Spacetime Geometry from Modular Theory

We prove that the spacetime metric of a globally hyperbolic Lorentzian manifold can be uniquely reconstructed from the modular data of a local quantum field theory. Starting from Tomita--Takesaki modular theory applied to wedge and double-cone algebras, we show that (i) modular flows of wedges generate the Poincaré group, (ii) modular flows of all regions generate the full diffeomorphism group, and (iii) relative modular invariants together with spectral functionals determine the metric tensor. Under mild regularity assumptions on the algebraic quantum field theory, we establish that the triple consisting of a type $\mathrm{III}_1$ factor with prescribed symmetry (e.g., $E_8$), its family of modular operators, and the net structure uniquely determines the four-dimensional Lorentzian geometry up to diffeomorphism. This provides a rigorous realization of the principle that spacetime geometry emerges from quantum entanglement.