The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set {X 1, X 2, X 3, X 4 } of four vector fields defined on all of M such that for every p ∈ M the set {X 1 (p), X 2 (p), X 3 (p), X 4 (p)} is a basis of T p M, where T p M denotes the fiber over p of the tangent vector bundle TM. Hence, the four-dimensional spacetime manifold M must be a parallelizable manifold. The tetrad field was introduced to allow the distant comparison of the direction of tangent vectors at different points of the manifold, hence the name distant parallelism. His attempt failed because there was no Schwarzschild solution in his simplified field equation.