orthonormal frame

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An orthonormal frame is a orthogonal basis of the tangent space T_p(M)\,, i.e., a set of tangent vector fields \hat{e}_i\, that span T_p(M)\, at every point p\,, and satisfy (\hat{e}_a)^\mu (\hat{e}_b)_\mu = \eta_{ab}\, where the components of \eta_{ab} = \operatorname{diag}(\pm,\pm,\pm...)\, depending on the signature of the metric g_{\mu\nu}\,.

Dropping indices, the frame \hat e_a\, has an associated coframe \tilde{e}^a\, defined through \tilde{e}^a(\hat e_b) = \delta^a_b\,. We therefore raise and lower "frame" indices with \eta_{ab}\,. Thus

(e^a)^\mu (e^{b})_\mu = \eta^{ab}\,,
(e_a)^{\mu} (e_b)_{\mu} = \eta_{ab}\,,
(e^a)^\mu (e_b)_\mu = \delta^a_b\,.

Because of completeness, we can also write

(e^a)^\mu (e_a)_\nu = \delta^\mu_\nu\,,

or

(e^a)_\mu (e_a)_\nu = g_{\mu\nu}\,,

in other words

\tilde{e}^a \otimes \hat e_a = Id\,,

the identity map on the tangent space. In other words,

(e^a)^\mu (e_a)_\nu = \delta^\mu_\nu\,, or
(e^a)_\mu (e_a)_\nu = g_{\mu\nu}\,.

It is useful to define a vector-valued oneform (\theta_\mu)^a = {e^a}_\mu\, as

\boldsymbol{\theta} = \mathbf{e}_\mu dx^\mu\,.

Internal SO(D) symmetry

The definition

g_{\mu\nu} = \eta_{ab} e^a_\mu e^b_\nu\,

is invariant under local transformations that preserve \eta_{ab}\,, which are exactly the group SO(D). In other words,

e^a_\mu \to {O^a}_b e^b_\mu\,,

leaves g_{\mu\nu}\, invariant provided that

{O^a}_b {O^c}_d \eta_{ac} = \eta_{bd}\,.

Such transformations act on \boldsymbol{\theta}\, via matrix multiplication as

\boldsymbol{\theta} \to \mathbf{O}\, \boldsymbol{\theta}\,.

Relation between coordinate and frame indices

Note that since (e^a)^\mu\, is just a vector with an internal label, expressions such as (e^a)^\mu X_\mu \,, where X_\mu\, is some oneform, are invariant under general coordinate transformations, i.e., they transform as a scalar. Define

X^a \equiv e^a_\mu X^\mu = (e^a)^\mu X_\mu \,.

Under local SO(D) transformations, X\, transforms as a vector:

X^a \to O^a_b X^b\,.

Nomenclature

The coframe is sometimes known as a vielbein (arbitrary dimensions), a vierbein or tetrad (four dimensions), dreibein (three dimensions), zweibein (two dimensions) and einbein (one dimension). The terms repère mobile, soldering form or orthonormal nonholonomic basis are also used. More esoteric are the elfbein (eleven dimensions) and the funfbein (five dimensions).

See also

Retrieved from "http://www.mathematics.thetangentbundle.net/wiki/Differential_geometry/orthonormal_frame"
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