Abstract:
Recent developments in soft theorems and the rise of Celestial Holography have rejuvenated the interest in the asymptotic structure of spacetimes. Bondi, van der Burg, Metzner, and Sachs's work and Ashtekar's notion of asymptotic flatness assume that the future null infinity in the conformal metric and the physical metric is $C^\infty$. Similarly, the absence of incoming radiation requires the past null infinity to be $C^\infty$. But this condition cannot be put on spatial infinity due to the probable presence of isolated sources, which are represented by an important class of solutions. Despite the lack of differentiability of spatial infinity extending to the metric, the assumption of $C^\infty$ of the manifold along with the null infinity leads to the peeling property, given by: $C_{\mu \nu \rho \sigma} = \mathcal{O}(\Omega)$. But Christodoulou and Klainerman argued that this peeling property is too restrictive and strong of a condition that eliminates plausible physical spacetime solutions. The shortcomings of the peeling property motivate the construction of the logarithmically asymptotic flat (LAF) spacetimes, which gives the relation: $C_{\mu \nu \rho \sigma} = \mathcal{O}(\Omega \log{\Omega})$ with Weyl tensor and its dual satisfying the relations: $C_{\mu \nu \rho \sigma} n^\mu n^\rho= \mathcal{O}(\Omega)$ and ${^*}C_{\mu \nu \rho \sigma} n^\mu n^\rho = \mathcal{O}(\Omega)$ where $n^\mu=g^{\mu \nu} \Omega_{,\nu}$. Upon taking this logarithmically asymptotic flat condition differentiability structure of the infinities change. In this thesis, we study the structure of future null infinity of asymptotically logarithmic flat (LAF) spacetime and if peeling is violated at $|u| \to \infty$ for massive particles, massless scalar, and massive scalar field on the 4-D Minkowski background.
