Similarity in the Arnowitt-Deser-Misner mass from Brill and Teukolsky initial datasets beyond the linear approximation

The Brill and Teukolsky gravitational waves are two of the most commonly used initial datasets for vacuum axisymmetric spacetimes. Their numerical evolution reveals distinct behaviors, particularly in the context of critical gravitational collapse. These differences can be traced back to the linear level, where Brill waves exhibit a multipolar structure, while Teukolsky waves are purely quadrupolar. Despite these structural disparities, we demonstrate an unexpected similarity between the two: both gravitational wave initial data follow a nonextensive scaling law for the distribution of the Arnowitt-Deser-Misner mass as a function of the initial wave amplitude, a remarkable feature of systems with long-range interactions.