what is the formula of spin connection in GR ?
can we show it in term of structure coefficients ?
If you want the explicit solution in terms of vielbeins, check e.g. Van Proeyen's textbook or notes on supergravity, or eqn.(2.7) of http://arxiv.org/abs/1011.1145. Section 2 of this article reviews GR as a gauge theory of the Poincare algebra.
Exactly i study Teleparallel gravity, that is a gauge theory for translation group,and i know how to derive spin connection in terms of vielbeins, but i want its definition in GR.
Generally, we only use the Leva-Civita connection in GR. (That's the unique torsion free connection that preserves the dot product of the metric). It's a classical theory, so it doesn't really need to incorporate quantum spin.
So what about spinors ?
we need spin connections, whenever we want to compare them.
You use the vielbein postulate. Conceptually, you state with it that the vielbein is just an inertial coordinate transformation. Algebraically, it allows you to solve the Gamma connection in terms of the spin connection and vielbein. See again the reference I gave.
As far as I remember tetrad formalism and spin connection are nicely explained in Nakahara's textbook.
If you want to learn about the differential geometry of spin connections, and of teleparallelism, I suggest you look at the book "Differential Geometry for Physicists" by Fecko.
You are right, But i found the explicit relation with use of the Koszul formula in a orthonormal frame.
Thank u, it was helpful
Really Thank u, Nice suggestion, I got the book !
This is only possible when a spacetime admits a spin structure. For a non-compact spacetime, a necessary and sufficient condition that it admits a spin structure is that it is parallelizable, i.e., that it admits a *global* tetrad field.
Spacetimes that are compact are possibly non-physical, as any compact spacetime admits closed timelike curves, so the above is probably a useful equivalence.
Hi george, I'm not familiar with these kind of technicalities, but I will look them up. Thanks!
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