Within general relativity, one of the most fundamental properties that any quasi-local energy is expected to satisfy in order to be deemed physically reasonable is the correct asymptotic behavior in the small sphere limit, first established by Horowitz and Schmidt for the Hawking energy in 1982. Given a spacetime $(M,\bar g)$, a point $p\in M$, and a future-directed timelike unit vector $e_0\in T_pM$, this limit is obtained by foliating the local future lightcone at $p$ by spacelike cross sections and evaluating the quasi-local energy along these sections as they approach the vertex.
Remarkably, although the small sphere limit has since been studied for many different notions of quasi-local energy, the underlying choice of background foliation has essentially always been the same: a foliation by constant-radius cuts in normal coordinates centered at the vertex. We refer to this distinguished choice as the natural background foliation. The purpose of this talk is to investigate the role of this foliation in more detail and study precisely how the small sphere limit of the Hawking energy depends on the choice of background foliation.
Surprisingly, this dependence turns out to be quite strong. After establishing the necessary null-geometric framework, we derive a curvature-dependent integral condition characterizing when, for a fixed tuple $(M,\bar g,p,e_0)$, a given background foliation yields the correct small sphere limit. However, if the small sphere limit is required to hold independently of the tuple $(M,\bar g,p,e_0)$, then the foliation is strongly constrained: it must, in a precise asymptotic sense, be $H^1$-close to the natural background foliation. In particular, among foliations whose leaves are obtained by linear rescalings along the null generators, the natural background foliation is the only one that yields the correct small sphere limit for all choices of $(M,\bar g,p,e_0)$.