Exploring the Time Evolution of Complexity in General Holographic Spacetimes

Following the proposal of the "complexity equals action" conjecture in 2015, interest arose in finding a boundary term to the Einstein-Hilbert action that provides a well-posed variational principle on boundaries of arbitrary signature. A boundary term satisfying this property exists, however, it is dependent on a highly non-unique auxiliary vector. We revisit this question and explore the difficulties in finding a boundary term independent of this auxiliary vector. Previous considerations of the complexity equal action conjecture have examined the time evolution of complexity in special cases, e.g. Schwarzchild-AdS black holes. Using the Iyer-Wald formalism, we apply the boundary term for arbitrary signature to explore the time evolution of complexity in general spacetimes.