Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge-Wheeler and Zerilli equations obtained recently by L. Anderson, P. Blue and J. Wang \cite{ABlu} is sufficient to obtain the conformal scattering.Comment: 20 pages, 3 firgure

Self-healing composites: A review

Self-healing composites are composite materials capable of automatic recovery when damaged. They are inspired by biological systems such as the human skin which are naturally able to heal themselves. This paper reviews work on self-healing composites with a focus on capsule-based and vascular healing systems. Complementing previous survey articles, the paper provides an updated overview of the various self-healing concepts proposed over the past 15 years, and a comparative analysis of healing mechanisms and fabrication techniques for building capsules and vascular networks. Based on the analysis, factors that influence healing performance are presented to reveal key barriers and potential research directions

Application of Current Algebra in Three Pseudoscalar Meson Decays of τ\tau Lepton

The decays of $\tau \to 3\pi \nu$ and $\tau \to \pi K^{*} \nu, K\rho \nu$ are calculated using the hard pion and kaon current algebra and assuming the Axial-Vector meson dominance of the hadronic axial currents. Using the experimental data on their masses and widths, the $\tau$ decay branching ratios into these channels are calculated and found to be in a reasonable agreement with the experimental data. In particular, using the available Aleph data on the $3\pi$ spectrum, we determine the $A_1$ parameters, $m_A=1.24\pm 0.02 GeV$, $\Gamma _A=0.43\pm 0.02$ GeV; the hard current algebra calculation yields a $3\pi$ branching ratio of $19 \pm 3 \%$.Comment: 14 pages, Tex, 6 included figure

Cauchy and Goursat problems for the generalized spin zero rest-mass fields on Minkowski spacetime

In this paper, we study the Cauchy and Goursat problems of the spin-$n/2$ zero rest-mass equations on Minkowski spacetime by using the conformal geometric method. In our strategy, we prove the wellposedness of the Cauchy problem in Einstein's cylinder. Then we establish pointwise decays of the fields and prove the energy equalities of the conformal fields between the null conformal boundaries \scri^\pm and the hypersurface $\Sigma_0=\left\{ t=0 \right\}$. Finally, we prove the wellposedness of the Goursat problem in the partial conformal compactification by using the energy equalities and the generalisation of H\"ormander's result.Comment: 42 pages, 3 figure

Conformal scattering theory for a tensorial Fackerell-Ipser equation on the Schwarzschild spacetime

In this paper, we prove that the existence of the energy and pointwise decays for the fields satisfying the tensorial Frackerell-Ipser equations (which are obtained from the Maxwell and spin $\pm 1$ Teukolsky equations) on the Schwarzschild spacetime is sufficient to obtain a conformal scattering theory. This work is the continuation of the recent work \cite{Pha2020} on the conformal scattering theory for the Regge-Wheeler and Zerilli equations arising from the linearized gravity fields and the spin $\pm 2$ Teukolsky equations.Comment: 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2005.1204

Conformal scattering theory for the Dirac field on Kerr spacetime

We investigate to construct a conformal scattering theory of the spin-$1/2$ massless Dirac equation on the Kerr spacetime by using the conformal geometric method and under an assumption on the pointwise decay of the Dirac field. In particular, our construction is valid in the exteriors of Schwarzschild and very slowly Kerr black hole spacetimes, where the pointwise decay was established.Comment: 39 pages, 2 figures. arXiv admin note: text overlap with arXiv:2106.0405

Three Essays on Inequality of Opportunity and Intergenerational Mobility

Senior Project submitted to The Division of Social Studies of Bard College

Peeling of Dirac fields on Kerr spacetimes

In a recent paper with J.-P. Nicolas [J.-P. Nicolas and P.T. Xuan, Annales Henri Poincare 2019], we studied the peeling for scalar fields on Kerr metrics. The present work extends these results to Dirac fields on the same geometrical background. We follow the approach initiated by L.J. Mason and J.-P. Nicolas [L. Mason and J.-P. Nicolas, J.Inst.Math.Jussieu 2009; L. Mason and J.-P. Nicolas, J.Geom.Phys 2012] on the Schwarzschild spacetime and extended to Kerr metrics for scalar fields. The method combines the Penrose conformal compactification and geometric energy estimates in order to work out a definition of the peeling at all orders in terms of Sobolev regularity near $\mathscr{I}$, instead of ${\mathcal C}^k$ regularity at $\mathscr{I}$, then provides the optimal spaces of initial data such that the associated solution satisfies the peeling at a given order. The results confirm that the analogous decay and regularity assumptions on initial data in Minkowski and in Kerr produce the same regularity across null infinity. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics.Comment: 29 page