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S10g. What is the tetrad formalism?
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A way of writing general relativity such that it can be
applied to a spinor (e.g. electron) field.
A tetrad is a set of four linearly independent
vector fields e_0, e_1, e_2, e_3.
Considering them orthonormal in the sense that
g(e_j,e_k)=eta_jk (*)
where eta is the Minkowski metric defines the
metric g uniquely; conversely, for any metric one can
choose (on any chart) such an orthonormal basis.
If the manifold is parallelizable then one can choose
the ONB even globally. In 4 dimensions, any manifold
which allows to define spinors consistently is
parallelizable (by a result of Geroch), hence reality
is most likely described by such a manifold.
Using (*), one can rewrite any formula involving the
metric into one involving instead tetrads, and many
things simplify - using tetrads is closer to the Cartan
formalism of differential geometry than using the metric
directly. E.g.,
sqrt(-det g) = det(e).
One has to be slightly careful not to confuse curved
and flat indices, but this is learnt very quickly.
Then one needs much less index shifting.
For gravitation coupled to a (classical) Dirac field,
the tetrad formalism is indispensable, since spinors
cannot be defined without a flat representation.