GR Tetrads & ZAMO example

Question

I am self-learning GR.

Intro: Tetrads are a way of representing general relativity in a coordinate-independent fashion.

I am having trouble understanding tetrad notations. Basically, I know that I can transform e.g. 4-velocities between tetrad frames by: $e^m_{ mu} x^mu=x^m$.

Problem: Most sources, however, give tetrads in some funny form by simply introducing vectors (example: equation 12 for Zero Angular Momentum Observer tetrad):

e.g. $gamma^{(t)}=|g_{tt}-omega^2 g_{phi phi}|^{1/2} dt$

Which is somehow related to the tetrad basis vectors.

Question: I) Is there any simple way to understand tetrad basis vectors, II) how can I relate the tetrad basis to vierbeins $e^m_{ mu}$ and III) are there any inherent symmetries in vierbeins e.g. $e^m_{ mu}$?

Note: I have mainly read physics books which did not deal with differential geometry

Ryan Unger · Accepted Answer

The tetrad and vielbein are exactly the same thing. The tetrad is just a set of four vectors $e_0,e_1,e_2,e_3$ which is orthonormal at each point:
$$g(e_i,e_j)=eta_{ij}=operatorname{diag}(-1,1,1,1)$$
Here $g$ is the metric tensor of spacetime. The vectors $e_i$ have components with respect to some coordinates $x^mu$, we denote these coordinate components by $e_i{}^mu$. Here $i$ denotes which one of the four vectors we are talking about and $mu$ which coordinate component. Since $mu$ and $i$ go over $4$ values, this is actually a matrix, and can be shown to be invertible. We denote its matrix inverse by $e^i{}_mu$.

The vectors $e_i$ have a "symmetry" in the following sense. If $Lambda_j{}^i$ is a Lorentz transformation, then the new set of vectors $Lambda_i{}^je_j$ is a tetrad:
$$g(Lambda_i{}^ke_k,Lambda_j{}^le_l)=Lambda_i{}^kLambda_j{}^lg(e_k,e_l)=Lambda_i{}^kLambda_j{}^leta_{kl}=eta_{ij}$$

GR Tetrads & ZAMO example

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