Relationship With Metric Tensor, in A Coordinate Basis
The vierbein field, has two indices: labels the general spacetime coordinate and labels the local lorentz spacetime or local laboratory coordinates.
The vierbein field or frame fields can be regarded as the square root of the metric tensor, since in a coordinate basis,
where is the Lorentz metric.
Local lorentz indices are raised and lowered with the lorentz metric in the same way as general spacetime coordinates are raised and lowered with the metric tensor. For example:
The vierbein field enables conversion between spacetime and local lorentz indices. For example:
The vierbein field itself can be manipulated in the same fashion:
- , since
And these can combine.
A few more examples: Spacetime and local lorentz coordinates can be mixed together:
The local lorentz coordinates transform differently from the general spacetime coordinates. Under a general coordinate transformation we have:
whilst under a local lorentz transformation we have:
- .
Read more about this topic: Frame Fields In General Relativity
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