Frame fields in general relativity

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A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by ${\displaystyle {\vec {e}}_{0}}$ and the three spacelike unit vector fields by ${\displaystyle {\vec {e}}_{1},{\vec {e}}_{2},\,{\vec {e}}_{3}}$. All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.

Frame fields were introduced into general relativity by Albert Einstein in 1928^[1] and by Hermann Weyl in 1929.^[2]

The index notation for tetrads is explained in tetrad (index notation).