiabatic approximation.

References

• Binnington, Taylor, and Eric Poisson, “Relativistic Theory of Tidal Love Numbers”, Physical Review D 80, 084018, 2009. Open PDF.
• Poisson, Eric, “Metric of a Tidally Distorted, Nonrotating Black Hole”, Physical Review Letters 94, 161103, 2005. Open PDF.
• Poisson, Eric, and Igor Vlasov, “Geometry and Dynamics of a Tidally Deformed Black Hole”, Physical Review D 81, 024029, 2010. Open PDF.

Further reading

• Porto, Rafael A., “The Effective Field Theorist’s Approach to Gravitational Dynamics”, Physics Reports 633, 1–104, 2016. Open PDF.