# Spin (Lorentz) connection

1. Jan 15, 2013

### Worldline

what is the formula of spin connection in GR ?
can we show it in term of structure coefficients ?

2. Jan 15, 2013

### haushofer

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.

3. Jan 15, 2013

### Worldline

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.

Last edited: Jan 15, 2013
4. Jan 15, 2013

### pervect

Staff Emeritus
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.

5. Jan 15, 2013

### Worldline

we need spin connections, whenever we want to compare them.

6. Jan 16, 2013

### haushofer

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.

7. Jan 16, 2013

### tom.stoer

As far as I remember tetrad formalism and spin connection are nicely explained in Nakahara's textbook.

8. Jan 16, 2013

### George Jones

Staff Emeritus
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.

9. Jan 17, 2013

### Worldline

You are right, But i found the explicit relation with use of the Koszul formula in a orthonormal frame.

Really Thank u, Nice suggestion, I got the book !

10. Jan 17, 2013

### George Jones

Staff Emeritus
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.

11. Jan 19, 2013

### haushofer

Hi george, I'm not familiar with these kind of technicalities, but I will look them up. Thanks!