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Rita Teixeira da Costa

Publications and source records attributed to Rita Teixeira da Costa.

9 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↗
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↗
arXiv2023

Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|<M$: physical space analysis

This paper concludes the study, initiated by the authors in arXiv:2007.07211, of the Teukolsky equation of spin $\pm 1$ and spin $\pm 2$ on Kerr backgrounds in the full subextremal range of parameters $|a| < M$. In our previous arXiv:2007.07211, we obtained uniform-in-frequency estimates for the ODEs governing separable solutions to the Teukolsky equation. In this paper, by adapting the techniques developed by the first author with Dafermos and Rodnianski for scalar waves, we show that our ODE estimates can be upgraded to estimates for the Teukolsky PDE. In particular, we conclude the proof that solutions of the Teukolsky equation on subextremal Kerr arising from regular initial data remain bounded and decay in time.

gr-qc↗
arXiv2023

Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|<M$: frequency space analysis

This paper is the first of a series regarding the Teukolsky equation of spin $\pm 1$ and spin $\pm 2$ on Kerr backgrounds in the full subextremal range of parameters $|a|<M$. In the present paper, we study fixed frequency solutions of the transformed system of equations introduced by Dafermos, Holzegel and Rodnianski, obtaining estimates which are uniform in the separation parameters. A corollary of our result, to be laid out in the second paper of the series, is that solutions of the Teukolsky equation on subextremal Kerr arising from regular initial data remain bounded and decay in time. This is a key step in establishing the full linear stability of Kerr under electromagnetic and gravitational perturbations. Our estimates can also be applied to understanding more delicate features of the Teukolsky equation, such as their scattering properties.

gr-qc↗
arXiv2021

Oscillations in wave map systems and homogenization of the Einstein equations in symmetry

In 1989, Burnett conjectured that, under appropriate assumptions, the limit of highly oscillatory solutions to the Einstein vacuum equations is a solution of the Einstein--massless Vlasov system. In a recent breakthrough, Huneau--Luk (arXiv:1907.10743) gave a proof of the conjecture in U(1)-symmetry and elliptic gauge. They also require control on up to fourth order derivatives of the metric components. In this paper, we give a streamlined proof of a stronger result and, in the spirit of Burnett's original conjecture, we remove the need for control on higher derivatives. Our methods also apply to general wave map equations.

math.AP↗
arXiv2021

The Teukolsky--Starobinsky constants: facts and fictions

The Teukolsky Master Equation describes the dynamics of massless fields with spin on a Kerr black hole. Under separation of variables, spin-reversal for this equation is accomplished through the so-called Teukolsky--Starobinsky identities. These identities are associated to the so-called Teukolsky--Starobinsky constants, which are spin-dependent. We collect some properties of the Teukolsky--Starobinsky constants and dispel some myths present in the literature. We show that, contrary to popular belief, these constants can be negative for spin larger than 2. Such fields thus exhibit a novel form of energy amplification which occurs for non-superradiant frequencies.

gr-qc↗
arXiv2019

Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes

We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on Kerr black hole spacetimes, both in the extremal and subextremal case. We also give a quantitative refinement of mode stability. As an immediate application, we show that the transmission and reflection coefficients of the scattering problem are bounded, independently of the specific angular momentum of the black hole, in any compact set of real frequencies excluding zero frequency and the superradiant threshold. While in the subextremal setting these results were known previously, the extremal case is more involved and has remained an open problem. Ours are the first results outside axisymmetry and could serve as a preliminary step towards understanding boundedness, scattering and decay properties of general solutions to the Teukolsky equation on extremal Kerr black holes.

gr-qc↗