arXiv subjects

Marc Casals

Publications and source records attributed to Marc Casals.

At least 19 recordsLinked to original sources

arXiv2026

Massive quantum divergence on the Cauchy horizon of a black hole

We investigate a quantum {\it massive} scalar field in the interior of a charged and spherically-symmetric (Reissner-Nordström) black hole. We examine the behaviour for varying values of the black hole charge, field mass and coupling constant when the field is in two quantum states: Hartle-Hawking (representing a black hole in thermal equilibrium) and Unruh (representing a black hole evaporating via the emission of Hawking radiation). We show that the vacuum polarization (in both states) generically {\it diverges} on the Cauchy horizon, in stark contrast to what happens for massless fields; also, the angular components of the quantum stress-energy tensor generically {\it diverge} on the Cauchy horizon and, in at least one of the two states, they do so more strongly than for massless fields. Furthermore, we calculate the energy fluxes in Eddington-Finkelstein coordinates $\{u,v\}$. We show that these fluxes in both states on the Cauchy horizon do not generically vanish. This implies, in particular, that in regular, Kruskal coordinates $\{U,V\}$ the ingoing flux diverges like $V^{-2}$ on the Cauchy horizon (where $V=0$). This divergence suggests that its backreaction via the semiclassical Einstein equations would yield a strong singularity, as opposed to its weaker, classical counterpart. Interestingly, there are exceptions, in which the energy fluxes in Eddington-Finkelstein coordinates vanish: (i) in the extremal limit (where the black hole is maximally charged); (ii) certain fine-tuned regions of parameter space, where the fluxes change sign.

gr-qc↗

arXiv2026

Metric reconstruction in a Bondi-like gauge

Metric perturbations obtained from a Hertz potential in an ingoing radiation gauge (IRG) generically grow in an outgoing coordinate $r$ at future null infinity, and are therefore not asymptotically flat in the Bondi sense. Hollands and Toomani showed that this growth can be removed, within the Green--Hollands--Zimmerman (GHZ) framework, by an IRG-compatible ``residual'' gauge transformation together with a carefully chosen choice of the undetermined integration ``constants'' of the Hertz potential. In this work we turn that proof into a practical algorithm. We show that this entire completion is generated by a single ``Held Master Scalar'' (HMS), which can be thought of as a function on future null infinity. The HMS is obtained from the leading asymptotic piece of the $0$-Weyl scalar through a Teukolsky--Starobinsky inversion. We give explicit formulas for the four integration constants of the Hertz potential associated with the $0$-Weyl scalar, and for the residual gauge vector in terms of the HMS, thereby bringing the metric perturbation into a Bondi-like gauge. In Schwarzschild the entire construction is diagonal in $(\ell, m)$-modes and every completion field is available in closed form; in Kerr the construction is tri-diagonal. We have numerically implemented the full scheme in $(\ell,m)$ and in $m$-modes, including the corrector piece of the GHZ framework. We illustrate our scheme numerically for a point-particle on a circular orbit in a puncture scheme in Schwarzschild and in Kerr, by demonstrating the Bondi-like fall-off. We also find agreement with the known Detweiler redshift computations in Schwarzschild and Kerr. Our method directly gives the Bondi news, mass-, and angular momentum aspects, for which we give exemplary results in Kerr.

gr-qc↗

arXiv2026

Dissection of a merger-ringdown waveform in the small-mass-ratio limit

Work over the past two decades has unveiled the rich phenomenology of black hole binary mergers and subsequent ringdowns, involving a tapestry of quasinormal modes (QNMs), nonlinearities, tails, transients, and secular effects including gravitational memory. Here we develop a framework for analyzing nonlinear merger-ringdown features in the small-mass-ratio limit, where individual effects can be cleanly isolated. Specializing to the case of a quasicircular, nonspinning black hole binary, we find the waveform sharply divides into a pre-merger extended inspiral phase, a merger phase lasting roughly half a cycle, and a post-merger ringdown dominated by QNMs. We show quadratic QNMs dominate over linear overtones in the ringdown phase for comparable-to-intermediate mass ratios, and we highlight nonlinear effects of gravitational-wave memory, including cubic wave-zone phenomena analogous to horizon absorption effects.

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arXiv2026

Gravitational Waves from Green's Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge

We present a decomposition of the Kerr Green's function in the time domain, motivated by the frequency-domain split previously studied in the Schwarzschild limit. We show that the identification of a quasinormal-mode contribution, a direct part, and a late-time tail is still available, where the split times are determined by the black hole spin and positions of the emitter and receiver. We have checked this Green's function with time-domain Teukolsky numerical simulations and find excellent agreement. We also apply this decomposed Green's function in the time domain to a model problem with a test particle plunging into a Kerr black hole. The dynamically excited direct wave and quasinormal modes are obtained by convoluting the Green's function with the particle's source term, which may be viewed as the first order in mass ratio of a spinning black hole ringdown.

gr-qc↗

arXiv2026

High-order gravitational late-time tails in Kerr spacetime

We calculate high-order late-time tails of the retarded Green function of the Teukolsky equation for linear field perturbations of (subextremal) Kerr spacetime. We calculate these tails at a fixed spheroidal harmonic $\ell$ and azimuthal number $m$ up to the first three orders for the field point: at finite radius (away from the event horizon) for large Boyer-Lindquist time $t$; along the future event horizon $\mathscr{H}^+$ for large ingoing Eddington-Finkelstein coordinate $v$; and along future null infinity $\mathscr{I}^+$ for large outgoing Eddington-Finkelstein coordinate $u$. We obtain the tail powers for generic integer field spin $s$ and the tail coefficients specifically for gravitational ($s=-2$) perturbations. Our asymptotics include the known leading power-law (generic) tails, respectively,$t^{-2\ell-3}$, $e^{imΩ_H v}v^{-2\ell-3-b}$ (where $b=1$ for $s>0, m=0$ and $b=0$ otherwise, and where $Ω_H$ is the angular velocity of the event horizon) and $u^{-\ell+s-2}$, as well as their higher-order logarithmic corrections: $t^{-2\ell-5}\ln t$, $e^{imΩ_H v}v^{-2\ell-5-b}\ln v$ and $u^{-\ell+s-3}\ln u$ (as well as $u^{-\ell+s-4}\ln^2 u$). Since we obtain the high-order expansions for modes for generic $\ell$ and $m$, we can readily infer the explicit expansions of the {\it full} retarded Green function for $s=-2$ (and its decay powers for generic integer $s$). We obtain the late-time asymptotics from small-frequency expansions of the Fourier modes of the retarded Green function in the frequency domain. Accordingly, we also provide small-frequency expansions of various quantities of interest in the scattering theory. We also attach two notebooks which provide expansions for specific values of $s$: one notebook provides them to the first three leading orders for generic $\ell$ and the other one to arbitrary order for specific values of $\ell$.

gr-qc↗

arXiv2026

Quantum fluxes and $\langle\hatΦ^2\rangle$ for a non-minimally coupled scalar field: ringdown and tail on approaching the polar Kerr inner horizon

We compute $\langle\hatΦ^{2}\rangle_\text{ren}$ as well as the energy fluxes $\langle \hat{T}_{uu}\rangle_\text{ren}$ and $\langle \hat{T}_{vv}\rangle_\text{ren}$ (where $u$ and $v$ are the standard Eddington-Finkelstein coordinates) associated with a quantum massless real scalar field $\hatΦ$, with a general curvature coupling constant $ξ$, near the inner horizon (IH) of a Kerr black hole, along the axis of rotation. The quantum field is in the Unruh state, corresponding to an evaporating black hole. We renormalize these quantities by the state-subtraction method. We drop the assumption of minimal coupling to the curvature, thereby generalizing the results of arXiv:2203.08502 for the fluxes at the IH. This requires understanding the asymptotic behavior of $\langle\hatΦ^{2}\rangle_\text{ren}$ neat the IH. State subtraction allows us to push the computation of $\langle\hatΦ^{2}\rangle_\text{ren}$ along the axis of rotation in the Kerr interior in arXiv:2409.17464 deeper into the near-IH region, exposing their final asymptotic behavior on approaching the IH. For $\langle\hatΦ^{2}\rangle_\text{ren}$ (a $ξ$-independent quantity in the Kerr case), we find that the approach to its finite asymptotic IH value is given, per $\ell$-mode, by a ringdown phase (namely exponentially damped oscillations), followed by an inverse-power tail, both in the tortoise coordinate $r_{*}$ (which diverges at the IH). Interestingly, in the regime where the ringing dominates, the ringing's complex frequencies are (numerically) found to match twice the well-known classical quasinormal-mode frequencies in Kerr, and the inverse-power tails are found to be $r_{*}^{-2\ell-3}$ (resembling Price's law in the classical black hole exterior, upon replacement $t\to r_*$). [Abridged]

gr-qc↗

arXiv2026

Waveform Modelling for the Laser Interferometer Space Antenna

LISA, the Laser Interferometer Space Antenna, will usher in a new era in gravitational-wave astronomy. As the first anticipated space-based gravitational-wave detector, it will expand our view to the millihertz gravitational-wave sky, where a spectacular variety of interesting new sources abound: from millions of ultra-compact binaries in our Galaxy, to mergers of massive black holes at cosmological distances; from the beginnings of inspirals that will venture into the ground-based detectors' view to the death spiral of compact objects into massive black holes, and many sources in between. Central to realising LISA's discovery potential are waveform models, the theoretical and phenomenological predictions of the pattern of gravitational waves that these sources emit. This white paper is presented on behalf of the Waveform Working Group for the LISA Consortium. It provides a review of the current state of waveform models for LISA sources, and describes the significant challenges that must yet be overcome.

gr-qc↗

arXiv2026

Calculation of a regularized Teukolsky Green function in Schwarzschild spacetime

We obtain exact expressions for various factors involved in the Hadamard form of the retarded Green function for the (Bardeen-Press-)Teukolsky equation on Schwarzschild spacetime. We use these to improve on previous results for the calculation of this Green function. We work in a spacetime $\mathcal{M}_2\times\mathbb{S}^2$ conformal to Schwarzschild, in which the metric takes a direct product form. This allows us to derive a separable form for the direct (i.e., singular) part of the Hadamard form of the retarded Green function. The angular factor in this quantity is calculated explicitly. This shows an interesting interplay between geodesics of $\mathbb{S}^2$, spin-weighted spherical harmonics, and Euler angles. The $\mathcal{M}_2$ factor equates to a spin-dependent factor that satisfies a transport equation along geodesics, times the square root of the van Vleck determinant. Both terms are calculated in an exact form for constant radius orbits (which includes the cases of circular timelike geodesics and static worldlines of Schwarzschild spacetime). This separable form also allows us to obtain the multipolar $\ell$-modes of the direct part for electromagnetic and gravitational field perturbations. We then use these $\ell$-modes to calculate, in the gravitational case, the retarded Green function minus its direct part: this is a better representation in practise of the retarded Green function for points near coincidence.

gr-qc↗

arXiv2026

Horizon Multipole Moments of a Kerr Black Hole

The horizon multipole moments of a Kerr black hole are computed from two distinct definitions that have been proposed in the literature. The first one [Ashtekar et al., Class. Quantum Grav. 21, 2549 (2004)] regards axisymmetric isolated horizons, while the second one [Ashtekar et al., J. High Energ. Phys. 2022, 28 (2022)] applies to generic (i.e., not necessarily axisymmetric) non-expanding horizons. We review these definitions in a common frame and perform a detailed study of the resulting multipole moments for the Kerr event horizon. The horizon multipoles are found to share several properties with the (Hansen) field multipoles, including parity constraints and the leading scaling behavior with respect to the Kerr spin parameter a in the regime of small a. For the axisymmetry-based definition, we have obtained a closed-form expression of the multipole moments in terms of a and the spherical harmonic degree l. For the generic definition, we have established closed-form expressions for the conformal unit round metric, the `electric' and `magnetic' potentials related to the multipoles, and the values of the multipoles in the small a limit. We show that the two definitions lead to different values of the Kerr horizon multipoles as soon as l >= 1 (generic nonzero value of a) or l >= 2 (small a limit).

gr-qc↗

arXiv2026

Decomposition of Schwarzschild Green's Function

We present a formulation of the spherically decomposed Green's function for a Schwarzschild black hole, based on a decomposition into two components, $G^+$ and $G^-$, based on their large-frequency behaviour. While similar decompositions have been considered previously, here we systematically apply it to Schwarzschild spacetime and analyze its implications for the analytic structure of the Green's function in the complex-frequency plane. We show that both $G^+$ and $G^-$ possess branch cuts along the imaginary axis, which give rise to the direct part and the late-time tail, while the poles of $G^+$ correspond to the quasinormal mode spectrum. This allows us to identify a $\textit{branch-cut direct part}$, a quasinormal-mode contribution, and a late-time tail through contours adapted to different causal spacetime regions. This is in sharp contrast to Leaver's original formulation, where the prompt response is tied to a technically difficult large-arc contribution. We validate our decomposition with independent time-domain Regge-Wheeler simulations finding excellent agreement. Our results provide a practical and physically transparent framework for disentangling the distinct pieces of the Schwarzschild response, and offer a natural starting point for extensions to Kerr perturbations and non-linear ringdown physics.

gr-qc↗

arXiv2026

Green functions of the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime

We present a calculation of the full retarded Green functions of the Regge-Wheeler and Teukolsky equations obeyed by gravitational field perturbations of Schwarzschild spacetime. We perform the calculations for spacetime points along: (i) a timelike circular geodesic (where null-separated points are not at caustics); and (ii) a static worldline (where null-separated points are at caustics). These Green functions show a 4-fold singularity structure away from caustics, and 2-fold at caustics (similarly to the case of scalar field perturbations, which we also reproduce). Physical oscillations near the singularities appear in the gravitational case, which were not present in the scalar case. We obtain our results by developing various numerical and analytical methods.

gr-qc↗

arXiv2025

Spin-2 Green's Functions on Kerr in Radiation Gauge

We construct retarded and advanced Green's functions for gravitational perturbations in Kerr in an ingoing radiation gauge. Our Green's functions have a frequency domain piece that has previously been obtained by Ori [Phys. Rev. D 67 (2003)] based on the Chrzanowski-Cohen-Kegeles metric reconstruction method. As is well known, this piece by itself is not sufficient to obtain an actual Green's function. We show how to complete it with a piece based on a method by Green et al. [Class. Quant. Grav. 37 (2020)]. The completion piece has a completely explicit form in the time-domain and is supported on pairs of points on the same outgoing principal null geodesic which are in the appropriate causal order. We expect our Green's functions to be useful for gravitational self-force calculations and other perturbation problems on Kerr spacetime.

gr-qc↗

arXiv2025

Computation of $\langle Φ^2\rangle$ and quantum fluxes at the polar interior of a spinning black hole

Renormalization of physical quantities for quantum field theories in curved spacetimes can be achieved via the consistent subtraction of counterterms within a regularization scheme such as a point-splitting method. Pragmatic mode-sum regularization (PMR) is a point-splitting method which is particularly suitable for rotating black hole spacetimes. We extend and tailor the t-splitting variant of PMR specifically for the interior of a Kerr black hole on the axis of rotation, focusing on a minimally-coupled massless scalar field in the physically-motivated Unruh state. The method addresses unique challenges in the black hole interior that do not occur outside. In particular, while the infinite sum over multipolar number l converges in the black hole exterior, it diverges in the interior, necessitating the subtraction of a so-called intermediate divergence which includes introducing an additional "small" split in the direction of the polar angle. This procedure is outlined and justified, along with the standard PMR method's counterterms subtraction. We apply this method to calculate the renormalized energy-momentum fluxes $\langle T_{uu}\rangle^U_\text{ren}$, $\langle T_{vv}\rangle^U_\text{ren}$ (where u and v are the standard Eddington coordinates) and the renormalized field square $\langle Φ^2\rangle^U_\text{ren}$ throughout the black hole interior, spanning from (just off) the event horizon to (just off) the inner horizon. Special emphasis is placed on the inner horizon vicinity, where our t-splitting results for the fluxes asymptote to those obtained directly at the inner horizon using a different method in a previous work. In an Appendix, we develop an alternative t-splitting PMR variant which does not include the intermediate divergence subtraction. We utilize it for independent computations that are used to verify the standard t-splitting variant presented in the main text.

gr-qc↗

arXiv2024

Implementation of a GHZ-Teukolsky puncture scheme for gravitational self-force calculations

Post-adiabatic models of extreme- and intermediate-mass-ratio inspirals will require calculations of second-order gravitational self-force effects in the spacetime of a spinning, Kerr black hole. We take a step toward such calculations by implementing the recently formulated Teukolsky puncture scheme with Green-Hollands-Zimmerman metric reconstruction [CQG 39, 015019 (2022)]. This scheme eliminates the critical obstacle of gauge singularities that arise in the standard no-string metric reconstruction. Our first proof-of-principle implementation is limited to the simple case of circular orbits in Schwarzschild spacetime, but the method also applies to generic orbits on a Kerr background. We conclude with a discussion of various approaches to the second-order self-force problem in Kerr.

gr-qc↗

arXiv2024

Infinite quantum twisting at the Cauchy horizon of rotating black holes

We present a numerical calculation of the expectation value of the quantum angular-momentum current flux density for a scalar field in the Unruh state near the inner horizon of a Kerr-de Sitter black hole. Our results indicate that this flux diverges as $V_-^{-1}$ in a suitable Kruskal coordinate such that $V_-=0$ at the inner horizon. Depending on the parameter values of the scalar field and black hole that we consider, and depending on the polar angle (latitude), this flux can have different signs. In the near extremal cases considered, the angle average of the expectation value of the quantum angular momentum current flux is of the opposite sign as the angular momentum of the background itself, suggesting that, in the cases considered, quantum effects tend to decrease the total angular momentum of the spheres away from the extremal value. We also numerically calculate the energy flux component, which provides the leading order divergence of the quantum stress energy tensor, dominant over the classical stress energy tensor, at the inner horizon. Taking our expectation value of the quantum stress tensor as the source in the semiclassical Einstein equation, our analysis suggests that the spheres approaching the inner horizon can undergo an infinite twisting due to quantum effects along latitudes separating regions of infinite expansion and contraction.

gr-qc↗

arXiv2023

Global Hadamard form for the Green Function in Schwarzschild space-time

The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic. The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point. In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood. We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution. (The term non-smooth applies modulo the causality-generating step functions that must appear in the retarded Green function.) The singularity structure is determined using two independent approaches, one based on a Bessel function expansion of the Green function, and another that exploits a link between the Green functions of Schwarzschild spacetime and Pleba{ń}ski-Hacyan spacetime (the latter approach also yields another representation for the {\it full} Schwarzschild Green function, not just for its non-smooth part). Our representation is not valid in a neighbourhood of caustic points. We deal with these points by providing a separate representation for the Green function in Schwarzschild spacetime which makes explicit its (different) singularity structure at caustics of this spacetime.

gr-qc↗

arXiv2023

Lensing of Vacuum Entanglement near Schwarzschild Black Holes

An important feature of Schwarzschild spacetime is the presence of orbiting null geodesics and caustics. Their presence implies strong gravitational lensing effects for matter and radiation, i.e., for excitations of quantum fields. Here, we raise the question whether the lensing manifests itself also in the vacuum of quantum fields, namely by lensing the distribution of vacuum entanglement. To explore this possibility, we use the method of entanglement harvesting, where initially unentangled localized quantum systems are temporarily coupled to the field at different locations. We find that for the Boulware, Hartle-Hawking and Unruh vacua in 3+1 dimensional Schwarzschild spacetime, the harvesting of vacuum entanglement is indeed greatly amplified near caustics. In particular, we establish that pre-existing vacuum entanglement can be harvested also for lightlike separations.

quant-ph↗

arXiv2023

Hidden spectral symmetries and mode stability of subextremal Kerr(-dS) black holes

We uncover hidden spectral symmetries of the Teukolsky equation in Kerr(-de Sitter) black holes, recently conjectured by Aminov, Grassi and Hatsuda (arXiv: 2006.06111 and arXiv: 2007.07906, 2020) in the zero cosmological constant case. Using these symmetries, we provide a new, simpler proof of mode stability for subextremal Kerr black holes. We also present a partial mode stability result for Kerr-de Sitter black holes.

gr-qc↗