Fundamental Complement of a Gravitating Region

Any gravitating region $a$ in any spacetime gives rise to a generalized entanglement wedge, the hologram $e(a)$. Holograms exhibit properties expected of fundamental operator algebras, such as strong subadditivity, nesting, and no-cloning. But the entanglement wedge EW of an AdS boundary region $B$ with commutant $\bar B$ satisfies an additional condition, complementarity: EW$(B)$ is the spacelike complement of EW$(\bar B)$ in the bulk. Here we identify an analogue of the boundary commutant $\bar B$ in general spacetimes: given a gravitating region $a$, its \emph{fundamental complement} $\tilde{a}$ is the smallest wedge that contains all infinite world lines contained in the spacelike complement $a'$ of $a$. We refine the definition of $e(a)$ by requiring that it be spacelike to $\tilde a$. We prove that $e(a)$ is the spacelike complement of $e(\tilde a)$ when the latter is computed in $a'$. We exhibit many examples of $\tilde{a}$ and of $e(a)$ in de Sitter, flat, and cosmological spacetimes. We find that a Big Bang cosmology (spatially closed or not) is trivially reconstructible: the whole universe is the entanglement wedge of any wedge inside it. But de Sitter space is not trivially reconstructible, despite being closed. We recover the AdS/CFT prescription by proving that EW$(B)=e($causal wedge of $B$).