General Relativity as a non-Abelian gauge theory

masudr

It occurred to me that I hadn't seen GR developed as a gauge theory in the same way QCD/electroweak are.

Are there any technical obstacles, or is it reasonably straightforward? And if it is well known, can someone please point me to a reference? Thanks.

lbrits

I think GR was actually one of the motivations behind Yang-Mills theories. I could wax poetically for hours about the subject, but any good "geometry for physicists" book will cover it. Basically you introduce an orthonormal frame which connects frame indices to tangent indices, and gauge those indices. So, for instance, your "field strength" is a Lie-algebra valued 2-form, namely the Riemann tensor. The method goes under the "principal bundle" formalism and can get quite heavy. But it is really simple, mathematicians just like to mathemagicate it.

The book by Nakahara is good, the book by Nash and Sen is good but contains a lot of typos. Anyway, it is more a thing about differential geometry than about GR. Ask if you need more.

masudr

I'll have a look at both. Thanks.

Haelfix

It works perfectly fine, especially in the Palatini formalism.

The main issue with the analogy is that the Riemann tensor contains second derivatives of the fundamental field 'entity' (the metric) whereas in the fibre bundle point of view, its really first derivatives