An Investigation Into the Principle of Symmetric Criticality

In physics, symmetries help us understand complex systems by revealing conserved quantities and simplifying equations. In the context of general relativity, where spacetime is curved and dynamic, these symmetries become especially important when trying to reduce or solve the equations that govern gravitational and matter fields. This dissertation investigates a mathematical principle known as the Principle of Symmetric Criticality (PSC), which helps determine when symmetries can be used to simplify physical theories without losing essential information. While often assumed to be valid, PSC does not always hold, and understanding when it does is critical for the reliable use of symmetry in theoretical physics. The first part of this work identifies all possible spacetime symmetries that allow such simplification for gravitational fields, specifically within Einstein’s theory. The second part introduces a new method for choosing coordinate conditions, or “gauge fixing,” when working with simplified versions of the equations. An important step for both theoretical understanding and computational modeling. Finally, the third part extends these ideas to spinor fields, which describe particles like electrons and are essential in quantum field theory. This extension uses a two-component spinor formalism adapted to curved spacetime, showing that the same symmetry criteria used for gravitational fields also apply to spinor fields. To support these results, custom computer code was written to automate the complex symbolic calculations involved. Together, the findings offer new tools and insights for researchers studying symmetry, gravity, and the mathematical foundations of field theory.