arXiv subjects

Dejan Gajic

Publications and source records attributed to Dejan Gajic.

At least 19 recordsLinked to original sources

arXiv2026

Charged scalar fields on Reissner--Nordstr\"om spacetimes II: late-time tails and instabilities

This is the second part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordstr\"om spacetimes via energy estimates. In this paper, we use purely physical-space based methods to establish the precise late-time behaviour of solutions to the charged scalar field equation in the form of oscillating and decaying late-time tails that satisfy inverse-power laws, assuming global integrated energy decay estimates, which are proved in the companion paper [Gaj26]. This paper provides the first pointwise decay estimates for charged scalar fields on black hole backgrounds without an assumption of smallness of the scalar field charge. We also prove the existence of asymptotic instabilities for the radiation field along future null infinity and, in the extremal case, also along the future event horizon. Both the energy methods and the precise late-time asymptotics derived in this paper are expected to play an important role in future nonlinear studies of black hole dynamics in the context of the spherically symmetric (Einstein--)Maxwell--charged scalar field equations, as well as in the context of extremal Kerr spacetimes.

gr-qc↗

arXiv2026

Charged scalar fields on Reissner--Nordstr\"om spacetimes I: integrated energy estimates

This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordstr\"om spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner--Nordstr\"om(--de Sitter) spacetimes. These estimates extend to Reissner--Nordstr\"om spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner--Nordstr\"om spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes.

gr-qc↗

arXiv2025

Linear and nonlinear late-time tails on dynamical black hole spacetimes via time integrals

We prove the global leading-order late-time asymptotic behaviour of solutions to inhomogeneous wave equations on dynamical black hole exterior backgrounds that settle down to Schwarzschild backgrounds with arbitrarily small decay rates. In particular, we show that for non-spherically symmetric solutions arising from compactly supported initial data, the late-time decay deviates from Price's law -- governing the decay for stationary black hole backgrounds -- by exhibiting slower time decay by exactly one additional power. Our proof is based around the observation that the emergence of late-time "tails", featuring inverse-polynomial decay in time, is intimately connected to conformal irregularity properties in space (towards future null infinity) of time integrals of the solutions. This relationship is exploited through a purely physical-space approach based around energy estimates, in which the dynamical wave operator is treated as a Schwarzschild wave operator with an inhomogeneous term. Going from almost-sharp decay to global asymptotics is achieved by exploiting this relation for the difference between the solution and a carefully chosen global tail function. We further apply our method to several examples of nonlinear wave equations and comment on its robustness to more general settings, such as dynamical spacetimes converging to sub-extremal Kerr spacetimes and higher-dimensional wave operators with even or odd spacetime dimensions.

gr-qc↗

arXiv2024

Quasinormal modes on Kerr spacetimes

We introduce a rigorous framework for defining quasinormal modes on stationary, asymptotically flat spacetimes as isolated eigenvalues of the infinitesimal generator of time translations. We consider time functions corresponding to a foliation of asymptotically hyperboloidal hypersurfaces and restrict to suitable Hilbert spaces of functions. These functions have finite Sobolev regularity in bounded regions, but need to be Gevrey-regular at null infinity. This framework is developed in the context of sub-extremal Kerr spacetimes, but also gives uniform-in-$Λ$ resolvent estimates on Kerr--de Sitter spacetimes with a small cosmological constant $Λ$. As a corollary, we also construct the meromorphic continuation (in a sector of the complex plane) of the cut-off resolvent in Kerr that is associated to the standard Boyer--Lindquist time function. The framework introduced in this paper bridges different notions of quasinormal modes found in the literature. As further applications of our methods, we prove stability of quasinormal frequencies in a sector of the complex plane, with respect to suitably small perturbations and establish convergence properties for Kerr--de Sitter quasinormal frequencies when the cosmological constant approaches zero.

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↗

arXiv2023

On the Relation Between Asymptotic Charges, the Failure of Peeling and Late-time Tails

The last few years have seen considerable mathematical progress concerning the asymptotic structure of gravitational radiation in dynamical, astrophysical spacetimes. In this paper, we distil some of the key ideas from recent works and assemble them in a new way in order to make them more accessible to the wider general relativity community. We also announce some new physical findings in this process. First, we introduce the conserved $f(r)$-modified Newman--Penrose charges on asymptotically flat spacetimes, and we show that these charges provide a dictionary that relates asymptotics of massless, general spin fields in different regions: Asymptotic behaviour near $i^+$ ("late-time tails") can be read off from asymptotic behaviour towards $\mathcal I^+$, and, similarly, asymptotic behaviour towards $\mathcal I^+$ can be read off from asymptotic behaviour near $i^-$ or $\mathcal I^-$. Using this dictionary, we then explain how: (I) the quadrupole approximation for a system of $N$ infalling masses from $i^-$ causes the "peeling property towards $\mathcal I^+$" to be violated, and (II) this failure of peeling results in deviations from the usual predictions for tails in the late-time behaviour of gravitational radiation: Instead of the Price's law rate $rΨ^{[4]}|_{\mathcal I^+}\sim u^{-6}$ as $u\to\infty$, we predict that $rΨ^{[4]}|_{\mathcal I^+}\sim u^{-3}$, with the coefficient of this latter decay rate being a multiple of the monopole and quadrupole moments of the matter distribution in the infinite past.

gr-qc↗

arXiv2023

Azimuthal instabilities on extremal Kerr

We prove the existence of instabilities for the geometric linear wave equation on extremal Kerr spacetime backgrounds, which describe stationary black holes rotating at their maximally allowed angular velocity. These instabilities can be associated to non-axisymmetric azimuthal modes and are stronger than the axisymmetric instabilities discovered by Aretakis in [Are15]. The existence of non-axisymmetric instabilities follows from a derivation of very precise stability properties of solutions: we determine therefore the precise, global, leading-order, late-time behaviour of solutions supported on a bounded set of azimuthal modes via energy estimates in both physical and frequency space. In particular, we obtain sharp, uniform decay-in-time estimates and we determine the coefficients and rates of inverse-polynomial late-time tails everywhere in the exterior of extremal Kerr black holes. We also demonstrate how non-axisymmetric instabilities leave an imprint on future null infinity via the coefficients appearing in front of slowly decaying and oscillating late-time tails.

gr-qc↗

arXiv2022

Late-time asymptotics for geometric wave equations with inverse-square potentials

We introduce a new, physical-space-based method for deriving the precise leading-order late-time behaviour of solutions to geometric wave equations on asymptotically flat spacetime backgrounds and apply it to the setting of wave equations with asymptotically inverse-square potentials on Schwarzschild black holes. This provides a useful toy model setting for introducing methods that are applicable to more general linear and nonlinear geometric wave equations, such as wave equations for electromagnetically charged scalar fields, wave equations on extremal Kerr black holes and geometric wave equations in even space dimensions, where existing proofs for deriving precise late-time asymptotics might not apply. The method we introduce relies on exploiting the spatial decay properties of time integrals of solutions to derive the existence and precise genericity properties of asymptotic late-time tails and obtain sharp, uniform decay estimates in time.

gr-qc↗

arXiv2021

Quasinormal modes in extremal Reissner-Nordström spacetimes

We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner-Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied cases of asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that $L^2$-based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce $L^2$-based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting.

gr-qc↗

arXiv2021

Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner-Nordstrom families of black holes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections were heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman-Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for compactly supported initial data (and hence with a vanishing Newman-Penrose constant) explicitly obtained here appear to be new.

gr-qc↗

arXiv2021

The trapping effect on degenerate horizons

We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner-Nordstrom up to and including the future event horizon. We show that such an estimate requires 1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and 2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a global trapping effect (in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces (MOTS). Our methods and results play a crucial role in our upcoming works on linear and non-linear wave equations on extremal black hole backgrounds.

gr-qc↗

arXiv2021

A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom

It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth exponentially decaying modes. We also derive scattering results in the black hole interior region.

math.AP↗

arXiv2021

Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere

In our previous work [Y. Angelopoulos, S. Aretakis, and D. Gajic, Late-time asymptotics for the wave equation on spherically symmetric stationary backgrounds, in Advances in Mathematics 323 (2018), 529-621] we showed that the coefficient in the precise leading-order late-time asymptotics for solutions to the wave equation with smooth, compactly supported initial data on Schwarzschild backgrounds is proportional to the time-inverted Newman-Penrose constant (TINP), that is the Newman-Penrose constant of the associated time integral. The time integral (and hence the TINP constant) is canonically defined in the domain of dependence of any Cauchy hypersurface along which the stationary Killing field is non-vanishing. As a result, an explicit expression of the late-time polynomial tails was obtained in terms of initial data on Cauchy hypersurfaces intersecting the future event horizon to the future of the bifurcation sphere. In this paper, we extend the above result to Cauchy hypersurfaces intersecting the bifurcation sphere via a novel geometric interpretation of the TINP constant in terms of a modified gradient flux on Cauchy hypersurfaces. We show, without appealing to the time integral construction, that a general conservation law holds for these gradient fluxes. This allows us to express the TINP constant in terms of initial data on Cauchy hypersurfaces for which the time integral construction breaks down.

gr-qc↗

arXiv2021

Nonlinear scalar perturbations of extremal Reissner-Nordström spacetimes

We present the first rigorous study of nonlinear wave equations on extremal black hole spacetimes without any symmetry assumptions on the solution. Specifically, we prove global existence with asymptotic blow-up for solutions to nonlinear wave equations satisfying the null condition on extremal Reissner-Nordström backgrounds. This result shows that the extremal horizon instability persists in model nonlinear theories. Our proof crucially relies on a new vector field method that allows us to obtain almost sharp decay estimates.

math.AP↗

arXiv2021

Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds

We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordström black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory, which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti-Murata-Reall-Tanahashi, Ori-Sela and Blaksley-Burko.

gr-qc↗

arXiv2021

Late-time tails and mode coupling of linear waves on Kerr spacetimes

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies $\ell\geq 1$ and $\ell\geq 2$. The $\ell$-dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting ("Price's law"). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing $\ell$-dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for resolving the conflicting numerology in Kerr late-time asymptotics that appears in the numerics literature.

gr-qc↗

arXiv2021

Price's law and precise late-time asymptotics for subextremal Reissner-Nordström black holes

In this paper, we prove precise late-time asymptotics for solutions to the wave equation supported on angular frequencies greater or equal to $\ell$ on the domain of outer communications of subextremal Reissner-Nordström spacetimes up to and including the event horizon. Our asymptotics yield, in particular, sharp upper and lower decay rates which are consistent with Price's law on such backgrounds. We present a theory for inverting the time operator and derive an explicit representation of the leading-order asymptotic coefficient in terms of the Newman-Penrose charges at null infinity associated with the time integrals. Our method is based on purely physical space techniques. For each angular frequency $\ell$ we establish a sharp hierarchy of $r$-weighted radially commuted estimates with length $2\ell+5$. We complement this hierarchy with a novel hierarchy of weighted elliptic estimates of length $\ell+1$.

gr-qc↗

arXiv2020

A model problem for quasinormal ringdown on asymptotically flat or extremal black holes

We consider a wave equation with a potential on the half-line as a model problem for wave propagation close to an extremal horizon, or the asymptotically flat end of a black hole spacetime. We propose a definition of quasinormal frequencies (QNFs) as eigenvalues of the generator of time translations for a null foliation, acting on an appropriate (Gevrey based) Hilbert space. We show that this QNF spectrum is discrete in a subset of $\mathbb{C}$ which includes the region $\{$Re$(s) >-b$, $|$Im $(s)|> K\}$ for any $b>0$ and some $K=K(b) \gg 1$. As a corollary we establish the meromorphicity of the scattering resolvent in a sector $|$arg$(s)| <φ_0$ for some $φ_0 > \frac{2π}{3}$, and show that the poles occur only at quasinormal frequencies according to our definition. This result applies in situations where the method of complex scaling cannot be directly applied, as our potentials need not be analytic. Finally, we show that QNFs computed by the continued fraction method of Leaver are necessarily QNFs according to our new definition. This paper is a companion to [D. Gajic and C. Warnick, Quasinormal modes in extremal Reissner-Nordström spacetimes, preprint (2019)], which deals with the QNFs of the wave equation on the extremal Reissner-Nordström black hole.

math.AP↗