- #1

haushofer

Science Advisor

- 2,262

- 617

## Main Question or Discussion Point

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.