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The polar vortices play a crucial role in the formation of the ozone hole and can cause severe weather anomalies. Their boundaries, known as the vortex "edges,'' are typically identified via methods that are either frame dependent or return nonmaterial structures and, hence, are unsuitable for assessing material transport barriers. Using two-dimensional velocity data on isentropic surfaces in the Northern Hemisphere, the authors show that elliptic Lagrangian coherent structures (LCSs) identify the correct outermost material surface dividing the coherent stratospheric vortex core from the surrounding incoherent surf zone. Despite the purely kinematic construction of LCSs, the authors find a remarkable contrast in temperature and ozone concentration across the identified vortex boundary. The authors also show that potential vorticity-based methods, despite their simplicity, misidentify the correct extent of the vortex edge.