Conservation Laws for Asymptotically Perfect Fluid Spacetimes

In physics, conservation laws, like those for energy or momentum, are powerful tools that help us understand how systems evolve and interact. In general relativity, where spacetime itself is curved by matter and energy, defining such conservation laws becomes especially subtle and complex. Traditional methods rely on idealized conditions like empty space or special symmetries, which limit their usefulness in more realistic, dynamic settings such as an expanding universe. This dissertation explores a modern mathematical framework, developed by Iyer and Wald, that allows conservation laws to be derived directly from the equations governing spacetime. Using this approach, I examine not only known solutions involving empty space (vacuum spacetimes) but also more realistic models that include matter, such as perfect fluids. Perfect fluids are often used to model stars, galaxies, and the large-scale behavior of the universe itself. The main achievement of this work is the development of a new conservation law tailored for spacetimes that resemble a perfect fluid at large distances—just like the universe described by the standard model of cosmology. This result is then tested using the McVittie solution, which describes a massive object embedded in an expanding universe. I also explore a two-dimensional spacetime model known as dilaton gravity, revealing conservation laws in that setting as well. In addition to the theoretical results, I created symbolic computer code to automate these calculations, making it easier for others to apply these methods to new models in the future. Overall, this work extends our ability to understand conserved quantities in more realistic and dynamic spacetimes, with applications ranging from astrophysics to cosmology.