arXiv subjects

Mohamed Ould El Hadj

Publications and source records attributed to Mohamed Ould El Hadj.

At least 19 recordsLinked to original sources

arXiv2026

Exceptional points and the breaking of Schwarzschild isospectrality under local potential perturbations

The odd- and even-parity gravitational perturbations of a Schwarzschild black hole are governed by the distinct Regge-Wheeler and Zerilli potentials, yet their quasinormal-mode (QNM) spectra are identical in vacuum. Building on our previous study of the odd-parity Regge-Wheeler spectrum [M. Ould El Hadj and S. R. Dolan, arXiv:2608.16283], we investigate the corresponding even-parity Zerilli problem and compare the two sectors under the same perturbation of amplitude $ε$ at radius $r_0$. Using the Chandrasekhar-Detweiler relation, we derive an exact relation between the normalized Wronskians and show that Schwarzschild QNM isospectrality is not preserved under an identical nonzero localized defect. We find that the even- and odd-parity repelling-point branches approach the same Schwarzschild QNM frequencies at large $r_0$, with their first parity-dependent separation appearing at order $1/\mathcal R^2$, where $\mathcal R=r_{0*}/M$ is the dimensionless tortoise-coordinate location of the defect. The corresponding exceptional points (EPs) form two shifted sequences with the same asymptotic spacing in the tortoise coordinate, while their relative displacement is fixed by the phase of the asymptotic Chandrasekhar-Detweiler factor. The critical perturbation amplitudes share the same exponential decay rate in the two sectors, whereas the ratio of their asymptotic prefactors is fixed by the modulus of the same factor. We further find that the even- and odd-parity EPs connect the same neighboring QNM overtones, with nearly identical arclengths for the corresponding spectral bridges. Finally, the exceptional degeneracy is parity selective: at identical values of $(r_0,ε)$, a pair of QNMs may coalesce at an EP in one parity sector while remaining distinct in the other, leading to a corresponding difference in the local pole structure of the ringdown response.

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arXiv2026

Exceptional points of the Schwarzschild spectrum under local potential perturbation: The Princess and the Pea

Black-hole quasinormal modes (QNMs) may coalesce at exceptional points (EPs) in the frequency spectrum. We study EPs and their phenomenology for gravitational perturbations of a Schwarzschild black hole whose Regge-Wheeler potential is perturbed by a localized delta-function of amplitude $ε$ at radius $r_0$. For real $ε$, we find infinite families of discrete EPs. Each family approaches a distinct unperturbed QNM overtone frequency as $r_0$ grows large. Under perturbation at fixed $r_0$, we find that neighboring QNM overtones migrate in the complex plane to meet at these EPs, with the higher overtone moving further. We derive large-$r_0$ asymptotics, finding that EPs have (asymptotically) equal spacing in the tortoise coordinate, and that EP amplitudes $ε$ can be exceedingly small, decaying exponentially according to a rule governed by the complex frequency of the unperturbed QNM. Using quadratic approximations around EPs and high-precision numerical methods, we investigate a range of phenomena in the model: square-root splittings, avoided crossings and hyperbolae, hysteresis and geometric phases generated by encircling an EP, adiabatic invariants, deformed Cassini ovals and cusp-like trajectories, as well as the lemniscate geometry of the excitation factors. We then replace the delta-function perturbation by a finite-width Gaussian, verify the persistence of the EP structure, and show that varying the width extends isolated EPs into exceptional lines in parameter space. Finally, we study the response to initial data and find that, although a weak perturbation can substantially reconfigure the QNM spectrum, the corresponding time-domain response remains perturbative. This contrast is captured by a re-interpretation of the tale of The Princess and the Pea.

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arXiv2026

Ringdown and echoes from compact objects: Debye series and Debye quasinormal modes

We introduce a new series decomposition of the waveform constructed in the spirit of Debye expansions in scattering theory, and we use this to analyse the time-domain response of compact, horizonless bodies to scalar-field perturbations on curved spacetimes. The Debye decomposition separates out direct exterior propagation, surface reflection, and successive transmissions through the interior of a compact body, and it provides an intuitive interpretation of the waveform in terms of geodesic trajectories. By analysing the quasinormal-mode (QNM) content of individual Debye terms, we set out a Debye-QNM description that is complementary to the standard QNM description. With this framework, we examine a scalar field propagating on two illustrative `Schwarzschild star' compact-body spacetimes: a neutron-star-like model $R>3M$ and an ultracompact object $R<3M$. We show that the Debye reconstruction matches well with the exact waveform, and that (unlike the standard QNM reconstruction) it converges even at early times, giving an accurate description of all waveform features including the prompt response. In the neutron-star case, the low-order Debye terms mainly describe the ringdown and a non-modal component associated with the sub-threshold branch cut. In the ultracompact case, the Debye series organizes the waveform into a prompt/ringdown contribution followed by a succession of individually resolved echo-like wavepackets. The new Debye-QNM expansion and the standard QNM expansion have complementary spectral interpretations: the former identifies modes in individual propagation channels, whereas the latter describes collective resonances that are resummations of the former. This distinction clarifies how echo-like structures emerge from repeated interior propagation, and how pole and branch-cut contributions enter the time-domain signal.

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arXiv2026

Scattering from compact objects: Debye series and Regge-Debye poles

We investigate elastic scattering by a compact, horizonless body in curved spacetime, considering a massless scalar wave incident on a static, spherically symmetric, uniform-density star of radius $R$ and mass $M$ with a Schwarzschild exterior. We introduce an exact Debye-series decomposition of the scattering matrix, in the spirit of Debye expansions in Mie scattering. This decomposition separates direct surface reflection from contributions involving transmission into the interior and subsequent propagation, and admits a natural trajectory interpretation. We then determine the associated Regge-Debye pole spectrum in the complex angular-momentum plane. For neutron-star-like tenuities ($R>3M$), the spectrum exhibits two pole families: a surface-wave branch associated with the surface matching condition and a broad-resonance branch associated with the interior regularity condition. For ultracompact objects ($R<3M$), the surface-wave branch persists, while the interior-resonance sector splits into broad- and narrow-resonance branches. We next reconstruct the scattering amplitude from the Debye partial-wave contributions and find excellent agreement with direct partial-wave calculations. Finally, we develop complex angular-momentum representations order by order in the Debye series, making explicit how the pole families and non-pole sectors contribute to each Debye term. In the neutron-star-like regime, we find a genuine competition between Regge-Debye pole sums and branch-cut contributions, and show that, at high frequency, the rainbow-like enhancement already arises from the first interior-transmission contribution and is dominated by the interior-resonance Regge-Debye poles. By contrast, in the ultracompact regime, the Debye amplitudes are overwhelmingly pole dominated.

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arXiv2025

Black hole absorption cross sections: Spin and Regge poles

We investigate the absorption of massless scalar, electromagnetic, and gravitational fields propagating in the Schwarzschild black hole geometry. Using complex angular momentum techniques, we first derive a representation of the absorption cross section that separates it into smooth background integrals and a discrete Regge pole series. This decomposition reveals the physical mechanisms underlying black hole absorption, including classical capture, surface wave interference near the photon sphere, and subleading background effects. We then construct a refined high-frequency analytical approximation that captures both the dominant oscillations and the fine structure of the absorption spectra for scalar, electromagnetic, and gravitational fields, incorporating spin-dependent phase corrections and higher-order effects. In addition, we provide a simplified expression that generalizes the sinc approximation to describe the leading oscillations for electromagnetic and gravitational fields. Our analysis offers a unified semiclassical interpretation of black hole absorption, combining geometric optics, surface wave dynamics, and resonant phenomena encoded by the Regge pole structure.

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arXiv2025

Gravitational waves in massive gravity: Waveforms generated by a particle plunging into a black hole and the excitation of quasinormal modes and quasibound states

With the aim of testing massive gravity in the context of black hole physics, we investigate the gravitational radiation emitted by a massive particle plunging into a Schwarzschild black hole from slightly below the innermost stable circular orbit. To do so, we first construct the quasinormal and quasibound resonance spectra of the spin-2 massive field for odd and even parity. Then, we compute the waveforms produced by the plunging particle and study their spectral content. This allows us to highlight and interpret important phenomena in the plunge regime, including (i) the excitation of quasibound states, with particular emphasis on the amplification and slow decay of the post-ringdown phase of the even-parity dipolar mode due to harmonic resonance; (ii) during the adiabatic phase, the waveform emitted by the plunging particle is very well described by the waveform emitted by the particle living on the innermost stable circular orbit, and (iii) the regularized waveforms and their unregularized counterparts constructed from the quasinormal mode spectrum are in excellent agreement. Finally, we construct, for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle, the regularized multipolar waveforms, i.e., the waveforms constructed by summing over partial waveforms.

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arXiv2023

Scattering and conversion of electromagnetic and gravitational waves by Reissner-Nordström black holes: The Regge pole description

We investigate the problem of scattering and conversion of monochromatic planar gravitational and electromagnetic waves impinging upon a Reissner-Nordström black hole using a Regge pole description, i.e., a complex angular momentum approach. For this purpose, we first compute numerically the Regge pole spectrum for various charge-to-mass ratio configurations. We then derive an asymptotic expressions for the lowest Regge poles, and by considering Bohr-Sommerfeld-type quantization conditions, obtain the spectrum of weakly damped quasinormal frequencies from the Regge trajectories. Next, we construct the scattering and conversion amplitudes as well as the total differential cross sections for different processes using both a complex angular momentum representation and a partial wave expansion method. Finally, we provide an analytical approximation of the scattering and conversion cross sections of different processes from asymptotic expressions for the lowest Regge poles and the associated residues based on the correspondence Regge poles, "surface waves" propagating close to the photon (graviton) sphere. This allows us to extract the physical interpretation encoded in the partial wave expansions in the high-frequency regime (i.e., in the short-wavelength regime), and to describe semiclassically with very good agreement both black hole glory and a large part of the orbiting oscillations, thus unifying these two phenomena from a purely wave point of view.

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arXiv2022

Regge pole description of scattering by dirty black holes

We study the problem of plane monochromatic scalar waves impinging upon a Schwarzschild dirty black hole -- a Schwarzschild black hole surrounded by a thin spherical shell of matter -- using the complex angular momentum approach. We first recall general results concerning the differential scattering cross section in the classical limit through a null geodesic analysis by exploring different configurations of the shell. In particular, we show that dirty black hole spacetimes may exhibit various critical effects for geometrical optics. We compute the Regge pole spectrum for various shell configurations and show that it exhibits two or three distinct branches of poles, labelled inner surface waves, broad resonances and outer surface waves. In the latter, two sub-families have been identified, the surface waves associated with the outer light-ring and the creeping modes associated with the surface of the shell. We show, using WKB analysis, that the position of the shell sets the real part of the broad resonances while its energy-momentum and the discontinuity of the potential at the shell's surface set their imaginary part. Next, we provide the complex angular momentum representation of the differential scattering cross section and examine the role of the different Regge pole branches. We compute the differential scattering cross section for various configurations at several frequencies and show a very good agreement with the partial-wave calculations. Finally, we highlight the role of the critical effects, i.e., orbiting, glory, grazing and rainbow scattering, and their impact on the differential scattering cross section.

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arXiv2021

Conversion of electromagnetic and gravitational waves by a charged black hole

In a strong electromagnetic field, gravitational waves are converted into electromagnetic waves of the same frequency, and vice versa. Here we calculate the scattering and conversion cross sections for a planar wave impinging upon a Reissner-Nordström black hole in vacuum, using the partial-wave expansion and numerical methods. We show that, at long wavelengths, the conversion cross section matches that computed by Feynman-diagram techniques. At short wavelengths, the essential features are captured by a geometric-optics approximation. We demonstrate that the converted flux can exceed the scattered flux at large scattering angles, for highly-charged black holes. In the short-wavelength regime, the conversion effect may be understood in terms of a phase that accumulates along a ray. We compute the scattering angle for which the converted and scattered fluxes are equal, as a function of charge-to-mass ratio. We show that this scattering angle approaches $90$ degrees in the extremal limit.

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arXiv2020

Electromagnetic radiation generated by a charged particle falling radially into a Schwarzschild black hole: A complex angular momentum description

By using complex angular momentum techniques, we study the electromagnetic radiation generated by a charged particle falling radially from infinity into a Schwarzschild black hole. We consider both the case of a particle initially at rest and that of a particle projected with a relativistic velocity and we construct complex angular momentum representations and Regge pole approximations of the partial wave expansions defining the Maxwell scalar $ϕ_2$ and the energy spectrum $dE/dω$ observed at spatial infinity. We show, in particular, that Regge pole approximations involving only one Regge pole provide effective resummations of these partial wave expansions permitting us (i) to reproduce with very good agreement the black hole ringdown without requiring a starting time, (ii) to describe with rather good agreement the tail of the signal and sometimes the pre-ringdown phase, and (iii) to explain the oscillations in the electromagnetic energy spectrum radiated by the charged particle. The present work as well as a previous one concerning the gravitational radiation generated by a massive particle falling into a Schwarzschild black hole [A. Folacci and M. Ould El Hadj, Phys. Rev. D$\textbf{98}$, 064052 (2018), arXiv:1807.09056 [gr-qc]] highlight the benefits of studying radiation from black holes in the complex angular momentum framework (they obviously appear when the approximations obtained involve a small number of Regge poles and have a clear physical interpretation) but also to exhibit the limits of this approach (this is the case when it is necessary to take into account background integral contributions).

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arXiv2020

Scattering from compact objects: Regge poles and the complex angular momentum method

We calculate the Regge poles of the scattering matrix for a gravitating compact body, for scalar fields and for gravitational waves in the axial sector. For a neutron-starlike body, the spectrum exhibits two distinct branches of poles, labeled surface waves and broad resonances; for ultracompact objects, the spectrum also includes a finite number of narrow resonances. We show, via a WKB analysis, that the discontinuity of the effective potential at the body's surface determines the imaginary component of the broad-resonance poles. Next, we examine the role of Regge poles in the time-independent scattering of monochromatic planar waves. We apply complex angular momentum techniques to re-sum the partial wave series for the scattering amplitude, expressing it as a residue series evaluated at poles in the first quadrant, accompanied by a background integral. We compute the scattering cross section at several frequencies, and show precise agreement with the partial-wave calculations. Finally, we show that compact bodies naturally give rise to orbiting, glory, and rainbow-scattering interference effects.

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arXiv2020

An alternative description of gravitational radiation from black holes based on the Regge poles of the ${\cal S}$-matrix and the associated residues

We advocate for an alternative description of gravitational radiation from black holes based on complex angular momentum techniques (analytic continuation of partial wave expansions, duality of the ${\cal S}$-matrix and effective resummations involving its Regge poles and the associated residues, Regge trajectories, semiclassical interpretations, etc.). Such techniques, which proved to be very helpful in various areas of physics to describe and analyze resonant scattering, were only marginally used in the context of black hole physics. Here, by considering the multipolar waveform generated by a massive particle falling radially from infinity into a Schwarzschild black hole, we show that they could play a fundamental role in gravitational-wave physics. More precisely, from the multipole expansion defining the Weyl scalar $Ψ_4$, we extract the Fourier transform of a sum over Regge poles and their residues which can be evaluated numerically from the associated Regge trajectories. This Regge pole approximation permits us to reconstruct, for an arbitrary direction of observation, a large part of the multipolar waveform $Ψ_4$. In particular, it can reproduce with very good agreement the quasinormal ringdown as well as with rather good agreement the tail of the signal. This is achieved even if we take into account only one Regge pole and, if a large number of modes are excited, the result can be improved by considering additional poles. Moreover, while quasinormal-mode contributions do not provide physically relevant results at "early times" due to their exponentially divergent behavior as time decreases, it is not necessary to determine from physical considerations a starting time for the Regge ringdown.

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arXiv2019

Regge pole description of scattering of gravitational waves by a Schwarzschild black hole

We revisit the problem of plane monochromatic gravitational waves impinging upon a Schwarzschild black hole using complex angular momentum techniques. By extending our previous study concerning scalar and electromagnetic waves [A. Folacci and M. Ould El Hadj, Phys.\, Rev.\, D {\bf 99}, 104079, (2019), arXiv:1901.03965], we provide complex angular momentum representations and Regge pole approximations of the helicity-preserving and helicity-reversing scattering amplitudes and of the total differential scattering cross section. We show, in particular, that for high frequencies (i.e., in the short-wavelength regime), a small number of Regge poles permits us to describe numerically with very good agreement the black hole glory and the orbiting oscillations and we then provide a semiclassical approximation that unifies these two phenomena.

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arXiv2019

Regge pole description of scattering of scalar and electromagnetic waves by a Schwarzschild black hole

We revisit the problem of scalar and electromagnetic waves impinging upon a Schwarzschild black hole from complex angular momentum techniques. We focus more particularly on the associated differential scattering cross sections. We derive an exact representation of the corresponding scattering amplitudes by replacing the discrete sum over integer values of the angular momentum which defines their partial wave expansions by a background integral in the complex angular momentum plane plus a sum over the Regge poles of the $S$-matrix involving the associated residues. We show that, surprisingly, the background integral is numerically negligible for intermediate and high frequencies and, as a consequence, that the cross sections can be reconstructed in terms of Regge poles with very good agreement. We show in particular that, for large values of the scattering angle, a small number of Regge poles permits us to describe the black hole glory and that, by increasing the number of Regge poles, we can reconstruct very efficiently the differential scattering cross sections for small and intermediate scattering angles and therefore describe the orbiting oscillations. In fact, in the short-wavelength regime, the sum over Regge poles allows us to extract by resummation the information encoded in the partial wave expansion defining a scattering amplitude and, moreover, to overcome the difficulties linked to its lack of convergence due to the long-range nature of the fields propagating on the black hole. As a consequence, from asymptotic expressions for the lowest Regge poles and the associated residues based on the correspondence Regge poles -- ``surface waves'' propagating close to the photon sphere, we can provide an analytical approximation describing both the black hole glory and a large part of the orbiting oscillations. We finally discuss the role of the background integral for low frequencies.

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arXiv2018

Multipolar gravitational waveforms and ringdowns generated during the plunge from the innermost stable circular orbit into a Schwarzschild black hole

We study the gravitational radiation emitted by a massive point particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole. We consider both even- and odd-parity perturbations and describe them using the two gauge-invariant master functions of Cunningham, Price, and Moncrief. We obtain, for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle, the regularized multipolar waveforms, i.e., the waveforms constructed by summing over of a large number of modes, and their unregularized counterparts constructed from the quasinormal-mode spectrum. They are in excellent agreement and our results permit us to especially emphasize the impact on the distortion of the waveforms of (i) the harmonics beyond the dominant $(\ell=2,m=\pm 2)$ modes and (ii) the direction of observation, and therefore the necessity to take them into account in the analysis of the last phase of binary black hole coalescence.

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arXiv2018

Electromagnetic radiation generated by a charged particle plunging into a Schwarzschild black hole: Multipolar waveforms and ringdowns

Electromagnetic radiation emitted by a charged particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole is examined. Both even- and odd-parity electromagnetic perturbations are considered. They are described by using gauge invariant master functions and the regularized multipolar waveforms as well as their unregularized counterparts constructed from the quasinormal-mode spectrum are obtained for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle. They are in excellent agreement and the results especially emphasize the impact of higher harmonics on the distortion of the waveforms.

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arXiv2016

Waveforms in massive gravity and neutralization of giant black hole ringings

A distorted black hole radiates gravitational waves in order to settle down in a smoother geometry. During that relaxation phase, a characteristic damped ringing is generated. It can be theoretically constructed from both the black hole quasinormal frequencies (which govern its oscillating behavior and its decay) and the associated excitation factors (which determine intrinsically its amplitude) by carefully taking into account the source of the distortion. In the framework of massive gravity, the excitation factors of the Schwarzschild black hole have an unexpected strong resonant behavior which, theoretically, could lead to giant and slowly decaying ringings. If massive gravity is relevant to physics, one can hope to observe these extraordinary ringings by using the next generations of gravitational wave detectors. Indeed, they could be generated by supermassive black holes if the graviton mass is not too small. In fact, by focusing on the odd-parity $\ell=1$ mode of the Fierz-Pauli field, we shall show here that such ringings are neutralized in waveforms due to (i) the excitation of the quasibound states of the black hole and (ii) the evanescent nature of the particular partial modes which could excite the concerned quasinormal modes. Despite this, with observational consequences in mind, it is interesting to note that the waveform amplitude is nevertheless rather pronounced and slowly decaying (this effect is now due to the long-lived quasibound states). It is worth noting also that, for very low values of the graviton mass (corresponding to the weak instability regime for the black hole), the waveform is now very clean and dominated by an ordinary ringing which could be used as a signature of massive gravity.

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arXiv2015

Waveforms produced by a scalar point particle plunging into a Schwarzschild black hole: Excitation of quasinormal modes and quasibound states

With the possibility of testing massive gravity in the context of black hole physics in mind, we consider the radiation produced by a particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole. In order to circumvent the difficulties associated with black hole perturbation theory in massive gravity, we use a toy model in which we replace the graviton field with a massive scalar field and consider a linear coupling between the particle and this field. We compute the waveform generated by the plunging particle and study its spectral content. This permits us to highlight and interpret some important effects occurring in the plunge regime which are not present for massless fields such as (i) the decreasing and vanishing, as the mass parameter increases, of the signal amplitude generated when the particle moves on quasicircular orbits near the innermost stable circular orbit and (ii) in addition to the excitation of the quasinormal modes, the excitation of the quasibound states of the black hole.

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