Research Themes

My main research works and interests lie in geometric and spectral analysis of PDEs in the field of Mathematical General Relativity. The topics I work on include geometric studies of black hole spacetimes &mdash interior and exterior regions, decay estimates, scattering theories, inverse problems, and quasi-normal modes/resonances. Such topics play major roles in understanding many of the important phenomena in General Relativity such as the Hawking effect and superradiance, and quantum fields on curved spacetimes. They are as well crucial steps towards solving some central problems like the stability of black holes and the cosmic censorship conjecture. Moreover, they are essential tools in observational studies of gravitational waves detection, black holes merging and ring-downs.

Keywords:

Differential Geometry ; Geometric Analysis ; PDE’s ; Mathematical Physics ; General Relativity ; Black holes; Decay of Fields; Analytic and Conformal Scattering Theories ; Inverse Problems ; Quasi-normal Modes (Resonances) ; Spectral Analysis ; Asymptotic Behavior ; Vector Field methods ; Maxwell, Dirac, Klein-Gordon, Schrödinger, and wave equations.

Award

In March 2022, I revieved the Atiyah Fellowship award from the London Mathematical Society to carry out my research projects. In particular, the award facilitated a scientific collaboration and visit to Queen Mary University of London, in addtion to interaction with mathematicians at the American University of Beirut by visiting the univeristy.

Some selected topics from my research

The Reissner-Nordström-de Sitter Black Hole

One of the first studies I did was a geometric study of the Reissner-Nordström-de Sitter (RNdS) black hole spacetime. Its metric is a spherically symmetric solution of the Einstein-Maxwell coupled system in the presence of a positive cosmological constant. It describes a spacetime with a massive, charged, but non-rotating black hole, and the de Sitter aspect means that there is a cosmological horizon as well. In this study, the goemtry of the RNdS spacetime and the structures in it (the photon sphere) was analysed with a detailed construction of the maximal analytic extension of such spacetimes in the most complete case of three horizons (diagram on the right). Locating the photon sphere, and therefore the trapping region, was important for decay estimates for the Maxwell's equations.

Electromagnetic Fields on Black Hole Spacetimes

In one study, I used Morawetz and geometric energy estimates —the so-called the vector field method — to prove decay results for Maxwell fields in the static outer region of a Reissner–Nordström-de Sitter (RNdS) black hole . We obtian two types of decay: The first is a uniform decay of the field's energy on achronal hypersurfaces near time-like infinities. The second decay result is a point decay in time which follows from the local decay of energy by Sobolev estimates. Both results are consequences of uniform bounds on the conformal energy/charge defined by the Morawetz conformal vector field. These bounds are obtained by wave analysis on the intermediate spin component of the field that uses Hardy-like estimates, giving a control on the trapping term that multiplies the angular derivates of the solution. Moreover, the dacay results hold for a more general class of spherically symmetric spacetimes.

These decay results were then used work to construct a complete conformal scattering theory for Maxwell fields outside a RNdS black hole background. I used decay results to show that the trace operators are injective and of closed images. Then, I solved the Goursat problem for Maxwell fields on the isotropic boundary which shows that the trace operators are surjective too. The resolution of the Goursat problem, which is a characteristic Cauchy problem, is the asymptotic completeness. In this article I developed what I call the waves re-interpretation method to rigorously solve the Goursat problem for Maxwell fields.

The method of waves re-interpretation

The idea of the waves re-interpretation method is the successive application of the (spin) operator in study to alternate between a system of wave equations and the original equations, possibly with perturbations. The transfer from the original Goursat problem to a wave Goursat problem is done by applying the studied operator a second time to the original equations to obtain a system of wave equations. We then use the already established theory for the well-posedness of the latter, including regular perturbations up to the first order. The Goursat data for the wave system is obtianed from the original data and the constraint equations. The second step is to reinterpret the wave solution as a solution of the original eqautions, which is done by applying our operator once more and using the well-posedness of the wave Goursat problem again. The method was first developped for Maxwell fields and later it was used by T.X. Pham for a general half integer spin field on Minkowski spacetime, and then by myself for Dirac fields inside black holes.

Scattering of Dirac fields

our articles on scattering for the Dirac equation are the first results of scattering for the Dirac field inside black holes. In this first article, with my collaborators, we have built a complete scattering theory for the charged and massive Dirac fields between the event horizon and the Cauchy horizon of the Reissner-Nordström type black hole. We show the asymptotic completeness of the massive and charged Dirac equation between the black hole horizon and the Cauchy horizon in the interior of the subextremal Reissner-Nordström(-(Anti-)de Sitter) black hole. In traditional language, this means the existence, uniqueness, and asymptotic completeness of scattering states. The result was first obtained analytically in the Hamiltonian formulation of the Dirac equation which shows that the wave operators and their inverses are bounded, in fact, unitary. Analytical scattering is proved by Cook's method. The result was then reinterpreted geometrically in terms of trace operators. Here, the treatment of charge in the Dirac equation was tricky and different gauge transformations were needed near each horizon.

In the second work on the scattering of Dirac fields aims to show that the wave re-interpretation method applies equally well in dynamic situations where the equations and operators depend on time, in particular inside holes black. Furthermore, this article also shows that the geometric results can be directly obtained using this method and that the scattering theory is therefore immediate, in other words, one completely avoids the need to pass by analytical scattering. I expect this method to be robust and general enough to be applied in different situations, in particular, in the asymmetric cases of Kerr black holes.

In a third work, we have obtained the scattering of Dirac fields inside rotating black holes. More precisely, we showed the existence, uniqueness, and asymptotic completeness of scattering data for massive and charges Dirac fields in the dynamic interiors Kerr, Kerr-Newman, Kerr-Newman-de Sitter and Kerr-Newman-Anti-de Sitter black holes, between the Cauchy and event horizons. As in the first work, our approach is based on the construction of wave operators where the Hamiltonian of the full dynamics depends on time. Here, we borrow techniques and tools from the theory of regularly generated dynamics which make it possible to deal adequately and without decomposition with the problematic terms arising from asymmetry. As expected, the Dirac operator on the sphere is the part that requires the most effort and dealing with it constitutes the main difficulty of this work. In Cook's method, most of the terms of the difference between the full and simplified Hamiltonians are directly controlled by the initial data and therefore it is clear that they decay at an exponential rate. The spherical term is however much more complicated since no simple commutation relation holds between it and the dynamics. To overcome this difficulty, we introduce a comparison operator whose role is to help prove the necessary control over the angular derivatives of the solution of the Dirac field in time.

Scattering Breakdown of Waves inside Black Holes

In this paper, we show that there is a breakdown of scattering of linear waves between the event horizon (or the Cauchy horizon) and an intermediate Cauchy hypersurface in the dynamic interior of a Reissner–Nordström-like black hole. More precisely, we show that the trace operators and their analytic counterparts, the inverse wave operators, do not have bounded inverses, even though these operators themselves are bounded. This result holds for the natural energy given by the energy–momentum tensor of the wave equation using the timelike vector field of the Regge–Wheeler variable, which asymptotically becomes normal to the horizons. The behaviour of solutions at low spatial frequencies and their behaviour at high angular momenta are the only obstructions causing this breakdown of scattering. The breakdown follows from an analysis of a 1+1-dimensional wave equation with exponentially decaying potential which we treat for general potentials, and we show that the breakdown is generic.

Most of the outcomes of this article are based on a main technical result requiring a fine frequency analysis which proves the existence of a sequence of initial data with non-zero constant energy, but whose sequence of corresponding solutions for the 1+1-dimensional wave equation has a limit of zero energy at infinite times. The breakdown result is interesting and curious because it shows the striking contrast between scattering in the interior and in the exterior of black holes, as well as between Dirac fields and waves. Outside the black hole, global scattering theories from the past boundary to the future boundary have always been obtained by concatenating the scattering theory of past states on the passed boundary to a Cauchy hypersurface with the scattering theory of the hypersurface of Cauchy to asymptotic future states on the future boundary. This work shows rigorously that inside the situation is different: although there is a global scattering theory from the event horizon to the Cauchy horizon, this scattering theory cannot be obtained from a concatenation of scattering theories from an intermediate Cauchy hypersurface. The result is also in contrast to the case of the Dirac equation as shown in my previous articles, where the construction using the intermediate scattering operators was possible in the interior regions.

Long term project: Geometric and Spectral Aspects of the Cosmic Censorship Conjecture

The Cosmic Censorship Conjecture (CCC), proposed in 1969 by Nobel Laureate in Physics (2020) Roger Penrose, is one of the most important and central open problems in General Relativity (GR). The conjecture has direct implications for the very foundation of physics, and determinism in GR will be lost if it is not true. At its heart is the Cauchy Horizon Instability (CHI), a mechanism widely believed to guarantee the validity of the CCC. The CHI is believed to be primarily due to gravitational blue-shifts (GBS) at the Cauchy horizon. Despite the importance of the GBS, its mathematical theory is still largely uncharted territory, but which is attracting considerable attention very quickly.

My project is a program using analytical and conformal approaches that aims at developping the mathematics of CHI by first constructing rigorous mathematical scattering theories for test fields, like waves, Dirac and Maxwell fields inside the black hole family of Kerr-Newman-(anti-)de Sitter (KN(A)dS), to formalize the GBS phenomenon and the CHI. Achieving the goals of this project may also help in the development of quantum field theories in curved space-times, such as the construction of Hadamard states.