4
$\begingroup$

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?

$\endgroup$
1
  • 2
    $\begingroup$ 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. $\endgroup$
    – peek-a-boo
    Oct 25, 2023 at 0:27

1 Answer 1

6
$\begingroup$

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.

This can be done locally over a set of coordinate patches, with one decomposition per coordinate patch. Extra conditions are required to be able to make it work globally, with a single vielbein used for the entire space. I don't remember what the relevant theorem is, and I can't find it. It might have been something proven by Thorne.

The matter's also been addressed here: Non-Coordinate Basis In GR; also here, How Do We Mathematically Define A Vielbein Field, where one of the replies provides a set of references, some of which may actually address the matter that I raise.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.