Spacetime geometry from Dirac spinor theory

The quintet of Dirac 4×4 matrices suggests that the fundamental dimension of the internal (spin) space is n=5, instead of the conventional dimension n=4. Then instead of the conventional 4×4 tetrads (vierbein), gravity is described in terms of the 5-bein (fünfbein or five legs). We discuss the properties of the spacetime geometry induced from this 5-leg Dirac spinor theory, where the spin connection contains 10×4=40 elements instead of 24 elements in the tetrad formulation of the general relativity theory.