Explore the fourth lecture in a series on self-force and radiation reaction in general relativity, delivered by Adam Pound at the ICTS Summer School on Gravitational Wave Astronomy. Delve into advanced topics such as the stress-energy tensor of gravitational waves, two-timescale expansion, and matched asymptotic expansions. Learn about general statements in general relativity, the adoption of Cartesian and scaled coordinates, matching conditions, and the consequences of inner and outer expansions. Examine strategies for adopting local coordinates and understanding the Riemann tensor. Gain insights into cutting-edge research in gravitational physics through this comprehensive 1 hour 34 minute lecture, which includes a Q&A session to address complex concepts.