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I have some vector components as measured in the comoving tetrad frame $V^{(\mu)}$. This vector exists at coordinates $x^{(\mu)}$, which is different from the origin of the tetrad coordinate system.

This tetrad frame has 4-velocity in the global coordinate frame $u^{\mu}$.

A transform of the vector components from the tetrad frame to the coordinate frame ($V^{(\mu)} \rightarrow V^{\mu} $) is a function of the 4-velocity and also the global coordinates (which feed into the metric).

My question is, do I consider the global coordinates to label the origin of the tetrad coordinate system, or the location of the vector? My confusion is due to the fact intuitively it seems to be the origin which has 4-velocity $u^{\mu}$, but I am transforming the vector which does not live at the origin?

Thanks in advance for any help

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    $\begingroup$ I have to say that this question is a bit confusing to me, because I dont' quite see what it means for the tetrad frame to "have a velocity" relative to the internal space. I typically just think of a tetrad frame as four orthonormal vector fields that I can then use to span the space of spacetime vector fields, with the orthonormality letting me take the complexity that arises from having a nontrivial metric and move it to complexity in manipulating the tetrad vectors. $\endgroup$ – Jerry Schirmer Jul 30 '19 at 14:55
  • $\begingroup$ For some context, I am considering the transformation as outline in Appendix B (page 11) of arxiv.org/abs/1102.0010 $\endgroup$ – user1887919 Jul 30 '19 at 14:58
  • $\begingroup$ I'll try to write a full answer later, but essentially: the tetrad is not a coordinate system. It's an orthonormal basis defined at each point of spacetime. $\endgroup$ – Javier Jul 31 '19 at 18:03
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The tetrad is not a coordinate system. It is a set of four (orthonormal) vectors defined at each point in spacetime. Since at each tangent space it works as a basis, you can define components of vectors $V^{(\mu)}$ with respect to it$^1$, but it makes no sense to talk of coordinates $x^{(\mu)}$. Whenever you need to convert a vector between "ordinary" components $V^\mu$ and tetrad components $V^{(\mu)}$, both the vector and the tetrad have to be evaluated at the same spacetime point, labeled by the coordinates $x^\mu$.

In your particular case, the tetrad is sort of determined by the fluid velocity $u^\mu$, with two caveats:

  • We don't have to define the tetrad $e_{(\mu)}$ with $e_{(0)}{}^\mu = u^\mu$ (though it's convenient to do so); we can choose any basis of the tangent spaces (which we usually want to be orthonormal, but not always)

  • The fluid velocity determines one of the tetrad vectors, but not the other three, which can be chosen freely as long as they're orthogonal to it.


$^1$ so I guess in this sense the tetrad does define a coordinate system, but of the tangent space, not of the manifold

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  • $\begingroup$ "Whenever you need to convert a vector between "ordinary" components 𝑉𝜇 and tetrad components 𝑉(𝜇), both the vector and the tetrad have to be evaluated at the same spacetime point. This is the crux of my misunderstanding. Thank you! $\endgroup$ – user1887919 Aug 1 '19 at 7:11

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