arXiv subjects

Gabriele Benomio

Publications and source records attributed to Gabriele Benomio.

6 recordsLinked to original sources

arXiv2025

The Maxwell equations on full sub-extremal and extremal Kerr spacetimes

We study the Cauchy problem for the Maxwell equations in the exterior region of Kerr black hole spacetimes. The equations are formulated for components of the Maxwell field relative to the algebraically special frame of Kerr, with the unknowns treated as tensorial quantities associated with a non-integrable horizontal distribution. The extremal Maxwell components decouple into Teukolsky equations, whereas the middle Maxwell components form a coupled system of transport and elliptic equations. Assuming control over the extremal components, we prove uniform boundedness (without loss of derivatives) and decay estimates for the middle components in the full |a|<=M range of spacetime parameters. Our analysis relies on (i) deriving a decoupled system of transport and elliptic equations for two modified middle Maxwell components and (ii) decomposing general solutions into a dynamical and stationary part, the latter determined by two real (electric and magnetic) charges which are entirely read off from the initial data at the event horizon. In the sub-extremal |a|<M case, works of Shlapentokh-Rothman and the second author provide the necessary control over the extremal components, yielding unconditional boundedness and decay results for all the unknowns in the equations. In the extremal |a|=M case, we formulate a conjectural boundedness and decay statement for the extremal components, motivated by work of Casals, Gralla and Zimmerman on fixed azimuthal mode solutions compactly supported away from the event horizon. Our boundedness and decay results for all the unknowns in the equations remain, therefore, conditional. We show that the complicated dynamics of the extremal components at the event horizon is inherited by the middle components; in particular, we uncover novel conservation laws for the middle components of axisymmetric solutions.

math.AP↗

arXiv2025

Elliptic curvature estimates for linearised gravitational perturbations of Kerr in the full sub-extremal range $|a|<M$

We consider the linearised vacuum Einstein equations around a Kerr exterior solution and present a scheme to prove elliptic $L^2(\mathbb{S}^2)$-estimates for the linearised curvature quantities in the equations. The scheme employs the linearised system of equations derived in the first author's doctoral thesis and applies to the full sub-extremal range of Kerr parameters $0\leq |a|<M$.

gr-qc↗

arXiv2024

On turbulence for spacetimes with stable trapping

Motivated by understanding the nonlinear gravitational dynamics of spacetimes admitting stably trapped null geodesics, such as ultracompact objects and black string solutions to general relativity, we explore the dynamics of nonlinear scalar waves on a simple (fixed) model geometry with stable trapping. More specifically, we consider the time evolution of solutions to the cubic (defocusing) wave equation on a four-dimensional static, spherically symmetric, and asymptotically flat (horizonless) spacetime admitting a stable photon sphere. Our study shows fundamental differences between linear and nonlinear scalar dynamics. The local energy, as well as all local higher-order energies, of solutions to the linear wave equation on our model spacetime can be rigorously proven to remain uniformly bounded and to decay uniformly in time. However, due to the presence of stable trapping, the uniform decay rate is slow. To help elucidate how the slow linear decay affects solutions to the nonlinear wave equation considered, we examine numerical solutions of the latter, restricting to axisymmetric initial data in this work. In contrast to the linear dynamics, we exhibit a family of nonlinear solutions with turbulent behaviour: Within the region of stable trapping, the slow linear decay allows local higher-order energies of the nonlinear solution to grow over the time interval that we numerically evolve, with the growth being induced by a direct energy cascade. As a complement to the numerical analysis, we provide a heuristic argument suggesting that, if a similar behaviour occurs for gravitational wave perturbations of the motivating spacetimes in general relativity, it would likely not generically lead to black hole or singularity formation.

gr-qc↗

arXiv2022

A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

We propose a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range $|a|<M$. Central to our framework is a new formulation of nonlinear gravitational perturbations of Kerr, whose two novel ingredients are the choice of a geometric gauge and non-integrable null frames both tailored to the outgoing principal null geodesics of Kerr. The vacuum Einstein equations for the perturbations are formulated in our gauge as a system of equations for the connection coefficients and curvature components relative to the chosen frames. When renormalised with respect to Kerr, the null structure equations with the form of outgoing transport equations do not possess any derivatives of renormalised connection coefficients on the right hand side. In this work, we derive the linearised vacuum Einstein equations around Kerr in the new framework. Our new framework is designed to effectively capture the stabilising properties of the red-shifted transport equations, thereby isolating one of the crucial structures of the problem. Such a feature is suggestive of future simplifications in the analysis. As an illustration, our companion work \cite{benomio_schwarzschild} employs the system of linearised gravity and its enhanced red-shifted transport equations, specialised to the $|a|=0$ case, to produce a new simplified proof of linear stability of the Schwarzschild solution. The full linear stability analysis in the full sub-extremal range $|a|<M$ is deferred to future work. As already apparent from \cite{benomio_schwarzschild}, our framework will allow to combine the new structure in the transport equations with the elliptic part of the system to establish a linear orbital stability result without loss of derivatives, indicating that the framework may be well suited to address nonlinear stability in the full sub-extremal range $|a|<M$.

gr-qc↗

arXiv2022

A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the $|a|=0$ case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin $\pm 2$ Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of \cite{DHR}, which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.

gr-qc↗

arXiv2018

The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves

The geometry of solutions to the higher dimensional Einstein vacuum equations presents aspects that are absent in four dimensions, one of the most remarkable being the existence of stably trapped null geodesics in the exterior of asymptotically flat black holes. This paper investigates the stable trapping phenomenon for two families of higher dimensional black holes, namely black strings and black rings, and how this trapping structure is responsible for the slow decay of linear waves on their exterior. More precisely, we study decay properties for the energy of solutions to the scalar, linear wave equation $\Box_{g_{\textup{ring}}} Ψ=0$, where $g_{\textup{ring}}$ is the metric of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class $\mathfrak{g}$ of black ring metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black ring exterior $(\mathcal{D},g_{\textup{ring}})$, with $g_{\textup{ring}}\in\mathfrak{g}$. The proof generalizes the perturbation argument and quasimode construction of Holzegel--Smulevici \cite{SharpLogHolz} to the case of a non-separable wave equation and crucially relies on the presence of stably trapped null geodesics on $\mathcal{D}$. As a by-product, the same logarithmic lower bound can be established for any five-dimensional black string. Our result is the first mathematically rigorous statement supporting the expectation that black rings are dynamically unstable to generic perturbations. In particular, we conjecture a new \textit{nonlinear} instability for five-dimensional black strings and thin black rings which is already present at the level of scalar perturbations and clearly differs from the mechanism driven by the well-known Gregory--Laflamme instability.

gr-qc↗