arXiv subjects

Maxime Van de Moortel

Publications and source records attributed to Maxime Van de Moortel.

16 recordsLinked to original sources

arXiv2026

Decay of weakly charged solutions for the spherically symmetric Maxwell-Charged-Scalar-Field equations on a Reissner-Nordström exterior space-time

We consider the Cauchy problem for the (non-linear) Maxwell-Charged-Scalar-Field equations with spherically symmetric initial data, on a sub-extremal Reissner--Nordström or Schwarzschild exterior space-time. We prove that the solutions are bounded and decay at an inverse polynomial rate towards time-like infinity and along the black hole event horizon, provided the charge of the Maxwell equation is sufficiently small. This condition is in particular satisfied for small data in energy space that enjoy a sufficient decay towards the asymptotically flat end. Some of the decay estimates we prove are arbitrarily close to the conjectured optimal rate in the limit where the charge tends to zero, according the heuristics present in the physics literature. Our result can also be interpreted as a first step towards the stability of Reissner--Nordström black holes for the gravity coupled Einstein--Maxwell-Charged-Scalar-Field model. This problem is closely connected to the understanding of strong cosmic censorship and charged gravitational collapse in this setting.

gr-qc↗

arXiv2026

Asymptotically flat black holes with a singular Cauchy horizon and a spacelike singularity

In our recent work [Van de Moortel, The coexistence of null and spacelike singularities inside spherically symmetric black holes], we analyzed the transition between null and spacelike singularities in spherically symmetric dynamical black holes and demonstrated that the spacelike portion is described by a Kasner metric with positive varying exponents that degenerate to $(1,0,0)$ near the null-spacelike transition. In the present paper, we provide examples of global spacetimes satisfying the assumptions of this previous result and apply its analysis to obtain a large class of asymptotically flat (spherically symmetric) black hole spacetimes that exhibit coexisting null and spacelike singularities. Our main results include: _The construction of one-ended asymptotically flat black hole spacetimes solving the Einstein-Maxwell-charged-scalar-field equations. The proof relies on a new spacelike-characteristic gluing method between any uncharged spherically symmetric solution and the event horizon of a charged dynamical black hole. _The construction of a large class of two-ended asymptotically flat black hole spacetimes solving the Einstein-Maxwell-(uncharged)-scalar-field equations. In both cases, we show that the terminal boundary in the black hole interior only has two distinct components: a weakly singular (null) Cauchy horizon $\mathcal{CH}_{i^+}$ where curvature blows up and a strong singularity $\mathcal{S}=\{r=0\}$. Our construction provides the first examples of black holes with coexisting null and spacelike singularities. These examples hold particular significance in the one-ended case as a model of gravitational collapse, where this phenomenon is conjecturally generic for the Einstein-scalar-field model, even beyond spherical symmetry.

gr-qc↗

arXiv2025

The Strong Cosmic Censorship Conjecture

In the wake of major breakthroughs in General Relativity during the 1960s, Roger Penrose introduced Strong Cosmic Censorship, a profound conjecture regarding the deterministic nature of the theory. Penrose's proposal has since opened far-reaching new mathematical avenues, revealing connections to fundamental questions about black holes and the nature of gravitational singularities. We review recent advances arising from modern techniques in the theory of partial differential equations as applied to Strong Cosmic Censorship, maintaining a focus on the context of gravitational collapse that gave birth to the conjecture.

gr-qc↗

arXiv2025

The coexistence of null and spacelike singularities inside spherically symmetric black holes

In our previous work [Van de Moortel, The breakdown of weak null singularities, Duke Mathematical Journal 172 (15), 2957-3012, 2023], we showed that dynamical black holes formed in charged spherical collapse generically feature both a null weakly singular Cauchy horizon and a stronger (presumably spacelike) singularity, confirming a longstanding conjecture in the physics literature. However, this previous result, based on a contradiction argument, did not provide quantitative estimates on the stronger singularity. In this study, we adopt a new approach by analyzing local initial data inside the black hole that are consistent with a breakdown of the Cauchy horizon. We prove that the remaining portion is spacelike and obtain sharp spacetime estimates near the null-spacelike transition. Notably, we show that the Kasner exponents of the spacelike portion are positive, in contrast to the well-known Oppenheimer-Snyder model of gravitational collapse. Moreover, these exponents degenerate to (1,0,0) towards the null-spacelike transition. Our result provides the first quantitative instances of a null-spacelike singularity transition inside a black hole. In our companion paper, we moreover apply our analysis to carry out the construction of a large class of asymptotically flat one or two-ended black holes featuring coexisting null and spacelike singularities.

gr-qc↗

arXiv2025

An extension of the $r^p$ method for wave equations with scale-critical potentials and first-order terms

The $r^p$ method, first introduced in [DR10], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Grönwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [VdM22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.

math.AP↗

arXiv2025

Kasner bounces and fluctuating collapse inside hairy black holes with charged matter

We study the interior of black holes in the presence of charged scalar hair of small amplitude $ε$ on the event horizon and show their terminal boundary is a crushing Kasner-like singularity. These spacetimes are spatially homogeneous and they differ significantly from the hairy black holes with uncharged matter previously studied in [M. Van de Moortel, Violent nonlinear collapse inside charged hairy black holes, Arch. Rational. Mech. Anal., 248, 89, 2024] in that the electric field is dynamical and subject to the backreaction of charged matter. This charged backreaction causes drastically different dynamics compared to the uncharged case that impact the formation of the spacelike singularity, exhibiting novel phenomena such as - Collapsed oscillations: oscillatory growth of the scalar hair, nonlinearly induced by the collapse. - A fluctuating collapse: The final Kasner exponents' dependency in $ε$ is via an expression of the form $|\sin\left(ω_0 \cdot ε^{-2}+ O(\log (ε^{-1}))\right)|$. - A Kasner bounce: a transition from an unstable Kasner metric to a different stable Kasner metric. The Kasner bounce occurring in our spacetime is reminiscent of the celebrated BKL scenario in cosmology. We additionally propose a construction indicating the relevance of the above phenomena -- including Kasner bounces -- to spacelike singularities inside more general (asymptotically flat) black holes, beyond the hairy case. While our result applies to all values of $Λ\in \mathbb{R}$, in the $Λ<0$ case, our spacetime corresponds to the interior region of a charged asymptotically Anti-de-Sitter stationary black hole, also known as a holographic superconductor, and whose exterior region was rigorously constructed in the recent mathematical work [W. Zheng, Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes, arXiv.2410.04758].

gr-qc↗

arXiv2024

Violent nonlinear collapse in the interior of charged hairy black holes

We construct a new one-parameter family indexed by $ε$ of two-ended, spatially-homogeneous black hole interiors solving the Einstein-Maxwell-Klein-Gordon equations with a (possibly zero) cosmological constant $Λ$ and bifurcating off a Reissner-Nordström-(dS/AdS) interior ($ε= 0$). For all small $ε\neq 0$, we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelike Kasner singularity foliated by spheres of zero radius $r$. Moreover, smaller perturbations (i.e. smaller $|ε|$) are more singular than larger one, in the sense that the Hawking mass and the curvature blow up following a power law of the form $r^{-O(ε^{-2})}$ at the singularity $\{r=0\}$. This unusual property originates from a dynamical phenomenon -- violent nonlinear collapse -- caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity $\{r=0\}$. This phenomenon was previously described numerically in the physics literature and referred to as "the collapse of the Einstein-Rosen bridge". While we cover all values of $Λ\in \mathbb{R}$, the case $Λ< 0$ is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes.

gr-qc↗

arXiv2024

Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity

We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential. We present a novel, geometric, physical-space approach for determining late-time asymptotics, based around embedding Minkowski spacetime conformally into the spacetime $AdS_2 \times \mathbb{S}^2$ (with $AdS_2$ the two-dimensional anti de-Sitter spacetime) to turn a global late-time asymptotics problem into a local existence problem for the wave equation in $AdS_2 \times \mathbb{S}^2$. Our approach is inspired by the treatment of the near-horizon geometry of extremal black holes in the physics literature. We moreover apply our method to another scale-invariant model: the (complex-valued) charged wave equation on Minkowski spacetime in the presence of a static electric field, which can be viewed as a simplification of the charged Maxwell-Klein-Gordon equations on a black hole spacetime.

math.AP↗

arXiv2024

The asymptotics of massive fields on stationary spherically symmetric black holes for all angular momenta

We study the massive scalar field equation $\Box_g ϕ= m^2 ϕ$ on a stationary and spherically symmetric black hole $g$ (including in particular the Schwarzschild and Reissner--Nordström black holes in the full sub-extremal range) for solutions $ϕ$ projected on a fixed spherical harmonic. Our problem involves the scattering of an attractive long-range potential (Coulomb-like) and thus cannot be treated perturbatively. We prove precise (point-wise) asymptotic tails of the form $t^{-5/6} f(t)+ O(t^{-1+δ})$, where $f(t)$ is an explicit oscillating profile. Our asymptotics appear to be the first rigorous decay result for a massive scalar field on a black hole. Establishing these asymptotics is also an important step in retrieving the assumptions used in work of the third author regarding the interior of dynamical black holes and Strong Cosmic Censorship.

gr-qc↗

arXiv2023

Polynomial time decay for solutions of the Klein--Gordon equation on a subextremal Reissner--Nordström black hole

We consider the massive scalar field equation $\Box_{g_{RN}} ϕ= m^2 ϕ$ on any subextremal Reissner--Nordström exterior metric $g_{RN}$. We prove that solutions with localized initial data decay pointwise-in-time at the polynomial rate $t^{-\frac{5}{6}+δ}$ in any spatially compact region (including the event horizon), for some small $ δ\leq \frac{1}{23} $. Moreover, assuming the validity of the Exponent Pair Conjecture on exponential sums in Number Theory, our result implies that decay upper bounds hold at the rate $t^{-\frac{5}{6}+ε}$, for any arbitrarily small $ε>0$. In our previous work, we proved that each fixed angular mode decays at the exact rate $t^{-\frac{5}{6}}$, thus the upper bound $t^{-\frac{5}{6}+ε}$ is sharp, up to a $t^ε$ loss. Without the restriction to a fixed angular mode, the solution turns out to have an unbounded Fourier transform due to discrete frequencies associated to quasimodes, and caused by the occurrence of stable timelike trapping. Our analysis nonetheless shows that inverse-polynomial asymptotics in $t$ still hold after summing over all angular modes.

gr-qc↗

arXiv2023

Nonlinear interaction of three impulsive gravitational waves II: the wave estimates

This is the second and last paper of a series aimed at solving the local Cauchy problem for polarized $\mathbb U(1)$ symmetric solutions to the Einstein vacuum equations featuring the nonlinear interaction of three small amplitude impulsive gravitational waves. Such solutions are characterized by their three singular "wave-fronts" across which the curvature tensor is allowed to admit a delta singularity. Under polarized $\mathbb U(1)$ symmetry, the Einstein vacuum equations reduce to the Einstein-scalar field system in $(2+1)$ dimensions. In this paper, we focus on the wave estimates for the scalar field in the reduced system. The scalar field terms are the most singular ones in the problem, with the scalar field only being Lipschitz initially. We use geometric commutators to prove energy estimates which reflect that the singularities are localized, and that the scalar field obeys additional fractional-derivative regularity, as well as regularity along appropriately defined "good directions". The main challenge is to carry out all these estimates using only the low-regularity properties of the metric. Finally, we prove an anisotropic Sobolev embedding lemma, which when combined with our energy estimates shows that the scalar field is everywhere Lipschitz, and that it obeys additional $C^{1,θ}$ estimates away from the most singular region.

gr-qc↗

arXiv2022

Strong Cosmic Censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry

Motivated by the Strong Cosmic Censorship Conjecture in the presence of matter, we study the Einstein equations coupled with a charged/massive scalar field with spherically symmetric characteristic data relaxing to a Reissner-Nordström event horizon. Contrary to the vacuum case, the relaxation rate is conjectured to be slow (non-integrable), opening the possibility that the matter fields and the metric coefficients blow up in amplitude at the Cauchy horizon, not just in energy. We show that whether this blow-up in amplitude occurs or not depends on a novel oscillation condition on the event horizon which determines whether or not a resonance is excited dynamically. If the oscillation condition is satisfied, then the resonance is not excited and we show boundedness and continuous extendibility of the matter fields and the metric across the Cauchy horizon. If the oscillation condition is violated, then by the combined effect of slow decay and the resonance being excited, we show that the massive uncharged scalar field blows up in amplitude. In our companion paper, we show that in the latter case a novel null contraction singularity forms at the Cauchy horizon, across which the metric is not continuously extendible in the usual sense. Heuristic arguments in the physics literature indicate that the oscillation condition should be satisfied generically on the event horizon. If these heuristics are true, then our result falsifies the $C^0$-formulation of Strong Cosmic Censorship by means of oscillations.

gr-qc↗

arXiv2022

The breakdown of weak null singularities inside black holes

It is widely expected that generic black holes have a non-empty but weakly singular Cauchy horizon, due to mass inflation. Indeed this has been proven by the author in the spherical collapse of a charged scalar field, under decay assumptions of the field in the black exterior which are conjectured to be generic. A natural question then arises: can this weakly singular Cauchy horizon close off the space-time, or does the weak null singularity necessarily "break down", giving way to a different type of singularity? The main result of this paper is to prove that the Cauchy horizon cannot ever "close off" the space-time. As a consequence, the weak null singularity breaks down and transitions to a different singularity for which the area-radius $r$ extends to $0$.

gr-qc↗

arXiv2021

Nonlinear interaction of three impulsive gravitational waves I: main result and the geometric estimates

Impulsive gravitational waves are (weak) solutions to the Einstein vacuum equations such that the Riemann curvature tensor admits a delta singularity along a null hypersurface. The interaction of impulsive gravitational waves is then represented by the transversal intersection of these singular null hypersurfaces. This is the first of a series of two papers in which we prove that for all suitable $\mathbb U(1)$-symmetric initial data representing three "small amplitude" impulsive gravitational waves propagating towards each other transversally, there exists a local solution to the Einstein vacuum equations featuring the interaction of these waves. Moreover, we show that the solution remains Lipschitz everywhere and is $H^2_{loc} \cap C_{loc}^{1, \frac 14-}$ away from the impulsive gravitational waves. This is the first construction of solutions to the Einstein vacuum equations featuring the interaction of three impulsive gravitational waves. In this paper, we focus on the geometric estimates, i.e. we control the metric and the null hypersurfaces assuming the wave estimates. The geometric estimates rely crucially on the features of the spacetime with three interacting impulsive gravitational waves, particularly that each wave is highly localized and that the waves are transversal to each other. In the second paper of the series, we will prove the wave estimates and complete the proof.

gr-qc↗

arXiv2020

Mass inflation and the $C^2$-inextendibility of spherically symmetric charged scalar field dynamical black holes

It has long been suggested that the Cauchy horizon of dynamical black holes is subject to a weak null singularity, under the mass inflation scenario. We study in spherical symmetry the Einstein-Maxwell-Klein-Gordon equations and \textit{while we do not directly show mass inflation}, we obtain a "mass inflation/ridigity" dichotomy. More precisely, we prove assuming (sufficiently slow) decay of the charged scalar field on the event horizon, that the Cauchy horizon emanating from time-like infinity is $\mathcal{CH}_{i^+}= \mathcal{D} \cup \mathcal{S}$ for two (possibly empty) disjoint connected sets $\mathcal{D}$ and $\mathcal{S}$ such that: _$\mathcal{D}$ (the dynamical set) is a past set on which the Hawking mass blows up (mass inflation scenario). _$\mathcal{S}$ (the static set) is a future set isometric to a Reissner--Nordström Cauchy horizon i.e.\ the radiation is zero on $\mathcal{S}$. As a consequence, we establish a novel classification of Cauchy horizons into three types: dynamical ($\mathcal{S}=\emptyset$), static ($\mathcal{D}=\emptyset$) or mixed, and prove that $\mathcal{CH}_{i^+}$ is globally $C^2$-inextendible. Our main motivation is the $C^2$ Strong Cosmic Censorship Conjecture for a realistic model of spherical collapse in which charged matter emulates the repulsive role of angular momentum: in our case the Einstein-Maxwell-Klein-Gordon system on one-ended space-times. As a result, we prove in spherical symmetry that: - two-ended asymptotically flat space-times are $C^2$-future-inextendible i.e. $C^2$ Strong Cosmic Censorship is true for Einstein-Maxwell-Klein-Gordon, assuming the decay of the scalar field on the event horizon at the expected rate. - In the one-ended case, the Cauchy horizon emanating from time-like infinity is $C^2$-inextendible. This result suppresses the main obstruction to $C^2$ Strong Cosmic Censorship in spherical collapse.

gr-qc↗

arXiv2017

Stability and instability of the sub-extremal Reissner-Nordström black hole interior for the Einstein-Maxwell-Klein-Gordon equations in spherical symmetry

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole -approaching a sub-extremal Reissner-Nordström background fast enough at infinity- in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting : 1. Stability : We prove that spherically symmetric characteristic initial data to the Einstein-Maxwell- Klein-Gordon equations approaching a Reissner-Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space-time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover if the decay is even stronger, we prove that the spacetime metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein-Maxwell-real-scalar-field in spherical symmetry. 2. Instability : We prove that for the class of space-times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L^2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein-Maxwell-real-scalar-field in spherical symmetry. This instability of the black hole interior can also be viewed as a step towards the resolution of the C^2 strong cosmic censorship conjecture for one-ended asymptotically initial data.

gr-qc↗