Black Hole Complementarity and the Island Paradigm

The black-hole information paradox highlights the tension between unitarity and semiclassical gravity. The island paradigm resolves this by reproducing the Page curve. This thesis argues the algebraic structure of the island paradigm realises Bohr's complementarity in quantum gravity. We evaluate the paradox and the island formula in Jackiw-Teitelboim gravity. In a fixed spacetime, local exterior algebras are type III$_1$ factors. Including the observer's clock (the ADM Hamiltonian) converts this into a type II$_\infty$ crossed product with a trace. We construct dressed observables for asymptotic and infalling observers. Since infalling observables are generated by a modular shift from the asymptotic clock, they form non-commuting subalgebras, suggesting complementarity. Finally, we introduce a relational framework using quantum reference frames. Gravitational entropy becomes observer-dependent: the asymptotic entropy features an island, while the infalling one does not. Therefore, the horizon, interior, and island emerge as relational structures.