[Paper Review] Dynamical Love Numbers for Kerr Black Holes

Malcolm J. Perry, M. Jimena Rodríguez|arXiv (Cornell University)|Oct 5, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper introduces dynamical tidal Love numbers for Kerr black holes, showing they are generically non-zero and exhibit logarithmic frequency dependence due to time-varying gravitational perturbations. Using analytic continuation and Teukolsky equation solutions, the authors derive frequency-dependent response coefficients that reveal a finite logarithmic contribution in the real part, contrasting with the vanishing static Love numbers.

ABSTRACT

While static Love number vanish identically for Kerr black holes, we show that the corresponding dynamical tidal coefficients are generically non-zero and exhibit logarithmic behavior. The computational method employs a related but simpler scheme consistent with CFT descriptions, low-frequency regimes and post-Newtonian results. These coefficients are illustrated with a numerical examples.

Motivation & Objective

• To resolve the long-standing puzzle of why static tidal Love numbers vanish for Kerr black holes while dynamical responses may not.
• To develop a consistent framework for computing frequency-dependent tidal response coefficients in Kerr spacetime.
• To establish a connection between the dynamical tidal response and conformal field theory (CFT) descriptions via hidden SL(2,R)×SL(2,R) symmetries.
• To provide a quantitative, analytic computation of the real and imaginary parts of tidal coefficients at finite frequency, including logarithmic corrections.

Proposed method

• Solving the Teukolsky equation for scalar perturbations in Kerr spacetime with ingoing boundary conditions at the horizon.
• Applying analytic continuation by treating the angular momentum quantum number ℓ as a non-integer parameter to access finite-frequency responses.
• Using transformation laws of hypergeometric functions to extract asymptotic behavior at spatial infinity and extract tidal response coefficients.
• Computing the response coefficients via the ratio of hypergeometric function parameters, identifying the real and imaginary parts of the dynamical Love numbers.
• Performing a perturbative expansion in ω around ω=0 to isolate the leading-order logarithmic frequency dependence.
• Relating the results to CFT via the near-horizon SL(2,R)×SL(2,R) symmetry and effective geometry constructions.

Experimental results

Research questions

• RQ1 Why do static tidal Love numbers vanish for Kerr black holes, yet dynamical responses remain non-zero under time-dependent perturbations?
• RQ2 What is the analytic structure of the frequency-dependent tidal response coefficients for Kerr black holes, particularly at low frequencies?
• RQ3 How do logarithmic terms emerge in the dynamical Love numbers, and what is their physical origin?
• RQ4 Can the dynamical tidal response be consistently computed using analytic continuation techniques and matched to CFT descriptions?
• RQ5 What is the role of frame-dragging and horizon structure in generating non-trivial real and imaginary parts in the response coefficients?

Key findings

• The real part of the dynamical Love number, κℓm(ω), exhibits a finite logarithmic frequency dependence, specifically proportional to ω log(r+−r−/r), indicating non-trivial tidal deformation under time-varying fields.
• The leading-order frequency dependence of the response coefficient is linear in ω, with the coefficient k(1)ℓm given explicitly in terms of ℓ, m, s, M, and γ.
• The imaginary part νℓm(ω) is non-zero at ω=0 due to frame-dragging effects, signaling dissipative processes even in the static limit.
• The dynamical response coefficients are derived analytically using hypergeometric function identities and analytic continuation, avoiding purely numerical methods.
• The results are consistent with post-Newtonian expectations, with tidal effects appearing at 5PN order in gravitational wave phase, and dissipation at 2.5PN for rotating bodies.
• The framework supports a CFT interpretation via near-horizon symmetries, linking the dynamical response to dual field theory descriptions.