Editing Teleparallelism (section)

==Teleparallel spacetimes==
The crucial new idea, for Einstein, was the introduction of a [[Cartan connection applications|tetrad]] field, i.e., a set {{math|{X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub>, X<sub>4</sub>}<nowiki/>}} of four [[vector field]]s defined on ''all'' of {{mvar|M}} such that for every {{math|''p'' ∈ ''M''}} the set {{math|{X<sub>1</sub>(''p''), X<sub>2</sub>(''p''), X<sub>3</sub>(''p''), X<sub>4</sub>(''p'')}<nowiki/>}} is a [[Basis (mathematics)|basis]] of {{math|''T<sub>p</sub>M''}}, where {{math|''T<sub>p</sub>M''}} denotes the fiber over {{mvar|p}} of the [[tangent vector bundle]] {{mvar|TM}}. Hence, the four-dimensional [[spacetime]] manifold {{mvar|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 '''[[Roland Weitzenböck|Weitzenböck]] connection''') {{math|{X<sub>''i''</sub>{{)}}}} to be the [[Connection (vector bundle)|linear connection]] {{math|∇}} on {{mvar|M}} such that<ref>{{cite book| last1=Bishop| first1=R. L. |last2=Goldberg|first2=S. I.|title=Tensor Analysis on Manifolds| url=https://archive.org/details/tensoranalysison0000bish| url-access=registration|date=1968|page=[https://archive.org/details/tensoranalysison0000bish/page/223 223]}}</ref>

<math display="block">\nabla_v\left(f^i\mathrm X_i\right)=\left(vf^i\right)\mathrm X_i(p),</math>

where {{math|''v'' ∈ ''T<sub>p</sub>M''}} and {{math|''f''{{isup|''i''}}}} are (global) functions on {{mvar|M}}; thus {{math|''f''{{isup|''i''}}X''<sub>i</sub>''}} is a global vector field on {{mvar|M}}. In other words, the coefficients of '''Weitzenböck connection''' {{math|∇}} with respect to {{math|{X<sub>''i''</sub>{{)}}}} are all identically zero, implicitly defined by:

<math display="block">\nabla_{\mathrm{X}_i} \mathrm{X}_j = 0,</math>

hence

<math display="block">{W^k}_{ij} = \omega^k\left(\nabla_{\mathrm{X}_i} \mathrm{X}_j\right)\equiv 0,</math>

for the connection coefficients (also called Weitzenböck coefficients) in this global basis. Here {{math|''ω<sup>k</sup>''}} is the dual global basis (or coframe) defined by {{math|''ω<sup>i</sup>''(X<sub>''j''</sub>) {{=}} ''δ''{{su|p=''i''|b=''j''}}}}.

This is what usually happens in {{math|'''R'''<sup>''n''</sup>}}, in any [[affine space]] or [[Lie group]] (for example the 'curved' sphere {{math|'''S'''<sup>3</sup>}} but 'Weitzenböck flat' manifold).

Using the transformation law of a connection, or equivalently the {{math|∇}} properties, we have the following result.

<blockquote>'''Proposition'''. In a natural basis, associated with local coordinates {{math|(''U'', ''x<sup>μ</sup>'')}}, i.e., in the holonomic frame {{math|∂<sub>''μ''</sub>}}, the (local) connection coefficients of the Weitzenböck connection are given by:

<math display="block">{\Gamma^{\beta}}_{\mu\nu}= h^{\beta}_{i} \partial_{\nu} h^{i}_{\mu},</math>

where {{math|X<sub>''i''</sub> {{=}} ''h''{{su|p=''μ''|b=''i''}}∂<sub>''μ''</sub>}} for {{math|''i'', ''μ'' {{=}} 1, 2,… ''n''}} are the local expressions of a global object, that is, the given tetrad.</blockquote>

The '''Weitzenböck connection''' has vanishing [[Riemann curvature tensor|curvature]], but – in general – non-vanishing [[torsion tensor|torsion]].

Given the frame field {{math|{X<sub>''i''</sub>{{)}}}}, 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 {{mvar|g}} of [[metric signature|signature]] (3,1) by

<math display="block">g\left(\mathrm{X}_i,\mathrm{X}_j\right)=\eta_{ij},</math>

where

<math display="block">\eta_{ij}=\operatorname{diag}(-1,-1,-1,1).</math>

The corresponding underlying spacetime is called, in this case, a [[Roland Weitzenböck|Weitzenböck]] spacetime.<ref>{{cite journal | doi=10.12942/lrr-2004-2 | doi-access=free | title=On the History of Unified Field Theories | date=2004 | last1=Goenner | first1=Hubert F. M. | journal=Living Reviews in Relativity | volume=7 | issue=1 | page=2 | pmid=28179864 | bibcode=2004LRR.....7....2G | pmc=5256024 }}</ref>

These 'parallel vector fields' give rise to the metric tensor as a byproduct.