# How to prove that vielbeins exist?

In Carroll's introductory book on General Relativity, he discusses the noncoordinate basis and how to construct the noncoordinate basis. When introducing this basis, he defines the vielbeins as the set of vectors comprising an orthonormal basis according to

$$g(\hat{e}_{(a)}, \hat{e}_{(b)}) = \eta_{ab}.$$

I have seen a construction of these via the exponential map and Riemann normal coordinates. However, I have not been able to find a demonstration that such a basis should exist at every point. How do we know that such a basis does exist at every point?

• Fix an arbitrary local basis of vector fields around a given point. Then, repeat the proof of the Gram-Schmidt orthogonalization process from linear algebra (suitably adapted for different metric signatures); it goes through almost verbatim; you don’t even need exponential maps or any other fancy Riemannian geometry techniques. Oct 25, 2023 at 0:27

Take the eigenvalue/eigenvector decomposition of the metric. Multiply each eigenvector by the square root of the absolute value of the corresponding eigenvalue. Then that gives you a vielbein, with metric coefficients $$±1$$, when referred to the frame comprising the eigenvectors. For $$4$$ dimensions, with Lorentzian signature that will give you a vierbein, with either signature $$(-1,+1,+1,+1)$$ or $$(+1,-1,-1,-1)$$, depending on which convention was used for the metric.