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Worldline
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what is the formula of spin connection in GR ?
can we show it in term of structure coefficients ?
can we show it in term of structure coefficients ?
Worldline said: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.
So what about spinors ?pervect said:It's a classical theory, so it doesn't really need to incorporate quantum spin.
George Jones said: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.
You are right, But i found the explicit relation with use of the Koszul formula in a orthonormal frame.haushofer said: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.
Thank u, it was helpfultom.stoer said:As far as I remember tetrad formalism and spin connection are nicely explained in Nakahara's textbook.
Really Thank u, Nice suggestion, I got the book !George Jones said: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.
haushofer said: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.
A Spin (Lorentz) connection is a mathematical tool used in theoretical physics and differential geometry to describe the interactions between spinor fields and spacetime. It is a generalization of the more commonly known metric connection, which describes the geometric properties of spacetime.
A Spin (Lorentz) connection takes into account the intrinsic angular momentum of particles, known as spin, while a metric connection only considers the geometric properties of spacetime. This allows for a more complete description of the interactions between particles and spacetime.
The Spin (Lorentz) connection is important in physics because it provides a framework for understanding the behavior of spinor fields, which are essential in describing the behavior of fundamental particles in quantum mechanics. It also plays a crucial role in theories such as general relativity and particle physics.
The Spin (Lorentz) connection is calculated using mathematical equations known as the spin connection equations, which are derived from the spinor representation of the Lorentz group. These equations describe the relationship between spinor fields and the curvature of spacetime.
No, the Spin (Lorentz) connection cannot be directly measured experimentally. It is a mathematical concept used in theoretical physics to describe the interactions between particles and spacetime. However, its effects can be observed through experiments and observations of spinor fields and their behavior in different spacetime environments.