Static electromagnetic Love tensors of 5-dimensional Myers-Perry black holes

Abstract: We study the separable master equations for the electromagnetic and gravitational perturbations in five-dimensional Myers-Perry black holes. In the static limit, while the master equation for the electric polarization of the Maxwell field reduces to that of a massless scalar field, the magnetic polarization and gravitational perturbation yield Heun equations for both its angular and radial components. Remarkably, these Heun equations fall into a special class that admits exact analytic solutions in terms of hypergeometric functions. We reconstruct the gauge field using master fields and study its asymptotic behavior. When expanding the result in the basis of modified spherical harmonics, we find modes with higher angular momentum arise in response to the excitation of sources with lower angular momentum. The static tidal Love tensor that characterizes such mixing structure of the response can be computed iteratively. We also discuss the possible near zone approximation of the master equations for the magnetic polarization.

• The paper develops an analytic approach to compute static electromagnetic Love tensors for 5D Myers-Perry black holes.
• It reduces complex Heun equations to hypergeometric sums under quantization conditions, enabling explicit evaluation of running Love numbers.
• The study highlights mode mixing and tensor generalizations in EFT, impacting tidal response modeling in higher dimensions.

Static Electromagnetic Love Tensors in Five-Dimensional Myers-Perry Black Holes

Introduction and Context

This paper develops an analytic approach to the computation of static electromagnetic Love tensors for five-dimensional Myers-Perry black holes. The primary motivation stems from recent interest in static response coefficients—most saliently, the family of tidal Love numbers that parameterize the multipolar responses of compact objects to external fields. In the context of both gravitational-wave physics and black hole effective field theory (EFT), understanding these response coefficients, their possible vanishing, and their tensorial structure is crucial for both astrophysical phenomenology and the deeper structure of black hole perturbation theory.

The study focuses on electromagnetic and gravitational perturbations in non-trivial higher-dimensional geometries, particularly the five-dimensional Myers-Perry solution, which generalizes the Kerr family to higher dimensions and admits two independent angular momenta. The primary technical challenge is the analysis of the relevant master equations governing these perturbations—specifically, their separability, analytic structure, and the implications for tidal response quantities.

Master Equations and Analytic Solutions

The investigation begins with a review of the Myers-Perry geometry and establishes the separability of wave equations for both the electromagnetic field and certain vector-type gravitational perturbations, primarily using the scalar master field reduction technique. This enables the derivation of decoupled ordinary differential equations (ODEs) for both the angular and radial channels of perturbations, depending on both polarization and geometry.

Static Limit and Special Structure

A critical aspect of the analysis is the static ($\omega=0$) regime, where the complexity of the equations is significantly reduced. For the electric polarization of the Maxwell field, the master equation reduces to the known massless scalar field equations. In contrast, the magnetic polarization and vector-type gravitational perturbations lead to Heun equations for both angular and radial parts.

A central technical advancement in the paper is the demonstration that, in the static limit, these Heun equations admit solutions that can be expressed as finite sums of hypergeometric functions. This occurs due to the presence of removable singularities under quantization conditions on the separation constants—a phenomenon previously observed only in specific cases and now shown to be generic in this context. This analytic tractability is highly nontrivial, given the general intractability of Heun functions, and opens up rigorous computation of response coefficients that are otherwise only accessible numerically.

Quantization and Mixing Structures

The quantization conditions on angular momentum and separation constants (such as $\tilde{\mu}$ for magnetic polarization) arise from regularity requirements at the poles of the $S^3$ angular coordinates. Notably, for the magnetic sector, the angular functions are not orthogonal for distinct eigenvalues, and the modes exhibit nontrivial mixing: excitations of lower angular momentum modes induce responses in higher angular momentum channels. This is in contrast to the scalar and electric polarizations and reflects the richer tensorial structure intrinsic to the electromagnetic sector in rotating backgrounds.

Asymptotic Analysis and Love Numbers

The analytic structure established in the preceding sections enables an explicit asymptotic expansion of the master fields at infinity, from which the static Love numbers and their tensor generalizations (“Love tensors”) can be computed.

• Running (Logarithmic) Love Responses: For integer quantized values of separation constants, the asymptotic expansion reveals logarithmic “running” behavior in the response coefficients, as opposed to pure power-law suppression. This is in parallel with previous studies in lower-dimensional black hole backgrounds and confirms the universal structure arising from the properties of the underlying ODEs.
• Conditions for Vanishing of Love Numbers: The vanishing of the static Love numbers (i.e., pure absorption with no induced multipole) is tied to the divergence of gamma function factors in the analytic expressions for the response, and occurs under tight constraints—for instance, when the two rotation parameters are equal, or for specific relations between the azimuthal quantum numbers.
• Tensorial Mixing: The full tidal response for the electromagnetic field cannot be encapsulated by scalar Love numbers due to the mode mixing: the electric static response is encoded in a lower-triangular Love tensor $\mathbf{k}^E_{jj'}$, mapping lower-$\ell$ sources to higher-$\ell$ responses, with explicit iterative formulas provided.

Effective Field Theory and Physical Implications

The analysis is placed within the effective field theory framework, where the black hole is described as a point particle with induced multipole couplings in the presence of external fields. The computed Love tensors correspond to Wilson coefficients in the EFT, quantifying finite-size corrections and clarifying the interplay between geometry, spin, and tidal response. This formalism unifies the calculation for both electromagnetic and gravitational perturbations and underlines the non-trivial necessity for generalized tensors (not just scalars) in higher-dimensional spinning geometries.

The methodology allows, for the first time in this context, analytic computation of the full set of static electromagnetic Love tensors, including explicit expressions for the first few matrix elements. These results provide essential input for black hole perturbation theory, the modeling of binary mergers with high spins in higher dimensions, and the exploration of symmetries underlying tidal phenomena.

Near Zone Approximation and Emergent Symmetries

A further section analyzes the near-zone limit (low frequency, near horizon), where the wave equations simplify and approximate symmetries may emerge. Here, for scalar fields, the reduction to hypergeometric form signals an emergent $\mathrm{SL}(2)$ symmetry (the so-called Love symmetry), suggesting deeper group-theoretic structures governing black hole tidal responses. For magnetic polarizations, a family of analytic near-zone approximations is constructed, admitting Heun-type ODEs with removable singularities and thus analytic solutions. This sets the stage for probing analogous hidden symmetries in the electromagnetic sector and motivates future work on symmetry-based constraints on black hole responses.

Conclusion

This work provides a comprehensive analytic treatment of the static electromagnetic and vector-type gravitational response of five-dimensional Myers-Perry black holes. The use of special properties of Heun equations, reduction to hypergeometric functions, and systematic asymptotic expansions yields explicit formulas for running Love numbers and demonstrates the necessity of generalized Love tensors to account for mode mixing in the electromagnetic case.

The results have immediate applications for the construction of black hole EFTs in higher dimensions, the classification of black hole microstructure through static responses, and the explicit computation of response coefficients appearing in gravitational wave modeling. Furthermore, the analytic techniques developed open avenues for analogous studies in yet higher-dimensional spacetimes, massive field perturbations, and the exploration of emergent symmetries in black hole physics.

Future directions include extension to other perturbing fields (e.g., higher spin), detailed categorization of hidden symmetries in various near-horizon and static limits, and the systematic evaluation of the impact of these tensors in higher-dimensional black hole merger scenarios and holographic setups.

In summary, this paper delivers an exact analytic characterization of static electromagnetic Love tensors in the rich setting of spinning higher-dimensional black holes, revealing both new mathematical structure and deepening the theoretical underpinnings of black hole tidal responses (2605.16792).