# Constructing vielbein from given metric: example: 2D spherical coordinates

Usual relation between metric and vielbein are given by \begin{align} g^{\mu\nu} = e^{\mu}{}_a e^{\nu}{}_{b} \eta^{ab} \end{align} where $\eta^{ab}$ is flat, $\mu,\nu$ is curved, ($i.e$, diffeomorphism index : index related with coordinate change) and $a,b$ are (Lorentz index).

I know that for given metric, vielbein form is not unique. Even though i want to construct vielbein and check above and inverse of above independently.

In the usual GR textbook, even though they mention vielbeins, for actual computation they just compute 1-forms (orthonormal basis) and obtain same results.

For example in sphere \begin{align} ds^2 = dr^2 + r^2 d\Omega^2 \end{align} and $e_r = dr$, $e_{\theta} = rd\theta$, $e_{\varphi} = r \sin(\theta) d\varphi$, then via cartan's formalism (via its structure equations), i can compute connection, Riemann tensor and so on.

What i want to know is process of computing vielbein of general given metric. (not necessarily be diagonal) Before generalizing i want to know some simple case.

• Note that the choice of vielbein is far from unique. – Qmechanic Apr 14 '17 at 8:43

## 1 Answer

The most general way is probably to look at it as a matrix equation, i.e.: $$g = e^T \eta \,e,$$

and then simply manipulate this to find the components of the vielbein. Remember that they are quadratic matrices.

• In terms of linear algebra: Finding the Eigensystem of $(g)$ – N0va Apr 14 '17 at 16:34