Leibniz indiscernibility argument: If the points of space or space-time are real, then shifting everything by a geometric symmetry transformation f (e.g., translation or rotation) produces a physically different situation, since the same bodies or events now occupy different points: whatever occupied point p now occupies the distinct point fp. This may violate identity of indiscernibles, or other principles of verifiability or theoretical parsimony.

Leibniz’s f was a symmetry of Euclidean space E, so the argument seems inapplicable to the inhomogeneous space-times of GTR. However, the underlying manifold M always possesses symmetries f, so one may take f also to move the geometric structure across M, along with the physical objects and events. The resulting variant models still represent physically distinct worlds, since the objects and geometric structure formerly at p are now at fp. Indeed there is more variability, since M has far more symmetries than E.