What is the Formula for Spin Connection in General Relativity?

[Worldline]

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

 

[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.

 

[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.

 

[pervect]

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.

 

[Worldline]

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.

 

[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.

 

[tom.stoer]

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

 

[George Jones]

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.

Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,

https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.

This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge field theories than traditionally is presented in physics courses.

Fecko has an unusual format. From its Preface,

A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).

The book is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.

From the review

There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...

A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.

Personal observations based on my limited experience with my copy of the book:

1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.

 

[Worldline]

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.

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

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

Thank u, it was helpful

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.

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

 

[George Jones]

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.

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.

 

[haushofer]

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