This book is devoted to researchers who would like to investigate interactions between gravitational waves and matter fields beyond the linear order, including the phenomena of memory effects, gravitational Faraday rotation, soft theorems, and formations of spacetime singularities due to the mutual focus of gravitational waves, just to name a few of them. Readers only require a basic understanding of general relativity to understand the materials. The book starts with an overview on the fundamentals of the Newman – Penrose formalism and a brief introduction to distribution theory, with which the author systematically develops a mathematical description of spacetimes of colliding plane waves. Then, the author presents a frame-independent definition of polarizations of a plane gravitational wave in a curved spacetime, studies in details the gravitational Faraday rotation of two plane gravitational waves, and shows that each of them can serve as a medium to the other precisely due to their nonlinear interactions. Exact solutions are also presented, which represent a variety of models including the collisions of two plane gravitational waves and the collisions of a plane gravitational wave with a matter shell, a massless scalar, an electromagnetic, or a neutrino wave. The formation of spacetime singularities due to the nonlinear interactions and the effects of polarizations of gravitational waves on the nature of singularities are also explored. Key Feature: ○The book provides a basic foundation for the studies of physical phenomena due to the nonlinear interactions of gravitational waves with themselves and/or with matter fields, which could be detected by current or forthcoming generation of gravitational wave detectors

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