arXiv subjects

Elena Giorgi

Publications and source records attributed to Elena Giorgi.

At least 19 recordsLinked to original sources

arXiv2026

Wave decay and horizon instability on strongly charged extremal Kerr-Newman black holes

We prove the first boundedness and pointwise decay result for the scalar wave equation on rotating extremal black holes without any symmetry assumptions. The result applies to slowly rotating (equivalently, strongly charged) extremal Kerr-Newman spacetimes. We establish uniform energy boundedness, integrated local energy decay, and a hierarchy of boundary-weighted estimates at the extremal horizon and at null infinity, from which inverse-polynomial pointwise decay follows in the entire exterior region. As a consequence, we also prove the expected Aretakis instability: for generic initial data, suitable transversal derivatives fail to decay along the event horizon, and higher transversal derivatives blow up asymptotically. The proof uses the $b$-structure of the wave operator near the two boundary hypersurfaces, together with a treatment of normally hyperbolic trapping on extremal Kerr--Newman.

math.AP↗

arXiv2025

Cauchy data for multiple collapsing boson stars

We construct Cauchy initial data for the Einstein-Maxwell-Klein-Gordon (EMKG) system, which evolves in finite time into spacetimes containing multiple trapped surfaces. From a physical perspective, this corresponds to preparing multiple well-separated boson stars, each of which collapses to form a spacelike black hole region. In particular, this extends the result of the second and third named authors on the formation of multiple trapped surfaces in vacuum to the EMKG system.

gr-qc↗

arXiv2025

Mass-Centered GCM Framework in Perturbations of Kerr(-Newman)

The nonlinear stability problem for black hole solutions of the Einstein equations critically depends on choosing an appropriate geometric gauge. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In this work, we develop an alternative GCM framework, that we call mass-centered, designed to overcome the breakdown of the standard GCM construction in the charged case, where electromagnetic-gravitational coupling destroys the exceptional behavior of the $\ell=1$ mode of the center-of-mass quantity used in the vacuum analysis. This construction is aimed at the nonlinear stability of Reissner-Nordstr\"om and Kerr-Newman spacetimes. Our approach replaces transport-based control of the center-of-mass quantity with a sphere-wise vanishing condition on a renormalized $\ell=1$ mode, yielding mass-centered GCM hypersurfaces with modified gauge constraints. The resulting elliptic-transport system remains determined once an $\ell=1$ basis is fixed via effective uniformization and provides an alternative construction in vacuum in the uncharged limit.

gr-qc↗

arXiv2025

Einstein-Maxwell Equations on Mass-Centered GCM Hypersurfaces

The resolution of the nonlinear stability of black holes as solutions to the Einstein equations relies crucially on imposing the right geometric gauge conditions. In the vacuum case, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has been successful in the proof of stability for slowly rotating Kerr spacetime. For the charged setting, our companion paper introduced an alternative mass-centered GCM framework, adapted to the additional difficulties of the Einstein-Maxwell system. In this work, we solve the Einstein-Maxwell equations on such a mass-centered spacelike GCM hypersurface, which is equivalent to solving the constraint equations there. We control all geometric quantities of the solution in terms of some seed data, corresponding to the gauge-invariant fields describing coupled gravitational-electromagnetic radiation in perturbations of Reissner-Nordstr\"om or Kerr-Newman, first identified by the second author and expected to be governed by favorable hyperbolic equations. This provides the first step toward controlling gauge-dependent quantities in the nonlinear stability analysis of the Reissner-Nordstr\"om and Kerr-Newman families.

gr-qc↗

arXiv2025

Conservation Laws and Boundedness for Linearised Einstein--Maxwell Equations on the Reissner--Nordström Black Hole

We study the linearised Einstein--Maxwell equations on the Reissner--Nordström spacetime and derive the canonical energy conservation law in double null gauge. In the spirit of the work of Holzegel and the second author, we avoid any use of the hyperbolic nature of the Teukolsky equations and rely solely on the conservation law to establish control of energy fluxes for the gauge-invariant Teukolsky variables, previously identified by the third author, along all outgoing null hypersurfaces, for charge-to-mass ratio $\frac{|Q|}{M} < \frac{\sqrt{15}}{4}$. This yields uniform boundedness for the Teukolsky variables in Reissner--Nordström.

gr-qc↗

arXiv2024

Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis \textit{exclusively in physical-space}. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.

math.AP↗

arXiv2024

Boundedness and Decay for the Teukolsky System in Kerr-Newman Spacetime II: The Case $|a| \ll M$, $|Q| <M$ in Axial Symmetry

We establish boundedness and polynomial decay results for the Teukolsky system in the exterior spacetime of very slowly rotating and strongly charged sub-extremal Kerr-Newman black holes, with a focus on axially symmetric solutions. The key step in achieving these results is deriving a physical-space Morawetz estimate for the associated generalized Regge-Wheeler system, without relying on spherical harmonic decomposition.

gr-qc↗

arXiv2023

Boundedness and Decay for the Teukolsky System in Kerr-Newman Spacetime I: The Case $|a|, |Q| \ll M$

We prove boundedness and polynomial decay statements for solutions of the Teukolsky system for electromagnetic-gravitational perturbations of a Kerr-Newman exterior background, with parameters satisfying $|a|, |Q| \ll M$. The identification and analysis of the Teukolsky system in Kerr-Newman has long been problematic due to the long-standing problem of coupling and failure of separability of the equations. Here, we analyze a system satisfied by novel gauge-invariant quantities representing gravitational and electromagnetic radiations for coupled perturbations of a Kerr-Newman black hole. The bounds are obtained by making use of a generalization of the Chandrasekhar transformation into a system of coupled generalized Regge-Wheeler equations derived in previous work. Crucial in our resolution is the use of a combined energy-momentum tensor for the system which exploits its symmetric structure, performing an effective decoupling of the perturbations. As for other black hole solutions, such bounds on the Teukolsky system provide the first step in proving the non linear stability of the Kerr-Newman metric to gravitational and electromagnetic perturbations.

gr-qc↗

arXiv2022

Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes

This is the last part of our proof of the nonlinear stability of the Kerr family for small angular momentum, i.e $|a|/m\ll 1$, in which we deal with the nonlinear wave type estimates needed to complete the project. More precisely we provide complete proofs for Theorems M1 and M2 as well the curvature estimates of Theorem M8, which were stated without proof in sections 3.7.1 and 9.4.7 of \cite{KS:Kerr}. Our procedure is based on a new general interest formalism (detailed in Part I of this work), which extends the one used in the stability of Minkowski space. Together with \cite{KS:Kerr} and the GCM papers \cite{KS-GCM1}, \cite{KS-GCM2}, \cite{Shen}, this work completes proof of the Main Theorem stated in Section 3.4 of \cite{KS:Kerr}.

math.AP↗

arXiv2021

The linear stability of Reissner-Nordström spacetime: the full subextremal range

We prove the linear stability of subextremal Reissner-Nordström spacetimes as solutions to the Einstein-Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations separately derived in previous works, where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range |Q|<M, completely in physical space. Such decay estimates, together with the estimates for the gauge-dependent quantities of the perturbations previously obtained, settle the problem of linear stability to gravitational and electromagnetic perturbations of Reissner-Nordström solution in the full subextremal range |Q|< M.

gr-qc↗

arXiv2021

The Carter tensor and the physical-space analysis in perturbations of Kerr-Newman spacetime

The Carter tensor is a Killing tensor of the Kerr-Newman spacetime, and its existence implies the separability of the wave equation. Nevertheless, the Carter operator is known to commute with the D'Alembertian only in the case of a Ricci-flat metric. We show that, even though the Kerr-Newman spacetime satisfies the non-vacuum Einstein-Maxwell equations, its curvature and electromagnetic tensors satisfy peculiar properties which imply that the Carter operator still commutes with the wave equation. This feature allows to adapt to Kerr-Newman the physical-space analysis of the wave equation in Kerr by Andersson-Blue, which avoids frequency decomposition of the solution by precisely making use of the commutation with the Carter operator. We also extend the mathematical framework of physical-space analysis to the case of the Einstein-Maxwell equations on Kerr-Newman spacetime, representing coupled electromagnetic-gravitational perturbations of the rotating charged black hole. The physical-space analysis is crucial in this setting as the coupling of spin-1 and spin-2 fields in the axially symmetric background prevents the separation in modes as observed by Chandrasekhar, and therefore represents an important step towards an analytical proof of the stability of the Kerr-Newman black hole.

gr-qc↗

arXiv2021

Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric

Motivated by gravitational wave observations of binary black hole mergers, we present a procedure to compute the leading order nonlinear gravitational wave interactions around a Kerr black hole. We describe the formalism used to derive the equations for second order perturbations. We develop a procedure that allows us to reconstruct the first order metric perturbation solely from knowledge of the solution to the first order Teukolsky equation, without the need of Hertz potentials. Finally, we illustrate this metric reconstruction procedure in the asymptotic limit for the first order quasi-normal modes of Kerr. In a companion paper, we present a numerical implementation of these ideas.

gr-qc↗

arXiv2021

Numerical computation of second order vacuum perturbations of Kerr black holes

Motivated by the desire to understand the leading order nonlinear gravitational wave interactions around arbitrarily rapidly rotating Kerr black holes, we describe a numerical code designed to compute second order vacuum perturbations on such spacetimes. A general discussion of the formalism we use is presented in (arXiv:2008.11770); here we show how we numerically implement that formalism with a particular choice of coordinates and tetrad conditions, and give example results for black holes with dimensionless spin parameters $a=0.7$ and $a=0.998$. We first solve the Teukolsky equation for the linearly perturbed Weyl scalar $Ψ_4^{(1)}$, followed by direct reconstruction of the spacetime metric from $Ψ_4^{(1)}$, and then solve for the dynamics of the second order perturbed Weyl scalar $Ψ_4^{(2)}$. This code is a first step toward a more general purpose second order code, and we outline how our basic approach could be further developed to address current questions of interest, including extending the analysis of ringdown in black hole mergers to before the linear regime, exploring gravitational wave "turbulence" around near-extremal Kerr black holes, and studying the physics of extreme mass ratio inspiral.

gr-qc↗

arXiv2020

Electromagnetic-gravitational perturbations of Kerr-Newman spacetime: the Teukolsky and Regge-Wheeler equations

We derive the equations governing the linear stability of Kerr-Newman spacetime to coupled electromagnetic-gravitational perturbations. The equations generalize the celebrated Teukolsky equation for curvature perturbations of Kerr, and the Regge-Wheeler equation for metric perturbations of Reissner-Nordström. Because of the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields", as put by Chandrasekhar, the stability of Kerr-Newman spacetime can not be obtained through standard decomposition in modes. Due to the impossibility to decouple the modes of the gravitational and electromagnetic fields, the equations governing the linear stability of Kerr-Newman have not been previously derived. Using a tensorial approach that was applied to Kerr, we produce a set of generalized Regge-Wheeler equations for perturbations of Kerr-Newman, which are suitable for the study of linearized stability by physical space methods. The physical space analysis overcomes the issue of coupling of spin-1 and spin-2 fields and represents the first step towards an analytical proof of the stability of the Kerr-Newman black hole.

math.AP↗

arXiv2020

Boundedness and decay for the Teukolsky system of spin $\pm2$ on Reissner-Nordström spacetime: the case $|Q| \ll M$

We prove boundedness and polynomial decay statements for solutions to the spin $\pm2$ generalized Teukolsky system on a Reissner-Nordström background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $α$. The second equation, coupled with the first one, is a new equation for a new gauge-invariant quantity involving the electromagnetic curvature components. The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild. These quantities verify a generalized coupled Regge-Wheeler system. These equations are the ones verified by the extreme null curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner-Nordström spacetime. Consequently, as in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of Reissner-Nordström metric for small charge to coupled gravitational and electromagnetic perturbations.

gr-qc↗

arXiv2020

The linear stability of Reissner-Nordström spacetime for small charge

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordström family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordström solution arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To obtain decay of the solution, one must add a residual pure gauge solution which is proved to be itself controlled from initial data. Our results rely on decay statements for the Teukolsky system of spin $\pm2$ and spin $\pm1$ satisfied by gauge-invariant null-decomposed curvature components, obtained in earlier works. These decays are then exploited to obtain polynomial decay for all the remaining components of curvature, electromagnetic tensor and Ricci coefficients. In particular, the obtained decay is optimal in the sense that it is the one which is expected to hold in the non-linear stability problem.

gr-qc↗

arXiv2020

A general formalism for the stability of Kerr

The goal of this paper is to provide a geometric framework for analyzing the uniform decay properties of solutions to the Teukolsky equation in the fully nonlinear setting of perturbations of Kerr. It contains the first nonlinear version of the Chandrasekhar transformation introduced in the linearized setting in \cite{D-H-R-Kerr} and \cite{Ma} with the intent to use it in our ongoing project to prove the full nonlinear stability of slowly rotating Kerr as solution to the Einstein vacuum equations.

math.AP↗

arXiv2019

Boundedness and decay for the Teukolsky equation of spin $\pm1$ on Reissner-Nordström spacetime: the $\ell=1$ spherical mode

We prove boundedness and polynomial decay statements for solutions to the spin $\pm1$ Teukolsky-type equation projected to the $\ell=1$ spherical harmonic on Reissner-Nordström spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor. This gives a first description in physical space of gauge-invariant quantities transporting the electromagnetic radiation in perturbations of a charged black hole. The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild by Dafermos-Holzegel-Rodnianski. The derived quantity verifies a Fackerell-Ipser-type equation, with right hand side vanishing at the $\ell=1$ spherical harmonics. The boundedness and decay for the projection to the $\ell\geq 2$ spherical harmonics are implied by the boundedness and decay for the Teukolsky system of spin $\pm2$ obtained in our previous work. The spin $\pm1$ Teukolsky-type equation is verified by the curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner-Nordström spacetime. Consequently, together with the estimates obtained in our previous work, these bounds allow to prove the full linear stability of Reissner-Nordström metric for small charge to coupled gravitational and electromagnetic perturbations.

gr-qc↗