Teleparallelism

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. The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set {X1, X2, X3, X4} of four vector fields defined on all of M such that for every p ∈ M the set {X1(p), X2(p), X3(p), X4(p)} is a basis of TpM, where TpM 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. In fact, one can define the connection of the parallelization (also called the Weitzenböck connection) {Xi to be the linear connection ∇ on M such that where v ∈ TpM and f^i are (global) functions on M; thus f^iXi is a global vector field on M. In other words, the coefficients of Weitzenböck connection ∇ with respect to {Xi are all identically zero, implicitly defined by: hence for the connection coefficients (also called Weitzenböck coefficients) in this global basis. Here ωk is the dual global basis (or coframe) defined by ωi(Xj) = δ. This is what usually happens in Rn, in any affine space or Lie group (for example the 'curved' sphere S3 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 (U, xμ), i.e., in the holonomic frame ∂μ, the (local) connection coefficients of the Weitzenböck connection are given by: where Xi = h∂μ for i, μ = 1, 2,.

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