A Classification of Spacetime Symmetries Compatible with a Generic Lagrangian

Symmetry reduction is an important tool for simplifying physical problems. But in some cases, symmetry reducing the Lagrangian may lead to incorrect equations. When this issue occurs, a symmetry is said to violate the principle of symmetric criticality (PSC). It's known that for PSC to hold in a local field theory, two conditions must be true (the Palais and Lie conditions), but for most spacetime symmetries, the validity of PSC has not been established. In this project, we provide a classification of spacetime symmetries based upon the validity of PSC for gravitational field theory. Using the DifferentialGeometry package in Maple, we tested both PSC conditions against a database listing all 92 possible spacetime symmetries. We found that over half of all spacetime symmetries violate PSC, with most obstructions coming from the boundary term of the first variational formula. The conditions used here are from a Lagrangian built locally from a metric field, but in the future, a similar analysis could be done for a Lagrangian built from additional fields. This classification will allow researchers to avoid pitfalls when using symmetry reduction in exotic gravitational problems.