Teleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field. Teleparallel spacetimes The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set of four vector fields defined on all of such that for every the set is a basis of , where denotes the fiber over of the tangent vector bundle . Hence, the four-dimensional spacetime manifold 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. In fact, one can define the connection of the parallelization (also called the Weitzenböck connection) to be the linear connection on such that where and are (global) functions on ; thus is a global vector field on . In other words, the coefficients of Weitzenböck connection with respect to are all identically zero, implicitly defined by: hence for the connection coefficients (also called Weitzenböck coefficients) in this global basis. Here is the dual global basis (or coframe) defined by . This is what usually happens in , in any affine space or Lie group (for example the 'curved' sphere but 'Weitzenböck flat' manifold). Using the transformation law of a connection, or equivalently the properties, we have the following result. Proposition. In a natural basis, associated with local coordinates , i.e., in the holonomic frame , the (local) connection coefficients of the Weitzenböck connection are given by: where for are the local expressions of a global object, that is, the given tetrad. The Weitzenböck connection has vanishing curvature, but – in general – non-vanishing torsion. Given the frame field , one can also define a metric by conceiving of the frame field as an orthonormal vector field. One would then obtain a pseudo-Riemannian metric tensor field of signature (3,1) by where The corresponding underlying spacetime is called, in this case, a Weitzenböck spacetime. It is worth noting to see that these 'parallel vector fields' give rise to the metric tensor