Teleparallelism - Teleparallel Spacetimes

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 fourdimensional 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 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 base. Here is the dual global base (or co-frame) defined by .

This is what usually happens in Rn, in any affine space or Lie group (for example the 'curved' sphere S3 but 'Weitzenböck flat' manifold).

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 as a by-product.