Title: The Einstein–Rosen Bridge: Mirror Symmetry and Non-Traversability in the Maximal Extension Author: Adam GableIndependent ResearcherPhoenix, Arizona, USAORCID: 0009-0003-2488-0612 Abstract The Einstein–Rosen bridge, frequently invoked in discussions of wormholes and the ER=EPR conjecture, is often described informally as a connection between separate universes. This paper provides a precise geometric analysis of the classical Einstein–Rosen bridge within the maximally extended Schwarzschild spacetime. Working in Kruskal–Szekeres coordinates, we explicitly demonstrate the reflection symmetry relating the two asymptotically flat exterior regions and show that the transformation (u, v) → (−u, v) leaves the spacetime metric invariant. We derive the induced three-metric on the time-symmetric hypersurface and analyze the geometry of the bridge throat, showing that it corresponds to the minimal two-sphere located at the bifurcation surface. We further establish that the Einstein–Rosen bridge is strictly non-traversable. Although the bridge forms a spatial connection between the two exterior regions on a single spacelike slice, any future-directed causal trajectory entering the black-hole interior inevitably terminates at the spacelike singularity. Consequently, no worldline from one exterior region can reach the other. These explicit coordinate-level derivations clarify several geometric properties that are commonly stated but rarely shown in detail in standard references. The results provide a self-contained technical treatment of the classical Einstein–Rosen bridge and help clarify the geometric structure underlying modern discussions of spacetime connectivity and entanglement geometry. This manuscript is archived on Zenodo to provide a permanent, citable record of the analysis presented here.
Adam V. Gable (Fri,) studied this question.