
Teleparallelism (also called teleparallel gravity), was an attempt by 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 curvaturefree 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 $\backslash $ of four vector fields defined on ''all'' of $M\backslash ,$ such that for every $p\backslash in\; M\backslash ,$ the set $\backslash $ is a basis of $T\_pM\backslash ,$, where $T\_pM\backslash ,$ denotes the fiber over $p\backslash ,$ of the tangent vector bundle $TM\backslash ,$. Hence, the fourdimensional spacetime manifold $M\backslash ,$ 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 Weitzenböck connection) $\backslash $ to be the linear connection $\backslash nabla\backslash ,$ on $M\backslash ,$ such that 〔 〕 :$\backslash nabla\_(f^\backslash mathrm\; X\_)=(vf^)\backslash mathrm\; X\_(p)\backslash ,$, where $v\backslash in\; T\_pM\backslash ,$ and $f^\backslash ,$ are (global) functions on $M\backslash ,$; thus $f^X\_\backslash ,$ is a global vector field on $M\backslash ,$. In other words, the coefficients of Weitzenböck connection $\backslash nabla\backslash ,$ with respect to $\backslash $ are all identically zero, implicitly defined by: :$\backslash nabla\_\; \backslash mathrm\_j\; =\; 0\; \backslash ,\; ,$ hence $W^k\_i\}\; \backslash mathrm\_j)\backslash equiv0\; \backslash ,\; ,$ for the connection coefficients (also called Weitzenböck coefficients) —in this global basis. Here $\backslash omega^k\backslash ,$ is the dual global basis (or coframe) defined by $\backslash omega^i(\backslash mathrm\_j)=\backslash delta^i\_j\backslash ,$. This is what usually happens in ''R''^{n}, in any affine space or Lie group (for example the 'curved' sphere ''S''^{3} but 'Weitzenböck flat' manifold). Using the transformation law of a connection, or equivalently the $\backslash nabla$ properties, we have the following result. Proposition. In a natural basis, associated with local coordinates $(U,\; x^)$, i.e., in the holonomic frame $\backslash partial\_$, the (local) connection coefficients of the Weitzenböck connection are given by: :$\backslash Gamma^\_\; \backslash partial\_h^\_\backslash ,$, where $\backslash mathrm\; X\_\; =\; h^\_\backslash partial\_\backslash quad\; =\; 1,2,\; \backslash dots\; n$ are the local expressions of a global object, that is, the given tetrad. Weitzenböck connection has vanishing curvature, but —in general— nonvanishing torsion. Given the frame field $\backslash $, one can also define a metric by conceiving of the frame field as an orthonormal vector field. One would then obtain a pseudoRiemannian metric tensor field $g\backslash ,$ of signature (3,1) by :$g(X\_,X\_)=\backslash eta\_\backslash ,$, where :$\backslash eta\_=\{\backslash mathrm\; \{diag\}\}(1,1,1,1)\backslash ,$. The corresponding underlying spacetime is called, in this case, a Weitzenböck spacetime.〔(On the History of Unified Field Theories )〕 It is worth noting to see that these 'parallel vector fields' give rise to the metric tensor as a byproduct. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Teleparallelism」の詳細全文を読む スポンサード リンク
