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Hi,
I have a question about gravity.
I think most of you know that we can obtain Einstein gravity by gauging the Poincaré algebra and imposing constraints. The Poincaré algebra consists of {P,M}. P describes translations, and M describes Lorentz rotations.
Gauging M gives us the so-called spin connection. The gauge field of the local translations is often taken (e.g. in a lot of supergravity texts) to be the vielbein e,
[tex]
\eta_{ab} e_{\mu}{}^a e_{\nu}{}^b = g_{\mu\nu}
[/tex]
What is the precise reason that this identification is justified? What do these "local translations" (I regard them as abstract internal transformations a la Yang-Mills) precisely have to do with the metric? Apart from the index structure I'm not really sure why this choice is justified.
I have a question about gravity.
I think most of you know that we can obtain Einstein gravity by gauging the Poincaré algebra and imposing constraints. The Poincaré algebra consists of {P,M}. P describes translations, and M describes Lorentz rotations.
Gauging M gives us the so-called spin connection. The gauge field of the local translations is often taken (e.g. in a lot of supergravity texts) to be the vielbein e,
[tex]
\eta_{ab} e_{\mu}{}^a e_{\nu}{}^b = g_{\mu\nu}
[/tex]
What is the precise reason that this identification is justified? What do these "local translations" (I regard them as abstract internal transformations a la Yang-Mills) precisely have to do with the metric? Apart from the index structure I'm not really sure why this choice is justified.